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The fattening phenomenon for level set solutions of the mean curvature flow Gavin, Colin Michael 2017

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The fattening phenomenon for level set solutionsof the mean curvature flowbyColin Michael GavinB.A., Lewis & Clark College, 2015a thesis submitted in partial fulfillmentof the requirements for the degree ofMASTER OF SCIENCEinthe faculty of graduate and postdoctoralstudies(Mathematics)The University of British Columbia(Vancouver)April 2017c© Colin Michael Gavin, 2017AbstractLevel set solutions are an important class of weak solutions to the meancurvature flow which allow the flow to be extended past singularities. Un-fortunately, when singularities do develop it is possible for the Hausdorffdimension of the level set solution to increase. This behaviour is referred toas the fattening phenomenon. The purpose of this thesis is to discuss thisphenomenon and to provide concrete examples, focusing especially on its re-lation to the uniqueness of smooth solutions. We first discuss the definitionof level set solutions in arbitrary codimension, due to Ambrosio and Soner.We then prove some technical results about distance solutions, a type ofset-theoretic subsolution to level set solutions. These include a new methodof gluing together distance solutions. Next, we present several known resultson the fattening phenomenon in the context of distance solutions. Finally,we provide a new example by proving that fattening occurs when immersedcurves in R3 develop self-intersections.iiPrefaceThe topic of this thesis was chosen in collaboration with the author’s su-pervisor, Dr. Jingyi Chen. A large portion of this thesis surveys existingresults. The organization and presentation of these results is unique to thiswork. Portions of Sections 3.3 and 4.2 present original results obtainedindependently by the author.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . vi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Mean Curvature Flow . . . . . . . . . . . . . . . . . . . . . . 42.2 Viscosity Methods . . . . . . . . . . . . . . . . . . . . . . . . 63 Level Set Flows and Distance Solutions . . . . . . . . . . . 113.1 Definition and Fundamentals . . . . . . . . . . . . . . . . . . 113.2 Equivalence with Smooth Flows . . . . . . . . . . . . . . . . . 163.3 Distance Solutions and Gluing . . . . . . . . . . . . . . . . . 204 The Fattening Phenomenon . . . . . . . . . . . . . . . . . . . 304.1 Definition and Previous Results . . . . . . . . . . . . . . . . . 304.2 Fattening of Immersed Curves . . . . . . . . . . . . . . . . . . 355 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53ivList of FiguresFigure 3.1 Gluing a segment between two circles . . . . . . . . . . . 29Figure 4.1 Fattening of a figure-eight curve . . . . . . . . . . . . . . 32Figure 4.2 A twisted curve in R3 . . . . . . . . . . . . . . . . . . . . 36Figure 4.3 Proof of the gradient bound . . . . . . . . . . . . . . . . . 43Figure 4.4 Proof of existence of connecting distance solutions . . . . 49vAcknowledgmentsI would like to thank my supervisor Dr. Jingyi Chen for suggesting thisfascinating topic. I have also benefited from many helpful and enjoyablediscussions with my colleague Jan Bohr.viChapter 1IntroductionThe mean curvature flow is a well studied geometric evolution equation forimmersed submanifolds of a Riemannian manifold. Briefly, it is a systemof parabolic PDE which moves an immersion in the direction of its meancurvature vector H (see Section 2.1 for a full definition). This process is ofinterest because it can be used to simplify the geometry of a submanifold,and also to find special submanifolds such as minimal surfaces (where H ≡0). Unfortunately, such applications are often impeded by the developmentof singularities which prevent flows from being extended for long times.In order to apply the mean curvature flow in cases when such singu-larities develop, there have been several attempts to define classes of weaksolutions. Broadly, there have been two major types of weak solutions.1 In1978, Bakke defined a generalized mean curvature flow in the language ofgeometric measure theory, which has become known as the Brakke flow [7].This flow has the advantage of being defined for a very broad class of initialdata (a large class of rectifiable varifolds). Furthermore, the well-developedregularity theory of geometric measure theory can be applied in this setting.However, Brakke’s definition allows for a great deal of non-uniqueness, andthere were certain gaps between the existence and regularity results that1There have also been several less well known formulations, such as the representationof mean curvature flow as a singular limit of Ginzberg-Landau type equations [8] or themethod of “ramps” applied in [1].1have proven difficult to address [15].The next wave of development in weak solutions of the mean curvatureflow occurred in the early 1990s. Two groups independently defined notionsof a weak mean curvature flow for codimension-1 surfaces represented bylevel sets of a scalar function, using the theory of viscosity solutions toanalyze the singular parabolic equation that defined the motion of theselevel sets [9, 12]. Critically, such level set flows can be defined for any closedinitial set and are unique. Subsequently, Ambrosio and Soner extended thisdefinition to higher codimension, and proved that Brakke flows are alwayscontained in the corresponding level set flow [2]. At nearly the same time,di Giorgi developed a purely geometric definition of weak solutions, basedon the use of classical solutions as barriers [4]. It was later shown that thisdefinition was, in fact, equivalent to the level set flows [5].In this thesis, we focus on the level set flow of Ambrosio and Sonerin arbitrary codimension. In contrast to Brakke flows, the uniqueness oflevel set flow is desirable, but also leads to some difficulties. In particular,when there is non-uniqueness among smooth mean curvature flows or Brakkeflows, the level set flow tends to develop a particular type of singularity called“fattening” in which the Hausdorff dimension of the solution can increase.For geometric applications, this can be quite undesirable unless fatteningis well understood. Our goal is to contribute to the understanding of thisphenomenon by summarizing some of the known results in the context ofAmbrosio and Soner’s work, and also providing a new example of fatteningfor immersed curves in R3.We begin by introducing the precise definition of the mean curvatureflow in Section 2.1 and summarizing the basics of viscosity solution the-ory necessary to understand Ambrosio and Soner’s work in Section 2.2. InChapter 3, we introduce level set flows and prove that they are well-definedand share some basic properties of smooth solutions of the mean curvatureflow. We then discuss equivalence of the level set flow with smooth solu-tions when the later exist, and present a recent result of Hershkovitz whichis stronger than that obtained in [2]. Next, we focus our attention on dis-tance solutions, a kind of “set-theoretic subsolution” to level set flows. A2highlight of this section is Theorem 3.16, which establishes a new techniqueto produce distance solutions by a gluing procedure. In Chapter 4, we in-troduce the fattening phenomenon and explain some existing results usingthe framework of distance solutions. Finally, we prove Theorem 4.17, whichestablishes the occurrence of fattening when curves evolving in R3 developtransverse self-intersections. The proof of this theorem makes crucial useof the gluing result for distance solutions, as well as a construction of suchsolutions satisfying a degenerate Dirichlet problem.3Chapter 2Preliminaries2.1 Mean Curvature FlowIn this thesis, we consider the evolution of immersed submanifoldsMk ↪−→Rn+kby the mean curvature flow. To define this process, first recall that ifF : M → Rn+k is an immersion, the Levi-Civita connection on M withrespect to the pullback g of the Euclidean metric is given by∇XY = (∇¯X¯ Y¯ )>where ∇¯ is the Euclidean connection on Rn+k and (·)> denotes projectiononto the tangent space of M . Then the difference between the Euclideanconnection and the induced connectionA(X,Y ) = ∇¯X¯ Y¯ −∇XYis a symmetric bilinear form with values in NxM called the second funda-mental form. We can then define the shape operator Aν : TxM → TxM fora unit normal vector ν ∈ NxM byg(Aν(X), Y ) = 〈A(X,Y ), ν〉.4For an orthonormal basis {ei} of TxM we defineH = trA =∑iA(ei, ei)to be the mean curvature vector of M at x. It can be verified that H isindependent of the choice of basis.On the other hand, we can define the area of the immersion F byA(F ) =∫Mdvolgwhere, as above, g is the pullback metric by F . Then if X is a compactlysupported normal vector field on M , the first variation of A with respect toX is given byδA(F )X = −∫M〈H,X〉 dvolg .From this formula, it is natural to consider the negative gradient flow of Afor immersions. This leads to the definition of the mean curvature flow forimmersions.Definition 2.1. A family of smooth immersions F : M × I → Rn+k on atime interval I ⊂ R is a smooth mean curvature flow if(∂F∂t)⊥= H(x, t) for x ∈M, t ∈ I (2.1)where H(x, t) is the mean curvature vector of the immersion F at x.Note that in the following, we will often refer a family of immersedmanifolds Mt which are the images of some smooth mean curvature flow assimply a smooth flow. Also, when M is a 1-manifold, the mean curvatureflow is traditionally known as the curve shortening flow.Classical solutions of (2.1) have been studied extensively in both thecases where the codimension k = 1 and for higher codimension. Shorttime existence and uniqueness of smooth flows is well known. Note thatby composing F with a diffeomorphism of M it is possible to produce non-identical solutions of (2.1) which have the same image Mt = F (M, t) ([21],5Proposition 3.1). Therefore, when discussing uniqueness of smooth solutionsto Equation (2.1), we will always consider whether the images Mt are unique,rather than the immersions F themselves.Proposition 2.2 ([21], Propositions 3.2 and 3.11). Suppose that F0 : M →Rn+k is a smooth immersion of a compact manifold M . Then there exists asmooth flow F : M × [0, T )→ Rn+k satisfying (2.1) such that F (·, 0) = F0.The images Mt = F (M, t) are uniquely defined by F0(M). Furthermore, forthe maximal such T , we havelim supt→TmaxMt|A|2 =∞where |A|2 = ∑ |A(ei, ei)|2 for an orthonormal basis {ei} of TxM .It is also well known that if M is compact the maximal time of existenceT to solutions of (2.1) will be finite. Therefore, a major goal of research onthe mean curvature flow has been to understand the formation of singulari-ties. In this area, codimension-1 flows are somewhat better understood, inpart due to the availability of stronger maximum principles in this setting.For example, the following avoidance property applies.Proposition 2.3 ([18], Theorem 2.2.1). Suppose that Mt and Nt are smoothmean curvature flows of n-dimensional submanifolds of Rn+1 on [0, T ) suchthat M0 and N0 are embedded and disjoint. Then Mt and Nt will eachremain embedded and the pair will remain disjoint for all t < T .When k > 1, Proposition 2.3 does not hold. This will become an impor-tant issue when we discuss weak solutions to (2.1) in higher codimension.2.2 Viscosity MethodsThe class of weak solutions to the mean curvature flow that we consider arebased on the theory of viscosity solutions to second order nonlinear ellipticand parabolic equations introduced by Crandall and Lions. In this section6we introduce this theory and prove some basic results about such solutions.The main reference here is the “User’s Guide to Viscosity Solutions” [10].Recall that a function u : Ω ⊂ Rd → R is said to be upper (resp.lower) semicontinuous if lim supx→x0 u(x) ≤ u(x0) (resp. lim infx→x0 u(x) ≥u(x0)). Importantly, upper and lower semicontinuous function achieve theirsuprema and infima on compact sets. If u is locally bounded we define theupper semicontinuous envelope of u byu∗(x0) = lim supx→x0u(x),and define the lower semicontinuous envelope u∗ analogously. Clearly, theupper semicontinuous envelope is an upper semicontinuous function satisfy-ing u∗(x) ≥ u(x), and likewise for the lower semicontinuous envelope.Our goal is to define a weak notion of sub- and supersolutions to equa-tions of the formut + F (u,∇u,∇2u) = 0 (2.2)which applies to functions u which are only semicontinuous. This will beaccomplished by requiring a local version of the maximum principle to holdfor such solutions. To this end, we make the following definition.Definition 2.4. A function F : R × Rd × Sd → R (where Sd is the set ofsymmetric d× d matrices) is called degenerate elliptic ifF (r, p,X) ≤ F (r, p, Y ) for Y ≤ Xwhere Y ≤ X if and only if X − Y is a positive semidefinite matrix. Fur-thermore, we say that F is proper if it is increasing in its first argument.If F is proper and degenerate elliptic, the problem (2.2) will be called de-generate parabolic. Now we can define our notion of sub- and supersolutionsto problems involving such operators.Definition 2.5. Suppose that F : R × Rd × Sd → R is proper, degenerateelliptic, and locally bounded. Let Ω ⊂ Rd be a locally compact, open set. We7say that an upper semicontinuous function u : Ω× (0, T )→ R is a viscositysubsolution of (2.2) if we haveφt(x0, t0) + F∗(u(x0, t0),∇φ(x0, t0),∇2φ(x0, t0)) ≤ 0 (2.3)for all φ ∈ C2(Ω× (0, T )) and (x0, t0) which are local maxima of u− φ.Likewise, we say a lower semicontinuous function u is a viscosity super-solution of (2.2) if the reverse inequality holds for F∗ at local minima ofu− φ.We will call u a viscosity solution of (2.2) if it is both a viscosity subsolu-tion and a viscosity supersolution, and when it is clear, the term “viscosity”will be omitted. Note that this definition is justified in that if u is a C2solution of ut = F (u,∇u,∇2u), it is easy to check by the weak maximumprinciple that u is a viscosity solution.We say that φ ∈ C2 touches u from above at (x0, t0) if φ(x0, t0) =u(x0, t0) and φ ≥ u on an open neighborhood of (x0, t0). Correspondingly,φ touches u from below at (x0, t0) if the opposite inequality holds. Thefollowing lemma gives a characterization of viscosity sub- and supersolutionsusing these conditions, which is often easier to use in practice than theoriginal definition.Lemma 2.6. A function u is a viscosity subsolution of (2.2) if and onlyif for every (x0, t0) ∈ Ω × (0, T ) and φ ∈ C2 touching u from above at(x0, t0), (2.3) holds. Likewise, u is a viscosity supersolution if and only ifthe corresponding condition holds for φ touching u from below.Proof. Suppose that (2.3) holds for test functions touching u above at (x0, t0).Let φ ∈ C2 be such that u − φ has a local maximum at (x0, t0). Thenφ˜ = φ− φ(x0, t0) + u(x0, t0) touches u from above at (x0, t0). Thus we haveφ˜t(x0, t0) + F∗(u(x0, t0),∇φ˜(x0, t0),∇2φ˜(x0, t0)) ≤ 0.Since the derivatives of φ˜ are equal to those of φ, this implies that u is aviscosity subsolution of (2.2).8On the other hand, suppose that u is a viscosity subsolution of (2.2). Ifthat φ touches u from above at (u0, t0), then u− φ has a local maximum at(x0, t0), so (2.3) holds.We now prove two important lemmas which allow us to construct newviscosity solutions from existing ones. First, we define the following weaklimit operations.Definition 2.7. Suppose that (un)∞n=1 : Ω ⊂ Rd → R, we definelim sup∗ u(x) = sup{lim supn→∞un(xn)∣∣ (xn)∞n=1 → x}andlim inf∗ u(x) = inf{lim infn→∞ un(xn)∣∣ (xn)∞n=1 → x} .These limits are called, respectively, the upper and lower half-relaxed limitsof un.Note that it is easy to check that the upper (resp. lower) half-relaxedlimit of upper (resp.) semicontinuous functions is upper (resp. lower) semi-continuous. It turns out that these operations are the correct limit underwhich viscosity sub- and supersolutions are preserved.Lemma 2.8 ([10], Lemma 6.1). If un is a sequence of viscosity subsolu-tions of (2.2), and if the upper half-relaxed limit u¯ = lim sup∗ un is boundedabove, it is also a viscosity subsolution. The same holds for sequences ofsupersolutions and their lower half-relaxed limit.Next, we consider a convolution-type operation which also preserves vis-cosity solutions.Definition 2.9. Let f, g : Rd → R and define the supremal convolution off and g byf ∗sup g = sup{f(y) + g(x− y) ∣∣ y ∈ Rn} .Likewise, define the infimal convolution of f and g byf ∗inf g = inf {f(y) + g(x− y) ∣∣ y ∈ Rn} .9As with the half-relaxed limits, these operations preserve upper and lowersemicontinuity respectively.Lemma 2.10. Suppose that u : Ω× (0, T )→ R is a viscosity subsolution ofut + G(∇u,∇2u) = 0, and g : Ω → R is an upper semicontinuous functionsatisfying g ≤ c. Letu˜(·, t) = u(·, t) ∗sup g.Then u˜ is also a viscosity subsolution. The same holds for viscosity super-solutions and infimal convolution with g ≥ c.Proof. (Following [20], Lemma 4.2.) Suppose that φ touches u˜ from aboveat (x0, t0). Because g is upper-semicontinuous and bounded above, thereexists y0 ∈ Ω such that u˜(x0, t0) = u(y0, t0) + g(x0 − y0). Let φ˜(x, t) =φ(x+ x0 − y0, t)− g(x0, y0). Thenφ˜(y0, t0) = φ(x0, t0)− g(x0, y0) = u(y0, t0).Furthermore, for x, t sufficiently close to (y0, t0) we haveφ˜(x, t) ≥ u˜(x+ x0 − y0, t)− g(x0, y0) ≥ u(x+ x0 − y0, t)by definition of u˜. Hence φ˜ touches u from above at (y0, t0) and so we haveφ˜t(y0, t0) +G(∇φ˜(y0, t0),∇2φ˜(y0, t0)) ≤ 0.This implies that the same equation holds for φ at (x0, t0), and thus u˜ isa subsolution. The same argument with inequalities reversed applies forsupersolutions.Note that the procedure of shifting test functions used in the proof ofLemma 2.10 is characteristic of the arguments which will be used later whenworking with viscosity solutions.10Chapter 3Level Set Flows and DistanceSolutions3.1 Definition and FundamentalsIn order to define a notion of weak solutions to mean curvature flow using themachinery of viscosity solutions, we must represent an embedded manifoldMn ↪−→Rn+k via a single function u. We then aim to write an equation ofthe form (2.2) for u which gives an equivalent evolution of the embedding.If k = 1, it is natural to represent Mt as a regular level set of a smoothfunction u : Rn+1 × (0, T ) → R. We begin this section by showing howthe corresponding evolution equation ought to be written. The situation issomewhat more complicated for k > 1, as Mt must be represented as a levelset of u at a singular value.First, consider a function u : Rn+1 × (0, T ) → R and a local parame-terization of the zero-level set by φ : Ω ⊂ Rn × (0, T ) → Rn+1 such thatu(φ(x, t), t) = 0. Differentiating givesut(φ, t) = −〈∇u(φ, t), φt〉.If the zero-level set is to move normally with speed v (to be determined11later), then we take φt = v∇u|∇u| , which givesut(φ, t) = −|∇u(φ, t)|v.We generalize this to hold on all points to obtainut = −|∇u|v. (3.1)In the case when k = 1, the mean curvature of a level set of u is simplygiven byH = −div( ∇u|∇u|).Therefore, using v = H in (3.1) the MCF equation for Mt becomesut = |∇u|div( ∇u|∇u|). (3.2)This equation was considered in context of viscosity solutions by Evans andSpruck [12] and Chen, Giga, and Goto [9]. It was shown that (3.2) corre-sponds to a degenerate parabolic problem which admits a unique viscositysolution for uniformly continuous initial data u0. Furthermore, the evolutionof the zero level set depends only on its initial geometry, not the choice ofu0, and this evolution agrees with a smooth mean curvature flow if it exists.Proceeding from this work, Ambrosio and Soner [2] generalized this ap-proach to the case when k > 1. This generalization will be the main class ofweak solution considered here, so we will explain it in detail. As above, thesurface Mt will be represented by the zero level set of u : Rn+k× (0, T )→ R.However, as noted above, the difficulty is that Mt must be represented bya singular level set of u, and therefore geometric quantities are difficult tocompute for this level set. Ambrosio and Soner instead consider regular ε-level sets for ε small. Such level sets are smooth hypersurfaces which “wraptightly” around Mt so we expect them to have k − 1 principle curvatureswhich are very large and n principle curvatures which closely approximatethose of Mt nearby. With this intuition, they design a flow which movesregular level sets by the sum of their smallest n principle curvatures. To do12so, we first represent the shape operator in terms of the level set function u.Lemma 3.1. Suppose that M is the zero level set of u : Rn+k → R and∇u 6= 0 on M . Then M is a smooth manifold with normal vector fieldν = ∇u|∇u| . At x ∈ M , let B = 1|∇u|P∇u∇2uP∇u where P∇u = I − ∇u⊗∇u|∇u|2 isthe orthogonal projection onto TxM . ThenB = Aν ⊕ 0where Aν is the shape operator on TxM and 0 acts on NxM .Proof. Note that Bη = 0 for η ∈ NxM , and BX ∈ TxM for X ∈ TxMby definition of P∇u so B splits into these subspaces. We can compute forX ∈ TxMBX =1|∇u|P∇u(∇2u)X =∇2u|∇u|X −1|∇u|3 〈∇u, (∇2u)X〉∇u.On the other hand, we haveAνX = ∇X ∇u|∇u| = 〈X,∇|∇u|−1〉∇u+ ∇2u|∇u|X=∇2u|∇u|X −1|∇u|3 〈(∇2u)X,∇u〉∇u.Thus we have BX = AνX.By Lemma 3.1, we see that the principle curvatures of a regular levelset of u are the eigenvalues of B with eigenvectors orthogonal to ∇u. Thismotivates us to define F : Rn+k \ {0} × Sn+k → R byF (p,X) = −n∑i=1λi(p,X) (3.3)where λi(p,X) are the ordered eigenvalues of PpXPp with eigenvector or-thogonal to p and Pp = I − p⊗p|p|2 as in (3.1). Then the equationut + F (∇u,∇2u) = 0 (3.4)13corresponds to the level set evolution equation (3.1) taking v to be the sumof the smallest n principle curvatures on regular level sets. The zero levelsets of viscosity solutions of this equation will be our weak solutions to meancurvature flow.Definition 3.2. Suppose that Γ∗ ⊂ Rd is closed and let u0 be any uniformlycontinuous function on Rd such that Γ∗ ={x∣∣ u0(x) = 0}. Suppose u ∈C(Rn+k × [0,∞)) is a viscosity solution of the problemut + F (∇u,∇2u) = 0 on Rn+k × (0,∞)u(x, 0) = u0(x, 0) for x ∈ Rn+k. (3.5)We letΓt ={x∣∣ u(x, t) = 0} (3.6)and call Γ =⋃t∈[0,∞) Γt × {t} the n-dimensional level set flow of Γ∗.Note that F is in fact a continuous degenerate elliptic operator awayfrom p = 0, so (3.4) can be considered in the viscosity sense. Furthermore,we recordF∗(0, A) = min|p|=1F (p,A) and F ∗(0, A) = max|p|=1F (p,A) (3.7)which will be used later. In [2] the following fundamental facts about solu-tions to (3.5) were proven.Proposition 3.3 ([2], Theorems 2.2-2.4). (a) (Comparison) Suppose thatu and v are sub- and supersolutions of (3.4) such that at least one ofu or v is uniformly continuous and there exists K > 0 such that|u(x, t)|+ |v(x, t)| ≤ K(1 + |x|)then u− v ≤ sup{u(x, 0)− v(x, 0) ∣∣ x ∈ Rn+k}.(b) (Existence) If u0 is uniformly continuous, there exists a unique uni-formly continuous solution u to (3.5) defined on Rn+k × [0,∞).14(c) (Relabeling) If u is a subsolution (resp. supersolution) of (3.4), andθ : R → R is uniformly continuous and nondecreasing, then θ ◦ u isalso a subsolution (resp. supersolution).As a demonstration of the methods used in the viscosity solution theory,we use Proposition 3.3 to prove that the n-dimensional level set flow of aset is well-defined.Lemma 3.4. Suppose that Γ∗ ⊂ Rn+k is closed and u0 and u˜0 are twouniformly continuous function which both have Γ∗ as their zero-level set.Let u and u˜ be the corresponding solutions of (3.5). Then the zero-level setof u is equal to that of u˜.Proof. (Following [2], Theorem 2.5) In particular, we will show that thelemma holds if u0(x) = dist(x,Γ), from which the full result follows bytransitivity. Let Γt and Γ˜t be the zero-level sets of u and u˜ respectively.(Γt ⊂ Γ˜t): Let ω(s) = sup{u˜0(y)∣∣ dist(y,Γ∗) ≤ s}. Then since u˜0 is uni-formly continuous, ω(s) is a non-decreasing uniformly continuous function,and thus by Proposition 3.3(c), ω ◦ u is a supersolution of (3.4). Now notethatω(u0(x)) = ω(dist(x,Γ)) ≥ u˜0(x).Hence by Proposition 3.3(a), u˜ ≤ ω ◦ u. Therefore, if x ∈ Γt, ω(u(x, t)) = 0so u˜(x, t) = 0, and so x ∈ Γ˜t.(Γ˜t ⊂ Γt): Let hε(s) be a sequence of non-decreasing cutoff functions withhε ≡ 0 on (−∞, 0] and hε ≡ 1 on [ε,∞). By Proposition 3.3(c), hε ◦u˜ is a supersolution of (3.4) for each ε. By Lemma 2.8, it follows thatlim inf∗ hε ◦ u˜ = 1− χΓ˜t is also supersolution. Finally, let v = min(u, 1), sothat v(x, 0) ≤ 1−χΓ and by comparison v ≤ 1−χΓ˜t for all t. This inequalityimplies that if x ∈ Γ˜t, then v(x, t) = 0, and so x ∈ Γt.It is an easy computation to check that F satisfiesF (λp, λX + σp⊗ p) = λF (p,X) (3.8)15for λ > 0 and σ ∈ R. This identity allows us to extend the usual scaling ofsolutions to mean curvature flow to level set flows.Lemma 3.5. Suppose Γ is the level set flow of Γ∗. Then for λ > 0, thelevel set flow Γ˜ of λΓ∗ is given by Γ˜t = λΓλ−1t.Proof. Let u be a solution to (3.5) with initial data u0 zero on Γ∗. Definev(x, t) = λ−1u(λ−1x, λ−1t). Note that the zero level set of v(·, 0) is λΓ∗. Weclaim that v solves (3.4). Suppose that φ touches v from below at (x0, t0).Define φ˜(x, t) = λφ(λx, λt). Then we haveφ˜(λ−1x0, λ−1t0) = λφ(x0, t0) = λv(x0, t0) = u(λ−1x0, λ−1t0)and for x and t sufficiently close to λ−1x0 and λ−1t0φ˜(x, t) = λφ(λx, λt) ≤ λv(λx, λt) = u(x, t).Hence φ˜ touches u from below at (λ−1x0, λ−1t0), and so at this point wehaveφ˜t + F (∇φ˜,∇2φ˜) ≥ 0.Applying the definition of φ˜ and (3.8) we haveλ2φt(x0, t0) + λ2F (∇φ(x0, t0),∇2φ(x0, t0)) ≥ 0.Therefore v is a viscosity supersolution of (3.4). The same argument withinequalities reversed shows that v is also a subsolution. Since v is a solutionof (3.4), Γ˜t is the zero-level set of v(·, t), which is exactly λΓλ−1t by thedefinition of v.3.2 Equivalence with Smooth FlowsAn important property of any formulation of weak solutions to the meancurvature flow problem is agreement with smooth flows when they exist.In the codimension-1 case, this equivalence was proven in [12] and [9]. In16their original paper on level set flows in arbitrary codimension, Ambrosioand Soner proved the following result.Proposition 3.6. If Mt ⊂ Rn+k is a smooth flow of embedded n-dimensionalsubmanifolds on [0, T ), then the n-dimensional level set flow Γ of M0 satis-fies Γt = Mt for 0 ≤ t < T .Ambrosio and Soner prove this result by considering the evolution of thedistance function δ(x, t) = dist(x,Mt). Recall that if, for all 0 < t < T ,Mt has a tubular neighborhood of radius ρ, the distance function δ(·, t) issmooth on U ={(x, t)∣∣ 0 ≤ t < T, 0 < δ(x, t) < ρ}. Ambrosio and Sonershow that on U we haveF (∇δ,∇2δ) ≤ δt ≤ F (∇δ,∇2δ) + Cδ (3.9)for some constant C independent of x and t. Using these bounds, theymodify δ (while maintaining the zero level set) to construct suitable sub-and supersolutions¯u and u¯ of (3.5), which are bounded above and belowby δ(·, 0) at t = 0. By the comparison principle of Proposition 3.3(a), if uis the solution of (3.5) with initial data δ(·, 0), we have¯u ≤ u and u¯ ≥ u.As in the proof of Lemma 3.4, these bounds show that Mt = Γt.While this is one of Ambrosio and Soner’s fundamental results and isimportant in justifying the definition of the level set flow, it is not entirelysatisfying because it says nothing about the case when Mt is a smooth flowon (0, T ) but does not extend smoothly to t = 0. This case is importantin applications of the weak solution theory to the question of solvabilityof the mean curvature flow equation for rough initial data. In particular,establishing equivalence of the level set flow and a smooth flow can be usedto prove uniqueness of the smooth flow.Hershkovits takes this approach in [14] when he considers the short timeexistence of smooth solutions of the mean curvature flow when the initialdata is an ε-Reifenberg set. Briefly, such sets are embedded topological17manifolds which have approximate tangent spaces at each point which areallowed to vary according to a scale parameter (the details are not relevanthere; see [19] for a complete definition). To apply the level set flow theoryof Ambrosio and Soner in this setting, Hershkovits proves a stronger versionof Proposition 3.6.Proposition 3.7. Suppose that Mt is a smooth flow of embedded n-dimensionalsubmanifolds of Rn+k on (0, T ) and M0 is a connected compact set. Supposefurther that for constants c21 ≤ 18 and 14c1 − c2 >√2n, Mt satisfies(i) supMt |A(t)| ≤ c1√t(ii) dH(Mt,M0) ≤ c2√t(iii) Mt has a tubular neighborhood of radius at least√t4c1where A(t) is the second fundamental form of Mt and dH is the Hausdorffdistance. Then the level set flow of M0 is equal to Mt on (0, T ).Note that while the constants in the statement of this result may seemsomewhat arbitrary, they cannot be easily scaled away (i.e. we cannot tradea worse bound on c1 for a better bound on c2) because (i-iii) are all scaleindependent with respect to the spacetime scaling of mean curvature flow.To prove this result, Hershkovits establishes a more precise version of(3.9). In particular for (x, t) in the same neighborhood U as above we haveδt = F (∇δ,∇2δ) + δn∑i=1〈A(vi, vi),∇δ〉21− δ〈A(vi, vi),∇δ〉 (3.10)where {vi} are principle directions for the shape operator A−∇δ at the closestpoint on Mt to x, and A is evaluated at the same point.We will also need the following lemma, which says that n-dimensionallevel set flows do not cross spheres evolving by the sum of their first nprinciple curvatures. Note that this is actually a very special property ofspheres, since in general the avoidance properties that hold in codimension-1(c.f. Proposition 2.3) do not apply in higher codimension. We will use thisresult to bound the rate at which the level set flow can move away from M0.18Lemma 3.8. Suppose that Γ∗ is a closed set and v : Rn+k × (0, T ) → R isa supersolution of (3.5), with v(x, 0) = dist(x,Γ∗). Thenv(x0, t) ≥ v(x0, 0)−√2nt (3.11)for all x0 ∈ Rn+k.Proof. We fix x0 ∈ Rn+k and defineu(x, t) = v(x0, 0)−√|x− x0|2 + 2kt.Note that u(x, 0) = dist(x0,Γ∗)− |x− x0|. Using the fact that the distancefunction to Γ∗ is 1-Lipschitz, we have u(x, 0) ≤ v(x, 0). Furthermore, it iseasy to check that u is in fact a solution to (3.4). (One can either computedirectly, or use the fact that the level sets of u are spheres moving by theirfirst n principle curvatures.) Therefore by Proposition 3.3(a), we haveu(x0, t) ≤ v(x0, t) =⇒ v(x0, 0)−√2nt ≤ v(x0, t)which proves the claim.Proof of Proposition 3.7. Following Herskovitz, Theorem 1.7 [14], let Γ bethe level set flow of M0 with level set function u solving (3.5) with initialdata dist(M0, ·). The argument of [2] that was discussed above still sufficesto show that Mt ⊂ Γt. We will only consider the more difficult problem ofshowing that Γt ⊂Mt.To do this, we consider v(x, t) = δ(x,t)√tdefined onN ={(x, t)∣∣ 0 ≤ t ≤ T, δ(x, t) < √t4c1}.Note that N ∩ (Rn+k × {0}) is empty, and so by (iii) v is smooth on N .Then, a computation using (3.10) and assumptions (i) and (ii) show that vis a classical subsolution of (3.4) on N . (Here is where the condition c21 <18is used.)19Now, if we can use the comparison principle to conclude that u ≥ v, wewill be done, since then x ∈ Γt would imply u(x, t) = 0 and so v(x, t) = 0which implies x ∈ Mt. A (slightly modified version) of the comparisonprinciple will apply if we can show that u ≥ v on the parabolic boundaryof N which consists of (x, t) such that t > 0 and δ(x, t) =√t4c1. Using thischaracterization along with (ii) we have√t4c1= δ(x, t) ≤ δ(x, 0) + dH(Mt,M0) ≤ δ(x, 0) + c2√t.This implies that δ(x, 0) ≥(14c1− c2)√t. Applying Lemma 3.8, we haveu(x, t) ≥ δ(x, 0)−√2nt ≥(14c1− c2 −√2n)√t = αv(x, t)where α is a positive constant by the constraints on c1 and c2. By the rela-beling result Proposition 3.3(c) for solutions of (3.4), αv is also a subsolutionon N , and by comparison we have u ≥ αv on N .3.3 Distance Solutions and GluingIn the previous section, (3.9) shows that the distance function to a familyof smooth manifolds evolving by mean curvature is a classical supersolutionof (3.4) within a tubular neighborhood. In fact, we will see below that thedistance function is actually a viscosity supersolution everywhere. From thisconclusion, Ambrosio and Soner extract the following definition, which canbe understood as an intrinsic characterization of “set-theoretic subsolutions”of the level set flow.Definition 3.9. Suppose that Γ ⊂ Rn+k × (0, T ) and for each t ∈ (0, T )supposed Γt = Γ ∩ Rn+k × {t} is closed. For any such set writeδΓ(x, t) = dist(x,Γt)for the spatial distance function. We call Γ an n-dimensional distance solu-20tion if δΓ is a supersolution of (3.5). Letlim inft→0Γt =⋂t∈(0,T )⋃s∈(0,t)Γs.Given a closed set Γ∗ ⊂ Rn+k, a distance solution is said to satisfy the initialinclusion Γ0 ⊂ Γ∗ if lim inft→0 Γt ⊂ Γ∗.The following lemma will be used extensively when considering distancesolutions in the remainder of this thesis. It greatly simplifies the process ofchecking whether a set Γ is a distance solution by restricting the class oftest functions that must be considered.Lemma 3.10. Suppose that Γ ⊂ Rn+k×(0, T ). Then Γ is a distance solutionif for all C2 test function φ touching δΓ from below at a point (y0, t0) on Γwe haveφt(y0, t0) + F∗(∇φ(y0, t0),∇2φ(y0, t0)) ≥ 0.Furthermore, if t 7→ Γt is continuous with respect to the Hausdorff distancedH, we need only consider φ and (y0, t0) such that ∇φ(y0, t0) 6= 0 and y0 ∈Γt0 is the minimizer in Γt0 of distance to some point outside Γ.Proof. Let φ be an arbitrary C2 test function which touches δΓ from belowat (x0, t0) ∈ Rn+k × (0, T ). Let y0 ∈ Γt0 be such that δΓ(x0, t0) = |x0 − y0|,and defineφ˜(x, t) = φ(x+ x0 − y0, t)− |x0 − y0|.Note that φ˜(y0, t0) = 0. Let ε be such that φ(z, t) ≤ δΓ(z, t) for (z, t) ∈B2ε((x0, t0)). Let y ∈ Bε(y0) and t ∈ (t0− ε, t0 + ε). Then (y+ x0− y0, t) ∈Bε((x0, t0)) and soφ˜(y, t) = φ(y + x0 − y0)− |x0 − y0| ≤ δΓ(y + x0 − y0, t)− |x0 − y0|.By the triangle inequality, we have δΓ(y + x0 − y0, t) ≤ |x0 − y0|+ δΓ(y, t),so we obtainφ˜(y, t) ≤ δΓ(y, t).21Thus φ˜ is a C2 function touching δΓ from below at (y0, t0). By our assump-tion, we haveφ˜t(y0, t0) + F∗(∇φ˜(y0, t0),∇2φ˜(y0, t0)) ≥ 0.But the derivatives of φ at (x0, t0) are equal to those of φ˜ at (y0, t0), so wehaveφt(x0, t0) + F∗(∇φ(x0, t0),∇2φ(x0, t0)) ≥ 0.From this it follows that Γ is a distance solution.Finally, if we only consider (x0, t0) 6∈ Γ in the proof above, we obtainthat δΓ is a supersolution of (3.4) on (Rn+k × (0, T )) \ Γ. For such points φin the above has ∇φ˜(x0, t0) 6= 0, since φ touches δΓ from below at a pointaway from its zero level set. Furthermore, by definition y0 is a minimizerin Γt0 of the distance to x0. To show that δΓ is a supersolution on all ofRn+k × (0, T ) we apply Lemma 3.11 below to δΓ. We note that by theassumption of dH-continuity, assumption (i) of the lemma is satisfied.Lemma 3.11 ([2], Lemma 3.11). Suppose that u : Rn+k × (0, T ) is lowersemicontinuous and(i) for (x, t) such that u(x, t) = 0 there exists a sequence (xn, tn)→ (x, t)with tn < t and u(xn, tn) = 0;(ii) u is a viscosity supersolution of (2.2) on{(x, t)∣∣ u(x, t) > 0};(iii) and u(·, t) is K-Lipschitz continuous with K independent of t.Then u is also a viscosity supersolution of (2.2) on Rn+k × (0, T ).Note that Lemma 3.10 shows that the distance function to a smooth flowis actually a viscosity supersolution globally since we only need to considertest functions touching δ from below at points onMt, and δ is a supersolutionnear Mt. Thus, smooth flows are distance solutions. (In fact, the main ideaof Lemma 3.10 is extracted from the proof of this fact in [2].)A distance solution Γ may be quite poorly behaved. For example, entireconnected components Γt may disappear instantaneously, and δ need only be22lower semicontinuous in time. However, it turns out that maximal distancesolutions (with respect to containment as sets) are exactly level set flows,which explains how distance solutions may be viewed as subsolutions.Proposition 3.12. Suppose that Γ∗ ⊂ Rn+k is closed and Γ is the n-dimensional level set flow of Γ∗. Then Γ is the maximal distance solutionsatisfying the initial inclusion Γ0 ⊂ Γ∗.Proof. (Following [2], Theorem 4.4) Let u be any solution of (3.5) with zerolevel set Γ. Let hε be as in the proof of Lemma 3.4. Let¯u = lim inf∗ hε ◦ u.As before¯u = 1 − χΓ is a supersolution by Lemma 2.8. Now let vK(·, t) =K(1− χΓ(·, t)) ∗inf g where g(x) = |x| and K > 0. By Lemma 2.10, vK is asupersolution. We claim thatvK(x, t) = min(δΓ(x, t), inf{K + |x− y| ∣∣ y 6∈ Γt}).The infimum in the definition of the infimal convolution must be attainedsince g becomes unbounded as x→∞. Hence there exists y such thatvK(x, t) = K(1− χΓ(y, t)) + |x− y|.If y ∈ Γt, then vK(x, t) = |x− y|. Otherwise, vK(x, t) = K + |x− y|. HencevK(x, t) = min(inf{|x− y| ∣∣ y ∈ Γt} , inf {K + |x− y| ∣∣ y 6∈ Γt})which is exactly the claim above. Finally, by Lemma 2.8 δΓ = lim inf∗ vKmust be a supersolution, and so Γ is a distance solution satisfying the initialinclusion Γ0 ⊂ Γ∗.On the other hand, if Γ˜ is any other distance solution satisfying Γ˜0 ⊂ Γ∗,we note that δΓ˜≤ u by Proposition 3.3(a). Hence Γ˜ ⊂ Γ, and so Γ is in factthe maximal distance solution satisfying Γ0 ⊂ Γ∗.As an example of the utility of this result, we consider the situation inwhich Γ∗ sits in an affine subspace Σ of Rn+k. If Γ∗ is a submanifold, itssmooth mean curvature flow clearly remains in Σ and is equivalent to that23obtained when Γ∗ is viewed as a subspace of Σ. The following propositiongeneralizes this fact to level set flows.Proposition 3.13. Let Σ ⊂ Rn+k be a d-dimensional affine subspace. Sup-pose that Γ∗ ⊂ Σ. Let Γ be the n-dimensional level set flow of Γ∗ in Rn+kand Γ˜ be the level set flow of Γ∗ in Σ. Then Γ = Γ˜.Proof. Without loss of generality, we identify Σ with the plane Rd × {0} ⊂Rn+k. We will write x = (y, z) ∈ Rd × Rn+k−d for coordinates in thisdecomposition.(Γ˜ ⊂ Γ): We will show that if Γ˜ is any distance solution as a subset ofΣ, then it is also a distance solution in Rn+k. By the characterization oflevel set flows as maximal distance solutions, this will prove Γ˜ ⊂ Γ. Let Fdand Fn+k refer to the operator F defined in (3.3) for ambient dimensionsd and n + k respectively. By Lemma 3.10, we consider a test functionφ : Rn+k × (0, T )→ R touching δΓ˜from below at ((y0, 0), t0) ∈ Γ˜. We thenneed to show thatφt(y0, 0, t0) + (Fn+k)∗(∇φ(y0, 0, t0),∇2φ(y0, 0, t0)) ≥ 0.By restricting φ to φ˜(y, t) = φ(y, 0, t), and using the fact that Γ˜ is a distancesolution in Σ, we haveφt(y0, 0, t0) + (Fd)∗(∇φ˜(y0, t0),∇2φ˜(y0, t0)) ≥ 0.Write A = ∇2φ(y0, 0, t0) and p = ∇φ(y0, 0, t0), and correspondingly A˜ =∇2φ˜(y0, t0) and p˜ = ∇φ˜(y0, t0). We will show that(Fd)∗(A˜, p˜) ≥ (Fn+k)∗(A, p) (3.12)which suffices to establish the necessary inequality. First we consider thecase when p 6= 0. Recall that the variational characterization of eigenvaluesgivesλj(PpAPp) = maxS⊂Rn+kdim(S)≥n+k−j+1minx∈S|x|=1〈x,p〉=0〈PpAPpx, x〉, (3.13)24where the condition 〈x, p〉 = 0 is due to the fact that we consider onlyeigenvalues with eigenvectors orthogonal to p. On the other hand, we haveλj(Pp˜A˜Pp˜) = maxS⊂Σdim(S)≥d−j+1minx∈S|x|=1〈x,p˜〉=0〈Pp˜A˜Pp˜x, x〉. (3.14)Letting q = (p˜, 0) ∈ Rn+k, we can write (3.14) in terms of vectors x ∈ Rn+kasλj(Pp˜A˜Pp˜) = maxS⊂Rn+kdim(S)≥d−j+1minx∈S∩Σ|x|=1〈x,q〉=0〈PqAPqx, x〉. (3.15)In (3.13) and (3.14) using the symmetry of Pp and Pq and the constraintson x, we have〈PpAPpx, x〉 = 〈Ax, x〉 and 〈PqAPqx, x〉 = 〈Ax, x〉.Finally, since d < n + k, if S is considered in the maximum in (3.13), thenit is considered in (3.15) and the corresponding minimum is over a largerset. Hence λj(Pp˜A˜Pp˜) ≥ λj(PpAPp). From the definition of Fd and Fn+k,(3.12) holds. Finally, note that having removed the normalization by |p| and|p˜|, Equations (3.13) and (3.14) are valid for all p, so the above argumentapplies in the case when p = 0 or p˜ = 0 as well.(Γ ⊂ Γ˜): Using the avoidance of spheres proven in Lemma 3.8, it is easy tosee that Γ remains within the subspace Σ. (Choose arbitrarily large spherestangent to each point on Σ.) We will apply Lemma 3.10 again to show thatΓ is a distance solution viewed as a subset of Σ. Therefore, we consider a C2test function φ on Σ× (0, T ) which touches δΓ|Σ from below at (x0, t0) ∈ Γ.Let λ be larger than all of the eigenvalues of P∇φ∇2φP∇φ at (x0, t0), anddefineφ˜(y, z, t) = φ(y, t) +12λ|z|2.Using the fact that δΓ(y, z, t) =√δΓ(y, 0, t)2 + |z|2, it is easy to check thatφ˜ touches δΓ from below at (x0, t0). Then, using the definition of φ˜, we can25compute thatφt(x0, t0) = φ˜t(x0, t0)≥ Fn+k(∇φ˜(x0, t0),∇2φ˜(x0, t0))= Fd(∇φ(x0, t0),∇2φ(x0, t0)).where the last equality is because our choice of λ ensures that any neweigenvalues of P∇φ˜∇2φ˜P∇φ˜ are large.From Proposition 3.12, we also see that distance solutions provide asimple method of proving “set theoretic lower bounds” on level set flows. Inparticular, if we can prove that Γ is a distance solution satisfying Γ0 ⊂ Γ∗,then the level set flow of Γ∗ must include Γ. We will take advantage ofthis fact in our discussion of the fattening phenomenon in Section 4.2. Asa preliminary, we now develop a method to glue together several distancesolutions into a new distance solution. Clearly by the properties of viscositysupersolutions, the union of two distance solutions is a distance solution,so instead we consider sets which are distance solutions apart from some“boundary.”Definition 3.14. Let Γ ⊂ Rn+k × (0, T ) and Σ ⊂ Γ be such that Γt andΣt are closed. The pair (Γ,Σ) will be called an interior distance solution ifevery point (x, t) ∈ Rn+k × (0, T ) such that x ∈ Γt \ Σt has a neighborhoodU on which δΓ is a viscosity supersolution of (3.4). The set Σ will be calledthe boundary of (Γ,Σ).The following proposition allows us to construct interior distance solu-tions by cutting subsets out of distance solutions. (This is also the justifi-cation for the terms interior distance solution and boundary.)Proposition 3.15. Let Γ be a distance solution and Ω ⊂ Γ be such that Ωtis compact and t 7→ Ωt is continuous with respect to the Hausdorff distance.Let ∂ΓΩ be the boundary of Ω relative to Γ. Then (Ω, ∂ΓΩ) is an interiordistance solution.26Proof. Let (x, t) be such that x ∈ Ωt \ ∂ΓΩt. Note that δΩ ≥ δΓ becauseΩt ⊂ Γt. Therefore, we will prove that there exists a neighborhood V 3 (x, t)on which δΩ ≤ δΓ. Then δΩ|V ≡ δΓ|V , and so δΩ is a supersolution on V .Since (x, t) 6∈ ∂ΓΩ, the definition of the relative boundary implies theexistence of a neighborhood U 3 (x, t) such that U ∩ Ω = U ∩ Γ. Letε > 0 be small enough that K = {x} × [t − ε, t + ε] ⊂ U . Define η =inf{|k − x| ∣∣ k ∈ K and x ∈ U c}. Note that by the dH-continuity of t 7→Ωt, Ω\U is closed, so η > 0. Furthermore, we can choose δ > 0 such that forall |t′ − t| < δ we have dH(Ωt,Ωt′) < η/4. Now let V = Bη/4(x)×Bmin(ε,δ)(t).Suppose to the contrary that there exists (y, s) ∈ V such that δΩ(y, s) >δΓ(y, s). Then we have dH(Ωt,Ωs) < η/4, so there exists x′ ∈ Ωs suchthat |x− x′| < η/4. On the other hand, there exists z ∈ Γs \ Ωs such thatδΓ(y, s) = |z − y|. Furthermore, we must have z ∈ U c, since if z ∈ U thenz ∈ U ∩ Γs = U ∩Ωs, but z 6∈ Ωs. Therefore |z − x| ≥ η. Also, by definitionof z, |z − y| < |x′ − y|. Then we can computeη ≤ |z − x| ≤ |z − y|+|y − x| < ∣∣x′ − y∣∣+|y − x| ≤ ∣∣x′ − x∣∣+2|x− y| ≤ 3η/4.This is a contradiction, so we must have δΩ|V ≤ δΓ|V .Now, our main result in this section describes how interior distance so-lutions may be glued to form distance solutions. The idea is that if thesolutions are joined so that each point which minimizes the distance to anexternal point is in the interior of one of the solutions, then the distancefunction will not be able to detect the boundaries.Theorem 3.16. Suppose that (Γi,Σi) for i = 1, . . . , N are interior distancesolutions. Let Γ = Γ1 ∪ · · · ∪ ΓN , and suppose that(i) if x ∈ Rn+k \ Γt and y ∈ Γt are such that δΓ(x, t) = |x− y|, theny ∈ Γjt \ Σjt for some j ∈ 1, . . . , N ;(ii) the mapping t 7→ Γt is continuous with respect to dH.Then Γ is a distance solution.27Proof. In order to appeal to Lemma 3.10, we consider a test functions φtouching δΓ from below at a point (y0, t0) ∈ Γ. By assumption (ii), thesecond part of the lemma allows us to assume further that there existsx0 ∈ Rn+k \ Ωt such that δΓ(x0, t0) = |x0 − y0|. By assumption (i) thisimplies that y0 ∈ Γjt \ Σjt for some j ∈ 1, . . . , N . By the definition of aninterior distance solution, there exists a neighborhood U 3 (y0, t0) on whichδΓj is a supersolution. But since Γj ⊂ Γ, we have δΓ ≤ δΓj . Hence φ alsotouches δΓj from below at (y0, t0). Since δΓj is a supersolution, this impliesthatφt(y0, t0) + F (∇φ(y0, t0),∇2φ(y0, t0)) ≥ 0.Thus the condition of Lemma 3.10 is satisfied, and so Γ is a distance solution.Note that, in fact, the finiteness of the collection of interior distancesolutions to be glued was not used in the proof of Theorem 3.16. Therefore,any collection of interior distance solutions satisfying assumptions (i) and(ii) may be glued in this fashion. Additionally, the following corollary toTheorem 3.16 will be useful in understanding level set flows in the casewhen k = 1.Corollary 3.17. Suppose that Γ is a distance solution with t 7→ Γt contin-uous with respect to dH and K is the closure of a connected component of(Rn+k × (0, T )) \ Γ. Then Γ ∪K is also a distance solution.Proof. Note that the entire space Rn+k × (0, T ) is a distance solution, sothe subset K satisfies the assumptions of Proposition 3.15. Hence, (K, ∂K)is an interior distance solution. Since ∂K ⊂ Γ, the pair of interior distancesolutions (Γ, ∅) and (K, ∂K) satisfies the assumptions of Theorem 3.16. ThusΓ ∪K is a distance solution.As an example of Theorem 3.16, consider two round circles C10 and C20in R2 incident at a point (see Figure 3.1). Each evolves by mean curvatureflow by shrinking about its center, giving two interior distance solutions(C1, ∅) and (C2, ∅). Let L be the line through the centers of C10 and C2028��� ���� ���� ������ > �Figure 3.1: Gluing a segment between two circlesand St ⊂ L be the segment between the intersections of L with C1t andC2t . Let p(t) and q(t) be the endpoints of St. Then by Proposition 3.15,(S, {p(t), q(t)}) is an interior distance solution. Theorem 3.16 implies thatΓt = C1t ∪ C2t ∪ St is a distance solution. Note that Γ0 = C10 ∪ C20 , soby Proposition 3.12, Γt is contained in the level set flow of C10 ∪ C20 . Thisshows that intersecting smooth manifolds may produce level set flows whichare distinct from the union of their smooth flows. This idea is explored ingreater detail in Chapter 4.29Chapter 4The Fattening Phenomenon4.1 Definition and Previous ResultsIn this chapter, we consider a particular type of singularity which can arise inthe level set flows considered in Section 3.1. As was noted in that section, thelevel set flow from an arbitrary closed initial set is well-defined and unique.Furthermore, as was shown in Section 3.3, the level set flow must contain alldistance solutions. In particular, if there are multiple smooth flows whoseimages approach the initial set at t = 0 (in e.g. Hausdorff distance), theymust all be contained in the level set flow. If such non-uniqueness occurs, weexpect the level set flow to become large in some sense. This is manifestedin the fattening phenomenon.Definition 4.1. Let Γ∗ ⊂ Rn+k be a closed set, and let Γ be its n-dimensionallevel set flow on a time interval I ⊂ R. Following [6], we will say that Γ∗develops α-dimensional fattening at time t∗ ∈ I ifHα(Γt) = 0 for t ≤ t∗ and Hα(Γt) > 0 for t ∈ (t∗, t∗ + ε)for some ε > 0 and α ∈ (n, n+ k].The occurrence of fattening for curves in R2 (with n = k = 1) is fullyunderstood and provides a prototype for understanding the relationship be-tween this phenomenon and uniqueness of classical solutions. First, recall30the following theorem of Lauer [17].Proposition 4.2 ([17], Theorem 1.2 and Corollary 9.3). Suppose that γ∗is the continuous image of S1 in R2 and H1(γ∗) < ∞. Let γ be the 1-dimensional level set flow of γ∗. There exists T > 0 such that at each time0 < t < T , the topological boundary ∂γt (viewed as a subset of R2) is thedisjoint union of N > 0 smooth closed curves, each of which evolve by meancurvature flow. Furthermore, γ∗ is a Jordan curve, then N = 1.Also recall the fact that for a simple smooth closed curve in R2 evolvingby curvature, the enclosed area A(t) satisfies∂A∂t= −2pi. (4.1)From these facts we summarize the characterization of fattening obtainedby Lauer.Proposition 4.3. If γ∗ is as in Proposition 4.2 and ∂γt has N componentsfor 0 < t < T , then either(i) N = 1, γ∗ never develops α-dimensional fattening for any α > 1, andthere is a unique smooth curve shortening flow of γ∗ on (0, T );(ii) or N > 1, γ∗ develops 2-dimensional fattening at t = 0, and there areat least two smooth curve shortening flows of γ∗ on (0, T ).Proof. First note that if N = 1, then for 0 < t < T , γt is a simple smoothclosed curve so no fattening occurs. By the results of [13], the unique clas-sical evolution of this curve will exist up until it shrinks to a round point.Now, we assume that N > 1. Then for any time 0 < t < T , ∂γtconsists of non-intersecting curves γ1t , . . . , γNt . Without loss of generality,assume that γ1t , . . . , γN−1t are contained in the region bounded by γNt . LetK be the closure of the connected component of R2 × (0, T ) \ ∂γ which isbounded between γN and γ0 = γ1 ∪ · · · ∪ γN−1 (see Figure 4.1). Then byCorollary 3.17, Ω = ∂γ ∪K is a distance solution. In fact, it is easy to see31γ*� �γ��γ�� γ����� < �Figure 4.1: Fattening of a figure-eight curvethat Ω must be the level set flow of γ∗ as Ω cannot be enlarged withoutenlarging its boundary. Note that the area of Ω is given byH2(Ωt) = 2pi(N − 1)tby evolution of enclosed area (4.1) for each of the boundary components ofΩ. Finally, further results of Lauer imply that γN and γ0 are both smoothcurve shortening flows of γ∗ on (0, T ).Proposition 4.3 shows that in the case of continuous curves of zeroLebesgue measure in R2, fattening of γ at time t = 0 is equivalent to thenon-uniqueness of the smooth curve shortening flow originating at γ∗. Suchnon-uniqueness can be attributed to the possibility of parameterizing theinitial data in at least two non-equivalent ways. For example, in Figure 4.1,the initial curve can be parameterized in at least three ways: smoothly bytraversing the self-intersection transversely, as a Lipschitz curve with twocorners at the self-intersection, or as a Lipschitz image of S1qS1. The laterparameterizations produce the outer and inner solutions γ3t and γ1t ∪ γ2t ,while the first parameterization produces a self-intersecting distance solu-tion contained in Kt.For codimension-1 surfaces Γ∗ ⊂ Rn+1, Ilmanen has proven that a resultsimilar to Proposition 4.3 holds. In particular, Theorems 11.4 and 12.9 of[15] imply that if the n-dimensional level set flow of Γ∗ does not develop32n-dimensional fattening, there is a unique “boundary motion” of Γ∗ whichcan be thought of as a kind of maximal Brakke flow. While the detailsof this result are outside the scope of this thesis, the tools that we havedeveloped allow us to prove that non-fattening implies uniqueness of smoothcodimension-1 flows. (Note that the while the statement of the this resultis not taken directly from any of the references, the principle is well-knownand not original to this work.)Proposition 4.4. Suppose Γ∗ ⊂ Rn+k has an n-dimensional level set flowΓ which does not develop n-dimensional fattening at t = 0. Then there is atmost one embedded smooth flow M of Γ∗ satisfying limt→0 dH(Mt,Γ∗) = 0.Proof. Suppose that there exist two different smooth flows M and N of Γ∗which approach Γ∗ in Hausdorff distance. Note that M and N are distancesolutions by the results of Section 3.2. Furthermore, it is easy to see that iflim inft→0Mt 6⊂ Γ∗, for some there would be points inMt at least ε away fromΓ∗ for arbitrarily small t. Hence we could not have limt→0 dH(Mt,Γ∗) = 0.Therefore, the initial inclusions M0 ⊂ Γ∗ and N0 ⊂ Γ∗ hold.As in the proof of Proposition 4.3, our approach is to glue in the regionbetween Mt and Nt. Let δM and δN be the signed distance functions toMt and Nt, defined so that{x∣∣ δM (x, t)} is compact, and likewise for δN .DefineKt ={x∣∣ δM (x, t)δN (x, t) ≤ 0} .It is easy to check that Kt satisfies the assumptions of Corollary 3.17, and soΩ = K ∪M ∪N is a distance solution. Since M and N are distinct smoothsolutions, by continuity Kt (and thus Ωt) has non-empty interior for t > 0.Finally, one can see thatdH(Kt,Γ∗) ≤ dH(Mt ∪Nt,Γ∗) ≤ max {dH(Mt,Γ∗), dH(Nt,Γ∗)} → 0so K satisfies the initial inclusion K0 ⊂ Γ∗. Hence Ω ⊂ Γ and Γ has n-dimensional fattening at t = 0.As an application of Proposition 4.4, we consider the case when Γ∗ isstar-shaped about a point x0 ∈ Rn+1. That is, each ray originating at33x0 intersects Γ∗ exactly once. Under this hypothesis, we will show thatthe level set flow of Γ∗ does not develop n-dimensional fattening for smallt > 0, and therefore Γ∗ admits at most one embedded smooth flow Mt withlimt→0 dH(Mt,Γ∗) = 0. Note that Soner has proven a similar result usingrather different methods [22].Proposition 4.5 ([22], Theorem 9.3). Let Ω ⊂ Rn+1 be a compact domainand assumed Γ∗ = ∂Ω is star-shaped about x0, then its level set flow Γ doesnot develop n-dimensional fattening at t = 0.Proof. Without loss of generality, suppose that x0 = 0. For x ∈ Rn+1 \ {0}define p(x) to be the unique intersection the ray from 0 through x with Γ∗.Let ρ(x) = |p(x)||x| . Now defineu0(x) =1/2 if ρ(x) ≤ 1/23/2 if ρ(x) ≥ 3/2ρ(x) otherwise.Note that Γ∗ is the 1-level set of u, and u0 is uniformly continuous. Notethat in codimension-1, Lemma 3.4 holds for every level set see e.g. [12].That is, the level set flow of the λ-level set of u0 is given by the λ-level setsof the solution u to (3.5) with initial data u0. In particular Γ is the 1-levelset of u.Now suppose to the contrary that Γ develops n-dimensional fatteningat t = 0. Let Γλ be the level set flow of λΓ∗. By Lemma 3.5 we haveΓλt = λΓλ−1t. On the other hand, for12 ≤ λ ≤ 34 , λΓ∗ is exactly the λ-levelset of u0. Therefore for12 < λ <32 , the λ-level sets of u are given by1λΓλt.Thus, if Hn(Γt) > 0 for 0 < t < t0, an uncountable number of level setsof u will have positive Hn-measure at some small positive time, which isimpossible.In the case when k > 1, much less is known about the fattening phe-nomenon. On one hand, the results of Herskovitz discussed in Section 3.2show that if the initial data Γ∗ is a ε-Reifenberg set with ε sufficiently small,34then Γ will not develop α-dimensional fattening at t = 0 for any α > n. (Infact even if dimH Γ∗ > n, then Γ will “thin” down to dimension n for somepositive time.) On the other hand, it is possible for sets which are initiallysmooth to develop self-intersections from which fattening occurs after finitetime. One known example of this phenomenon was produced by Bellettini,Novaga, and Paolini. Using methods based on the geometric barrier formu-lation of di Giorgi, they showed in some special cases that disjoint curvesin R3 develop 3-dimensional fattening at the time of their first transverseintersection [6]. This suggests that, even in the case of curves, there maynot be a characterization of fattening as simple as that in Proposition Fattening of Immersed CurvesIn this section we extend the example of Bellettini et al. of fattening ofcurves in R3 mentioned above. To introduce our results, we first give amore detailed description of this example. Bellettini et al. consider a pairof embedded closed curves in R3 which lie in distinct planes, and whichare initially linked. Up until the time at which they intersect, these curvesevolve smoothly by curve shortening flow. It is shown that from the time ofthe intersection onward, there is (in the language of this thesis) a distancesolution which remains connected. There is also a disconnected distancesolution consisting of the smooth evolutions of the original curves. Theauthors then use a similar method to that in Corollary 3.17 to prove fatteningby joining together these distance solutions. The proof of the existence ofthe connected distance solution relies heavily on the fact that the initialcurves are planar. Hence, the method is not applicable to single curveswhich develop a self-intersection, such the one shown in Figure 4.2.Our aim is generalize the example of linked planar curves by provingthat if any smooth curve in R3 (possibly non-compact or with multiple com-ponents) evolving by curve shortening flow has a transverse self-intersection,then the corresponding level set flow develops fattening at the time of theself intersection. This result was expected by Bellettini et al. and broadensthe cases in which fattening is known to occur. As with previous examples35Figure 4.2: A twisted curve in R3of fattening, the method will be to apply Theorem 3.16 to prove that thelevel set flow contains a large set after the time of the intersection. In par-ticular, we consider suitably chosen planes Σ through the intersection point.We show that Σ contains two distinct intersections with the evolving curveafter the time of the self-intersection. We then construct a distance solutioncontained in Σ which connects these two intersections. Finally, we showthat all of these distance solutions must be contained in the level set flow.In this section, M will be a 1-manifold (possibly disconnected) andγ : M × I → R3 will be a smooth immersed curve evolving by the curveshortening flow on some time interval I. That is∂∂tγ = κN (4.2)where κ is the curvature of γ and N is the Frenet unit normal vector. Notethat the combination κN is always well-defined even though N may not be.The unit tangent vector of γ will be denoted by T .To set up our construction, we first define the type of self-intersectionswhich we consider. The following definition ensures that a self-intersectionwill immediately break apart and that the curve does not lie in a singleplane.Definition 4.6. A self-intersection γ(p1, t) = γ(p2, t) for p1 6= p2 will be36called strongly transverse if〈κ(p1, t)N(p1, t), κ(p2, t)N(p2, t)〉 < 0 (4.3)anddim span {N(p1, t), T (p1, t), N(p2, t), T (p2, t)} = 3. (4.4)Note that the normal vectors in (4.4) are well-defined because (4.3) ensuresthat κ(p1, t) and κ(p2, t) are non-zero.From now on, we will assume without loss of generality that γ has astrongly transverse self-intersection at t = 0 such that γ(p1, 0) = γ(p2, 0) =0. We proceed by constructing a large family of planes which have well-controlled intersections with γ(·, t) for t > 0.Lemma 4.7. With the above assumptions, for all ε > 0 there exists ν ∈ S2such that ∣∣∣∣〈κ(pi, 0)N(pi, 0), ν〉〈T (pi, 0), ν〉∣∣∣∣ < ε for i = 1, 2.and 〈T (pi, 0), n〉 6= 0 for i = 1, 2.Proof. For brevity, we let Ni = N(pi, 0), Ti = T (pi, 0) and κi = κ(pi, 0) fori = 1, 2.There are two cases to consider. First, if N1 and N2 are linearly depen-dent, then we can write ∣∣∣∣〈κ2N2, ν〉〈T2, ν〉∣∣∣∣ = ∣∣∣∣〈λκ2N1, ν〉〈T2, ν〉∣∣∣∣for some λ, and choose ν such that 〈N1, ν〉 = 0 while 〈Ti, ν〉 6= 0 for i = 1, 2.On the other hand, suppose that N1 and N2 are linearly independent.Then N1 ×N2 6= 0. We claim that one of 〈N1 ×N2, T1〉 or 〈N1 ×N2, T2〉 isnon-zero, for if both were zero, then the setsA = {N1, N2, T1}B = {N1, N2, T2}37would both be linearly dependent. Thus the nullspace of the matrix X withA∪B as columns would have dimension at least 2, and X would have rankat most 2, contradicting (4.4). Now, without loss of generality, suppose that|〈N1 ×N2, T1〉| > 0. Then we can choose ν such that 〈κ2N2, ν〉 = 0 and ν isclose enough to N1 ×N2 that ∣∣∣∣〈κ1N1, ν〉〈T1, ν〉∣∣∣∣ < εbecause the norm of the denominator is bounded below as ν → N1×N2.Lemma 4.7 now allows us to apply the implicit function theorem tochoose planes Σ through the origin for which the intersections of Σ andγ move away from each other after t = 0 and the curves traced by theintersection points have bounded gradient (in a sense made precise below).Proposition 4.8. Suppose that γ is as above. For ν ∈ S2, let Σ(ν) bethe plane through 0 ∈ R3 with normal vector ν. There exists a closed ballN ⊂ S2, a time T > 0, and δ0 > 0 such for any ν ∈ N , there exist smoothαi : [0, T ]→ Σ(ν) (i = 1, 2) such that(i) αi(0) = 0 and αi(t) ∈ γ(M, t);(ii) 〈α′1(t), α′2(t)〉 < 0;(iii) and∣∣∣∣〈 α′i(t)|α′i(t)| , αˆ〉∣∣∣∣ ≥ δ0 for t ∈ [0, T ]where αˆ is a unit vector in the direction α′1(0)− α′2(0).Proof. Let ε > 0 and choose ν0 according to Lemma 4.7. Choose a neigh-borhood U 3 p1 on which we can parameterize γ by arc length s such thatat t = 0, s = 0 corresponds to p1. Consider the function F : S2×U × I → Rgiven byF (n, s, t) = 〈γ(s, t), n〉.Note that we have F (ν0, 0, 0) = 0 and Fs(ν0, 0, 0) = 〈T (p1, 0), ν0〉 6= 0 by 4.7.Hence, by the implicit function theorem, there exists a neighborhood V ⊂38S2× I containing (ν0, 0) and a function s1 : V →M such that s1(n0, 0) = 0and F (ν, s1(ν, t), t) = 0. Let α1(ν, t) = γ(s1(ν, t), t). Define α2(ν, t) in thesame way, using p2 in place of p1.By definition of F , we have αi(ν, t) ∈ Σ(ν)∩ γ(M, t) hence condition (i)is satisfied. The implicit function theorem and the evolution of γ also giveα′i(ν, 0) = κ(pi, 0)N(pi, 0)−〈κ(pi, 0)N(pi, 0), ν〉〈T (pi, 0), ν〉 T (pi, 0).Using the fact that 〈κ(p1, 0)N(p1, 0), κ(p2, 0)N(p2, 0)〉 < 0, we can ensurethat the choice of ε above is small enough that 〈α′1(ν0, 0), α′2(ν0, 0)〉 is nega-tive. (Note also that choosing such ε depends only |κ(pi, 0)|.) Furthermore,a simple computation shows that 〈α1(ν0, 0), α2(ν0, 0)〉 < 0 implies that αˆexists and ∣∣∣∣〈 α′i(ν0, 0)|α′i(ν0, 0)| , αˆ〉∣∣∣∣ > 0.Using the fact that conditions (ii) and (iii) hold for ν = ν0 and t = 0,by continuity of α1 and α2 we can choose a cylinder Bρ(ν0)× [0, T ] ⊂ V onwhich they hold uniformly.Note that because the smooth solution γ exists past the time T given inProposition 4.8, all of the time derivatives of the αi are uniformly boundedon [0, T ]. In particular |α′i| ≤ κ0 where κ0 is the upper bound on thecurvature of γ on [0, T ].Our goal is now to find, for each ν ∈ N ⊂ S2, a connected set ην ⊂Σ(ν)× [0, T ] such that(ην ,{αi(ν, t)∣∣ i = 1, 2}) is an interior distance solution(ην)0 = {0}(4.5)Ideally, such a solution would simply be a curve evolving by curve shorteningflow with endpoints α1(ν, t) and α2(ν, t); however, the problem of findingsuch a curve is ill-posed since the nominal initial curve is not regular. Toavoid this problem, we will find such a curve on the time interval [t0, T ]and show that we can obtain a solution to (4.5) by taking a weak limit39as t0 ↘ 0. It turns out that this limit may not be a curve, but this isacceptable for our purposes. For the time being, we fix a particular ν ∈ Nand let αi(t) = αi(ν, t). The gradient bounds above let us represent αi as agraph over αˆ.Lemma 4.9. Let {e1, e2} be orthonormal vectors in Σ(ν) with e1 = αˆ, andlet {x, y} be the corresponding coordinates of Σ(ν). For i = 1, 2, there existfunctions βi : [0, Bi] → R and xi : [0, T ] → [0, Bi] such that xi(0) = 0 andαi(t) = xi(t)e1 + βi(xi(t))e2. Furthermore, |β′i(x)| < δ−10 , the functions xiare monotonic, and B1 < 0 < B2.Proof. Let xi(t) = 〈αi(t), e1〉. Then using (ii) and (iii) from Proposition 4.8,we see that xi(t) is monotonic and therefore invertible. Thus we can defineβi(x) = 〈αi(x−1i (x)), e2〉. Clearly this gives αi(t) = xi(t)e1 +βi(xi(t))e2. Wecan computeβ′i(x) =〈α′i(x−1i (x)), e2〉〈α′i(x−1i (x)), e1〉and using condition (iii) again we have∣∣β′i(x)∣∣ < 1δ0∣∣∣∣〈 α′i(t)|α′i(t)| , e2〉∣∣∣∣ ≤ δ−10 .Finally, we have Bi = xi(T ). Using the monotonicity of the xi along with thefact that 〈α′1(0), α′2(0)〉 < 0 the Bi must have opposite sign. By relabelingif necessary we have B1 < 0 < B2.Using the notation from Lemma 4.9, for a time interval I, letΩI ={(x, t)∣∣ t ∈ I and x1(t) < x < x2(t)} (4.6)as usual ΩI will be the closure of ΩI and ∂PΩI will be the parabolic bound-ary. We may now consider the restricted graphical problemut =uxx1+u2xin Ω[t0,T ]u(xi(t), t) = βi(xi(t)) for i = 1, 2u(x, t0) = u0(x) for x ∈ [x1(t0), x2(t0)].(4.7)40By the usual graphical formulation of the curve shortening flow problem, itis clear that the graphs of solutions to (4.7) are solutions to (4.5) on thetime interval [t0, T ].Proposition 4.10. Let x0i = xi(t0) for i = 1, 2. Suppose that u0 : [x01, x02]→R is a smooth function which satisfies the compatibility conditionsu0(x0i ) = βi(x0i ) (4.8)(u0)x(x0i ) = (βi)x(x0i ) (4.9)(u0)xx(x0i ) = 0. (4.10)Then there exists a smooth solution of (4.7) on Ω¯[t0,t1] for some t1 > t0.Proof. We first need to construct a smooth function G : Ω[t0,T ] → R suchthat G(xi(t), t) = βi(xi(t)) for i = 1, 2 and G(x, t0) = u0(x), andGt(x, t0) =Gxx(x, t0)1 +Gx(x, t0)2. (4.11)The only thing that needs to be checked is that (4.11) holds at x = x0i , asG can be always be chosen so (4.11) holds for x ∈ (x01, x02). To do this, wecan computeGx(xi(t), t)x′i(t) +Gt(xi(t), t) = (βi)x(xi(t))x′i(t)which implies thatGt(x0i , t0) = ((βi)x(x0i )−Gx(x0i , t0))x′i(t0) = ((βi)x(x0i )−(u0)x(x0i ))x′i(t0) = 0using the compatibility condition on u0. On the other hand,Gxx(x0i , t0) = (u0)xx(x0i ) = 041so (4.11) holds at x = x0i . Thus, (4.7) reduces tout = uxx1+u2x in Ω[t0,T ]u|∂PΩ[t0,T ] = G|∂PΩ[t0,T ] . (4.12)Standard theory on quasilinear parabolic equations ([16], Theorem VI.4.1)now implies the existence of a smooth solution on Ω[t0,t1] for some t1 >t0.In order to complete the construction of solutions to (4.5), we must showthat the short time solutions given by (4.10) can be extended to [t0, T ], andthat we can obtain a solution in the limit as t0 ↘ 0. The following uniformgradient bound will serve both of these needs.Proposition 4.11. Suppose that u is a smooth solution to (4.7) on Ω[t0,t1)with |(u0)x| < δ−10 . Then |ux| ≤ δ−10 on Ω¯[t0,t1).Proof. First we show that |ux(xi(t), t)| ≤ δ−10 for t ∈ [t0, t1). To simplifymatters, we will prove ux(x2(t), t) ≤ δ−10 . The same proof will work for theother cases. Suppose toward a contradiction that at t′ ∈ [t0, t1) we haveux(x2(t′), t′) > δ−10 . Let `(x) = δ−10 (x− x2(t′)) + β2(x2(t′)) describe the linethrough α2(t′) with slope δ−10 . We claim that there exists x′ ∈ (x1(t′), x2(t′))such that `(x′) = u(x′, t′). In particular, note that for some ε > 0 we have`(x2(t′) − ε) − u(x2(t′) − ε, t′) > 0 by our assumption on ux. A simplecomputation using the bound (βi)x < δ−10 and the fact that x1(t′) < 0 showsthat `(x1(t′))− u(x1(t′), t′) < 0 (see Figure 4.3). Hence by the intermediatevalue theorem there exists x′ as claimed above. Note that the graphs of uand ` must cross transversely at (x′, u(x′, t′)), and thus by continuity thiscrossing must have existed on some time interval before t′.Let G(u) and G(`) denote the graphs of u and `. Both G(u) and G(`)move by curve shortening flow on their interiors. Thus the intersectionprinciple of Angenent ([3], Section 5), implies that the number of crossingsbetween these curves is non-increasing as long as the endpoints of G(u)remain disjoint from G(`). It is easy to see that t′ is the first time that anendpoint of G(u) intersects G(`). Furthermore, G(`) is disjoint from G(u0)42α� α��� ℓ� ��α� α��ℓ� �′Figure 4.3: Schematic of the proof of the gradient bound in Proposition 4.11.The shaded region on the left shows the area that u0 may lie in. Note thatit must be disjoint from `.by the assumption that |ux| ≤ δ−10 . Hence we have obtained a contradictionas the number of interior intersections of G(u) and G(`) must be zero up totime t′. This completes the proof that |ux(xi(t), t)| ≤ δ−10 for t ∈ [t0, t1).Now, we apply the maximum principle to prove the result on the entiredomain. Denote by ∂∂s the operator which gives the derivative with respectto arc length. It is well known (see, e.g. [1]) that for a curve moving bycurve shortening flow with tangent vector T and V a fixed vector, we have∂∂t〈T, V 〉 = ∂2∂s2〈T, V 〉+ 2κ2〈T, V 〉. (4.13)In our case, ∂∂s =1√1+u2x∂∂x and we take V = e2 to obtain∂∂t(ux1 + u2x)=∂2∂s2(ux1 + u2x)+ 2κ2(ux1 + u2x). (4.14)Thus 〈T, e2〉 = ux1+u2x satisfies the parabolic maximum principle. Combiningthe assumptions and the first part of this proof give bounds on ∂PΩ[t0,t1),which then extend to the interior. Thus |〈T, e2〉| < δ−101+δ−20, from which theconclusion follows.It is well known that it is possible to obtain strong curvature bounds43for graphical solutions of mean curvature flow satisfying a uniform gradientbound.Proposition 4.12. Suppose that u is a solution to (4.7) with |ux| ≤ δ−10 .Let κ = uxx(1+u2x)3/2 . Recall that κ0 is the uniform bound on the curvature ofthe original evolving curve γ. We have(a)∣∣∣∂lκ∂sl ∣∣∣ ≤ Cl(u0, δ0, ‖α1‖Cl , ‖α2‖Cl) for l = 0, 1, 2, . . .;(b) and (t− t0)κ(x, t)2 ≤ C(δ0, κ0) for (x, t) ∈ Ω(t0,t1).Proof. These bounds follow from well-known results of Ecker and Huisken[11]. Proposition 4.12(a) is proven via an inductive application of the max-imum principle to the evolution equations satisfied by the derivatives of κ([11], Proposition 4.3). For Proposition 4.12(b), see [11], Proposition 4.4.The only modification necessary is the incorporation of the contribution ofthe derivatives of the boundary data αi into the constants.These bounds allow us to extend the solution up to time T using theusual long-time existence procedure (see, e.g. [11], Theorem 4.6).Lemma 4.13. With the same notation as Proposition 4.10, if |(u0)x| ≤ δ−10 ,there exists a smooth solution of (4.7) on Ω[t0,T ] satisfying the bounds ofProposition 4.12.Proof. Let A ⊂ [t0, T ] be the set of times up to which the maximal solutionof (4.7) is defined. By Proposition 4.10, A is open and non-empty. Sinceu0 satisfies the hypothesis of Proposition 4.11, the maximal solution hasbounded gradient for all time, and therefore Proposition 4.12(a) and theArzela-Ascoli theorem imply that A is closed. Therefore A = [t0, T ].In order to make use of the long-time existence theory above, we needto construct appropriate initial data for each time t0 > 0. Note that we donot assert any curvature bounds on our choice of initial data, and in fact,none are possible if α′1(0) 6= −α′2(0). Therefore, the curvature independentbound of Proposition 4.12(b) is crucial in obtaining convergence below.44Lemma 4.14. For each t0 > 0, there exists an initial function u0 : [x01, x02]→R, satisfying the compatibility condition of Proposition 4.10 and with |(u0)x| <δ−10 .Proof. Let ε > 0 and defineu˜0(x) =β1(x01) + (β1)x(x01)(x− x01) if x01 ≤ x ≤ x01 + εβ2(x02) + (β2)x(x02)(x− x02) if x02 − ε ≤ x ≤ x02ξ(x) otherwisewhere ξ(x) is the linear function which makes u˜0 continuous. By taking εsmall enough, we can see that u˜0 is Lipschitz with constant less than δ−10 byusing the bounds on the βi and their derivatives. Thus, we can obtain u0by smoothing u˜0 slightly near the two corners at x01 + ε and x02 − ε.To obtain weak limits of the solutions to (4.7) on [t0, T ] from which wecan construct the desired interior distance solution, we need to ensure thatthese limits will have sufficient regularity. In particular, we show that halfrelaxed limits preserve local Lipschitz constants.Lemma 4.15. Let U be a connected domain in Rd. Suppose that un : U → Ris a sequence such that for each K ⊂⊂ U , un|K is L-Lipschitz for someconstant L depending only on K. The upper and lower half relaxed limitsu¯ = lim sup∗ un and¯u = lim inf∗ un also satisfy this property with the sameLipschitz constants.Proof. We prove the result for the upper half relaxed limit. Let x, y ∈ Uand K ⊂ U be a compact set containing x and y. Then un|K is Lipschitzwith constant L. Suppose xn → x and yn → y. Without loss of generalitywe can assume that xn ⊂ K and yn ⊂ K. Then we have|un(xn)− un(yn)| ≤ L|xn − yn| ⇒ un(xn) ≤ L|xn − yn|+ un(yn).Taking the lim sup of both sides giveslim supun(xn) ≤ L|x− y|+ lim supun(yn).45Taking the supremum over such sequences xn and yn givesu¯(x)− u¯(y) ≤ L|x− y|.To obtain the lower bound, the argument can be repeated, switching theroles of x and y. For the lower half-relaxed limit, the same proof applies.Finally, we can put together the results above to obtain existence ofsolutions to (4.5).Proposition 4.16. There exists a solution η to (4.5) such that ηt consists ofthe region in Σ(ν) bounded by two Lipschitz curves joining α1(t) and α2(t).Proof. By Lemma 4.13, for each n > 1, we obtain a solution un of (4.7) onΩ[T/n,T ] with initial data un0 given by Lemma 4.14. If K ⊂⊂ Ω(0,T ], thenfor some N and all n > N , un will be defined on K. The uniform gradientbound of Proposition 4.11 combined with the interior time-derivative boundof Proposition 4.12(b) shows that un|K is L-Lipschitz for some constantdepending L on δ0, κ0, and K only. Thus, by Lemma 4.15,u¯ = lim sup∗ un and¯u = lim inf∗ unare locally Lipschitz with u¯ ≥¯u. Furthermore, by Lemma 2.8, u¯ (resp.¯u)is a subsolution (resp. supersolution) of (4.7) on Ω(0,T ]. We defineηt ={xe1 + ye2∣∣ x1(t) ≤ x ≤ x2(t) and¯u(x) ≤ y ≤ u¯(x)}for t > 0 and η0 = {0} and∂η ={(x, t)∣∣ t ∈ [0, T ], x = α1(ν, t) or x = α2(ν, t)} .It remains to be shown that (η, ∂η) is an interior distance solution. Supposethat (x, t) ∈ η \ ∂η. By a similar argument to that in the proof of Proposi-tion 3.15, there exists a neighborhood (x, t) ∈ U ⊂ Σ(ν) × I such that for(x0, t0) ∈ U and y0 ∈ ηt with δη(x0, t0) = |x0 − y0|, we have y0 ∈ ηt\∂ηt∩U .We will show that δη|U is a supersolution of (3.4) with n = k = 1. The proof46of Proposition 3.13 will then imply that δη|U×R is a supersolution of (3.4)with k = 2.The strategy to do this is to convert test functions for η into test functionsfor¯u and u¯, and then apply the equations obtained using the fact that¯uand u¯ are supersolutions and subsolutions of (4.7) (see Figure 4.4). We firstconsider a point (p0, t0) ∈ U such that p0 is given in the coordinates usedabove as (x0, y0) and satisfies¯u(x0, t0) > y0.Let φ : Σ(ν) × I → R be a C2 function touching δ from below at (p0, t0).Defineφ˜(q0, t) = φ(x+ p0 − q0, t)− |p0 − q0|where q0 ∈ η \ ∂η is such that δη(p0, t0) = |p0 − q0|. LetZt ={φ˜(x, t) = 0∣∣ x ∈ Σ(ν)} .We claim(i) there exists a neighborhood V 3 (q0, t0) such that Zt ∩ Vt lies belowthe graph of¯u(·, t);(ii) and φ˜y(q0, t0) < 0.To see (i), let W 3 (p0, t0) be such that φ ≤ δη on W , and define V ={(x− p0 + q0, t)∣∣ (x, t) ∈W}. Let (z˜, t) ∈ V ∩Z. Then there exists (z, t) ∈W such that z˜ = z − p0 + q0. We then have0 = φ˜(z˜, t) = φ(z˜ + p0 − q0, t) = φ(z, t)− |p0 − q0|.By definition of W , this impliesδ(z, t) ≥ φ(z, t) = |p0 − q0|.Since |z˜ − z| = |p0 − q0| either z˜ lies outside ηt or on the boundary of ηt.Therefore z˜ must lie below the graph of¯u.47To see (ii), note that ∇φ˜(q0, t0) = ∇φ(p0, t0) 6= 0, since φ touches adistance function from below away from its zero set. Thus, if φ˜y(q0, t0) = 0,the implicit function theorem would give w such that φ˜((w(y, t), y), t) = 0 ina neighborhood of (q0, t0). Unless the graph of¯u had a vertical tangent atq0, this would contradict (i). Since¯u is locally Lipschitz, this is impossible.Furthermore, note that for ε small, we haveφ((x0, y0 + ε), t0) ≤ δη((x0, y0 + ε), t0) ≤ δη((x0, y0), t0) = φ(p0, t0)since p0 lies below the graph of¯u. Hence φy(p0, t0) ≤ 0. Combining thesefacts gives φ˜y(q0, t0) < 0.From (i) and (ii), the implicit function theorem gives v such thatφ˜((x, v(x, t)), t) = 0 (4.15)and v(x, t) ≤¯u(x, t) (4.16)for (x, t) in a neighborhood of ((q0)x, t0). Note also that v((q0)x, t0) = (q0)y,so v touches¯u from below at ((q0)x, t0). Therefore, by the fact that¯u is asupersolution of (4.7), we havevt ≥ vxx1 + v2x. (4.17)at ((q0)x, t0). Differentiating (4.15) and substituting into (4.17) gives− φ˜tφ˜y− 1φ˜yF (∇φ˜,∇2φ˜) ≥ 0at (q0, t0). Using (ii) and the definition of φ˜, this givesφt + F (∇φ,∇2φ) ≥ 0at (p0, t0), which shows that δη is a supersolution of (3.4) on the portion of Ulying below the graph of¯u. The same argument, with inequalities reversed,and using the fact the u¯ is a supersolution, implies that δη is likewise a48α�α����(�)��Figure 4.4: Schematic of the proof of Proposition 4.16. The shaded regionis the interior of (ην)t.supersolution on the portion of U lying above the graph of u¯. Finally, weapply Lemma 3.11 to conclude that δη is a supersolution on all of U , andtherefore (η, ∂η) is an interior distance solution.Theorem 4.17. Suppose that γ : M × [−ε, ε) → R3 is a smooth immersedcurve evolving by curve shortening flow which has a strongly transverseself-intersection at time 0. Then the level set flow Γ of γ−ε develops 2-dimensional fattening at time 0.Proof. Let N ⊂ S2 and 0 < T < ε be as in Proposition 4.8. Then byProposition 4.16, for each ν ∈ N , there exists a solution ην of (4.5) definedon [0, T ]. By the construction of these solution, Theorem 3.16 shows thatthe setΩ = γ(M, [−ε, ε)) ∪⋃νην49is a distance solution on [−ε, T ). (Note that γ(M, [−ε, ε]) is a distancesolution because it is locally a union of embedded smooth flows.) ThereforeΩt ⊂ Γt for −ε ≤ t < T .At time 0 < t < T , there must be at least one curve in N on whichthe maps ν 7→ αi(ν, t) are injective and the segment `ν joining α1(ν, t) toα2(ν, t) does not pass through the origin. Then each ην for ν along this curvecontains at least one Lipschitz curve (say the graph of¯u). Furthermore, bythe condition on `ν , this curve has a segment with length bounded belowwhich lies only in Σ(ν). This is enough to show that H2(Γt) > 0.As an example of the type of case which Theorem 4.17 addresses whichis not covered by [6], consider the twisted curve depicted in Figure 4.2.By symmetry, it is easy to see that the small central twist must develop astrongly transverse self intersection after a finite amount of time. At thispoint, we conclude that the level set flow will develop (at least) 2-dimensionalfattening.50Chapter 5ConclusionIn this thesis, we have studied generalized evolutions of submanifolds bymean curvature flow, focusing on the level set solutions of Ambrosio andSoner, and the fattening phenomenon which occurs with such solutions.Apart from presenting some of the existing results on this phenomenon, ourmain contributions have been to prove Theorem 3.16, a new gluing resultfor distance solutions, and to use this result to prove Theorem 4.17 whichdemonstrates the occurrence of fattening when immersed curves developself-intersections.Theorem 4.17 may help to understand the fattening phenomenon inhigher codimension in several ways. First, it verifies a case of fatteningwhich was suspected, but not proven to occur except in some very specialcases [6]. Second, it provides an interesting piece of information when consid-ering the relationship between fattening and non-uniqueness. In particular,the existence results for smooth flows with rough initial data in higher codi-mension generally require a smallness assumption (on e.g. the Lipschitz orReifenberg constants of the initial data [14]). On the other hand, such anassumption may not be satisfied by a parameterization of a self-intersectingcurve. Thus, it seems plausible that the fattening proven in Theorem 4.17is not directly due to the existence of multiple smooth solutions, in con-trast to the case of curves in R2 (Proposition 4.3). Finally, we note thatwhile some of results in Section 4.2 were specific to curves, the basic method51of constructing multiple codimension-1 distance solutions confined to affinesubspaces in order to prove fattening may be applicable in other situations.For example, the same method could feasibly be used to demonstrate fatten-ing when a 2-dimensional surface in R4 develops a self-intersection along acurve. Constructing such examples of fattening in higher codimension mayfurther illuminate our understanding of this phenomenon, and subsequentlyallow for more applications of the mean curvature flow.52References[1] Steven J. Altschuler and Matthew A. Grayson. “Shortening spacecurves and flow through singularities”. In: J. Differential Geom. 35.2(1992), pp. 283–298.[2] Luigi Ambrosio and Halil Mete Soner. “Level set approach to meancurvature flow in arbitrary codimension”. In: J. Differential Geom.43.4 (1996), pp. 693–737.[3] Sigurd Angenent. “Nodal properties of solutions of parabolicequations”. In: Rocky Mountain J. Math. 21.2 (1991). Currentdirections in nonlinear partial differential equations (Provo, UT,1987), pp. 585–592.[4] Giovanni Bellettini. Lecture notes on mean curvature flow, barriersand singular perturbations. Vol. 12. Appunti. Scuola NormaleSuperiore di Pisa (Nuova Serie) [Lecture Notes. Scuola NormaleSuperiore di Pisa (New Series)]. Edizioni della Normale, Pisa, 2013,pp. xviii+325.[5] Giovanni Bellettini and Matteo Novaga. “Comparison resultsbetween minimal barriers and viscosity solutions for geometricevolutions”. In: Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26.1(1998), pp. 97–131.[6] Giovanni Bellettini, Matteo Novaga, and Maurizio Paolini. “Anexample of three-dimensional fattening for linked space curvesevolving by curvature”. In: Comm. Partial Differential Equations23.9-10 (1998), pp. 1475–1492.[7] Kenneth A. Brakke. The motion of a surface by its mean curvature.Vol. 20. Mathematical Notes. Princeton University Press, Princeton,N.J., 1978, pp. i+252.53[8] Lia Bronsard and Robert V. Kohn. “Motion by mean curvature asthe singular limit of Ginzburg-Landau dynamics”. In: J. DifferentialEquations 90.2 (1991), pp. 211–237.[9] Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto. “Uniquenessand existence of viscosity solutions of generalized mean curvatureflow equations”. In: J. Differential Geom. 33.3 (1991), pp. 749–786.[10] Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions. “User’sguide to viscosity solutions of second order partial differentialequations”. In: Bull. Amer. Math. Soc. (N.S.) 27.1 (1992), pp. 1–67.[11] Klaus Ecker and Gerhard Huisken. “Mean curvature evolution ofentire graphs”. In: Ann. of Math. (2) 130.3 (1989), pp. 453–471.[12] L. C. Evans and J. Spruck. “Motion of level sets by mean curvature.I”. In: J. Differential Geom. 33.3 (1991), pp. 635–681.[13] Matthew A. Grayson. “The heat equation shrinks embedded planecurves to round points”. In: J. Differential Geom. 26.2 (1987),pp. 285–314.[14] Or Hershkovits. Mean Curvature Flow of Arbitrary Co-DimensionalReifenberg Sets. 2015. eprint: arXiv:1508.03234.[15] Tom Ilmanen. “Elliptic regularization and partial regularity formotion by mean curvature”. In: Mem. Amer. Math. Soc. 108.520(1994), pp. x+90.[16] O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Ural’tseva. Linearand Quasi-linear Equations of Parabolic Type. Translations ofMathematical Monographs. American Mathematical Society, 1988.[17] Joseph Lauer. “A new length estimate for curve shortening flow andlow regularity initial data”. In: Geom. Funct. Anal. 23.6 (2013),pp. 1934–1961.[18] Carlo Mantegazza. Lecture notes on mean curvature flow. Vol. 290.Progress in Mathematics. Birkha¨user/Springer Basel AG, Basel,2011, pp. xii+166.[19] E. R. Reifenberg. “Solution of the Plateau Problem form-dimensional surfaces of varying topological type”. In: Acta Math.104 (1960), pp. 1–92.[20] Luis Silvestre. Viscosity Solutions of Elliptic Equations. 2015.54[21] Knut Smoczyk. “Mean curvature flow in higher codimension:introduction and survey”. In: Global differential geometry. Vol. 17.Springer Proc. Math. Springer, Heidelberg, 2012, pp. 231–274.[22] Halil Mete Soner. “Motion of a set by the curvature of itsboundary”. In: J. Differential Equations 101.2 (1993), pp. 313–372.55


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