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Inradius bounds for stable, minimal surfaces in 3-manifolds with positive scalar curvature Richardson, James 2012

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Inradius Bounds for Stable, Minimal Surfaces in 3-Manifolds with Positive Scalar Curvature by James Richardson B.Sc., The University of Western Australia, 2008 B.Sc. (Hons), Monash University, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) May 2012 c James Richardson 2012Abstract Concrete topological properties of a manifold can be found by examining its geometry. Theorem 17 of this thesis, due to Myers [Mye41], is one such example of this; it gives an upper bound on the length of any minimizing geodesic in a manifold N in terms of a lower positive bound on the Ricci curvature of N , and concludes that N is compact. Our main result, Theorem 40, is of the same  avour as this, but we are instead concerned with stable, minimal surfaces in manifolds of positive scalar curvature. This result is a version of Proposition 1 in the paper of Schoen and Yau [SY83], written in the context of Riemannian geometry. It states: a stable, minimal 2- submanifold of a 3-manifold whose scalar curvature is bounded below by  > 0 has a inradius bound of q 8 3  p  , and in particular is compact. iiTable of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The Minimal Submanifold Technique . . . . . . . . . . . . . 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The Submanifold has Non-empty Boundary . . . . . . . . . . 7 2.3 The Submanifold Has No Boundary . . . . . . . . . . . . . . 11 3 Inradius Bounds for Minimal Submanifolds . . . . . . . . . 18 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Inradius Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Appendices A The First and Second Variation Formulae . . . . . . . . . . 38 B Warped Product Metrics . . . . . . . . . . . . . . . . . . . . . 45 iiiChapter 1 Introduction A major area of study in Riemannian geometry is understanding the re- lationship between curvature and topology of Riemannian manifolds. An important and widely used tool in this area is the minimal submanifold technique. The application of this technique is the main theme of this the- sis. The main theorem in this thesis, Theorem 40 is as follows: Theorem 40. Let M be a stable, minimal, orientable, complete 2- dimensional submanifold of a 3-dimensional Riemannian manifold (N; g). Suppose that there is a globally de ned unit length normal vector e3 on M . Suppose that on M we have RN   > 0. 1. If M has no boundary then: diam (M)  r 8 3  p  : 2. If M has non-empty boundary then: dM (p; @M) := inffdM (p; q) 2 R : q 2 @Mg  r 8 3  p  for all p 2M . This theorem is a version of Proposition 1 in Schoen and Yau [SY83] but stated in the context of Riemannian geometry. In this context, Theorem 40 can be seen as an example of a theorem within the framework of the minimal submanifold technique. Chapter 2 gives an expose of this technique, starting with a discussion of the typical conditions we look for in the submanifoldM . In this technique we are concerned with submanifolds M that are minimal and stable. Roughly speaking, these are submanifolds which locally minimize volume inside the ambient space N . That is, if M is deformed slightly inside N it will not decrease in volume. To give a precise de nition of this we need two formulae, known as the  rst and second variations of volume, derived in Appendix A. 1Chapter 1. Introduction Section 2.1 introduces the  rst and second variations of the submanifold M and gives precise de nitions of minimality and stability. As discussed in Section 2.1 the minimal submanifold technique consists of two main types of theorems. The  rst assumes something about the topology of a manifold and tries to prove the existence of a stable, minimal submanifold inside the ambient space. The second assumes some curvature conditions and the existence of a particular stable, minimal submanifold and examines the second variation of the submanifold. The technique combines the two theorems together to relate the curvature of the manifold to its topology. Theorem 40 can be seen to be a theorem of the second type. Theorem 40 has two cases; the submanifold M has no boundary and the submanifold M has non-empty boundary @M 6= ;. We divide our expose into these two cases; Section 2.2 deals with the case where M has non- empty boundary @M 6= ;, and Section 2.3 deals with the case where M has no boundary. Chapter 3 gives a proof of our main theorem, Theorem 40. Section 3.1 gives a brief discussion of the conditions of the theorem and Section 3.2 gives proofs of two inradius bounds, including our main theorem, Theorem 40. The proof uses the second variation formula. Associated to this formula is a Jacobi operator L acting on functions on M . The condition of stability implies the second variation is non-negative which in turn implies L has non-negative eigenvalue. Letting f be an eigenfunction for L, the idea is then to consider a warped product manifold with warping function f and using further arguments conclude the inradius bounds in Theorem 40. The warped product manifold is a generalization of the usual product manifold and arises naturally in the study of general relativity. Appendix B gives a de nition for this warped space and some curvature calculations for warped product manifolds. The concluding chapter, Chapter 4, gives a discussion of Theorem 40 in a wider context. It begins with a discussion of how it relates to the result of Schoen and Yau [SY83], and rephrases it in their language - in terms of a bound on the \2-dimensional diameter" or \ ll radius" of the ambient space. We then give a discussion of possible extensions of this result to the case where the ambient manifold is of higher dimension. This includes two conjectures under stronger curvature assumptions. The chapter concludes with recent work that shows how these conjectures, if they are true, would give strong conclusions about the topology of the manifold. 2Chapter 2 The Minimal Submanifold Technique 2.1 Introduction The main theme in this thesis is the minimal submanifold technique. In this chapter we give an exposition of some successful applications of this technique. We begin with the basic de nitions used in the technique; speci cally, the de nition of a minimal submanifold, what it means for a minimal sub- manifold to be stable, and the de nition of the index - which gives a measure of the stability of a submanifold. A submanifold which is minimal and stable is a submanifold which, in some sense, locally minimizes volume inside the ambient space. Let (N; g) be a Riemannian manifold and M a submanifold of N . To obtain a concrete de nition for minimality and stability,  rst consider a deformation or variation of M inside N indexed by a variable t 2 (  ;  ), which leaves M unchanged when t = 0. We may write this as a map f : (  ;  )  M ! N , with f(0;M) = f0(M) = M . Then we can consider t! vol(ft(M)) as a function from R to R. If f0(M) = M locally minimizes volume we know from single variable calculus that ddt vol(ft(M))jt=0 = 0 and d2 dt2 vol(ft(M))jt=0  0. We shall derive formulae for these expressions soon but  rst we need to de ne something called the variation vector  eld which measures the direction in which M changes under a variation f . Given a variation f we de ne a variation vector  eld V := df( @@t)jt=0. This is a vector  eld on M that describes the direction ft(M) varies inside N at each point of f0(M) = M as t increases. Since we are interested in the derivative of t! vol(ft(M)) we only need to look at what ft(M) looks like locally around t = 0 and thus it will su ce to look only at the variation vector  eld V , rather than the entire function f . Variation vector  elds are sections of the bundle TN ! M . Equivalently every section of the bundle TN ! M de nes a variation vector  eld. We use the notation  (M;TN) 32.1. Introduction for these sections and the notation  c(M;TN) for V 2  (M;TN) with compact support and we may give these spaces a vector space structure in the obvious way. Variation vector  elds encode all the information we need to know about the variation so these spaces are important in deciding whether a submanifold M is locally volume minimizing. A variation vector  eld V can be split into a normal V N 2  (M;TMN ) and a tangential component V T 2  (M;TM) to M , as V = V N + V T . A variation vector  eld V tells us the direction that M varies inside N with re- spect to a variation f , so if we look only at the tangential component V T we are not gaining any information. A variation with a tangential variation vec- tor  eld amounts to a re-parameterization of M and ft(M) is unchanged as t varies. Thus is makes sense to restrict ourselves only to normal variations. Given a normal variation vector  eld V 2  (M;TN), we now derive forumulae for ddt vol(ft(M))jt=0 and d2 dt2 vol(ft(M))jt=0, known as the  rst and second variation respectively. Here M may have possible boundary @M , and if so we require that the boundary is  xed in the variation. This is equivalent to saying V = 0 on @M . We state both formulae here, the proofs can be found in Appendix A. The second variation formula depends on the vector  eld V , but also involves the curvature of the ambient manifold N . This is where the power of the minimal submanifold technique lies; if we can guarantee the existence of a stable, minimal submanifold M of N then we may be able to gain information about the curvature of N by looking at the second variation of M . Given a submanifold M of N and a variation f : (  ;  ) M ! N of M in N with normal variation vector  eld V we now state the formula for the  rst variation ddt vol(ft(M))jt=0. We assume M is compact and orientable with possible boundary @M . Proposition 1. The  rst variation of a compact, orientable submanifold M (with possible boundary @M), of a Riemannian manifold (N; g) with normal variation vector  eld V (with V = 0 on @M), is given by: d dt vol(ft(M))jt=0 =  Z M hV;HM idv where HM is the mean curvature of M in N and dv is the Riemannian volume form associated with the pullback metric f 0 g. Proof. This is Proposition 54 of Appendix A. Remark 2. We need M to be compact and orientable in order to de ne the integral in Proposition 1. However, as we shall see, we can generalize the 42.1. Introduction condition ddt vol(ft(M))jt=0 = 0 to non-compact, non-orientable submani- folds. From examining the integral in Proposition 1 we can see that the con- dition that ddt vol(f(M))jt=0 = 0 for any normal variation vector  eld V is equivalent to saying that HM is zero. This motivates the next de nition. De nition 3. A submanifold M inside N is minimal if HM = 0 on M . Remark 4. HM is de ned for non-compact and non-orientable manifolds so this is a generalization of the condition that the  rst variation is zero. Remark 5. We call 1-dimensional minimal submanifolds geodesics. The geodesic equation for such a submanifold  be recovered from the condition of minimality via a computation of the mean curvature; 0 = H = r _ _ . Given a submanifold M and a variation f : (  ;  ) M ! N with normal variation vector  eld V we derive an expression for the second variation d2 dt2 vol(ft(M))jt=0. We assume M is compact and orientable with possible boundary @M . Proposition 6. Assuming M is minimal, i.e. HM = 0, the second variation of a compact, orientable submanifold M , (with possible boundary @M), in N with normal variation vector  eld V (with V = 0 on @M), is given by: d2 dt2 vol (ft (M)) jt=0 = Z M  krV k2  X i=1 RN (ei; V; V; ei) 2k V k2 ! dv = Z M  kDV k2  X i=1 RN (ei; V; V; ei) k V k2 ! dv where D is the normal connection on TN jM !M , i.e. DXY = (rXY ) N . Proof. This is Proposition 57 of Appendix A. There is a curvature term in Proposition 6. This allows us to make topo- logical statements about the ambient manifold N under particular curvature assumptions, usually some form of positive curvature assumption. If M is a local minimum for the volume we must have the inequality d2 dt2 vol(ft(M))jt=0  0. We assumed that M was compact and orientable so that the integral in Proposition 6 would make sense. However if V were com- pactly supported it would still be well-de ned. This motivates the following de nition. 52.1. Introduction De nition 7 (Stability). An orientable submanifold M inside N is stable if it is minimal and the integral in Proposition 6 is non-negative for all compactly supported normal variation vector  elds V 2  c(M;TN). A non-orientable submanifold is stable if its orientable double cover is stable. Remark 8. The above de nition extends the condition d2 dt2 vol(ft(M))jt=0  0 to non-compact and non-orientable manifolds. De nition 9. We de ne an index form I on compactly supported V 2  c(M;TN) I(V; V ) = Z M  krV k2  X i=1 RN (ei; V; V; ei) 2k V k2 ! dv = Z M  kDV k2  X i=1 RN (ei; V; V; ei) k V k2 ! dv: Then our condition that d 2 dt2 vol(ft(M))jt=0  0 is equivalent to saying the quadratic form I is positive semi-de nite on  c(M;TN). We may use this de nition of the index form I to de ne a number as- sociated to the submanifold M which measures its stability. Known as the index this is de ned as follows: De nition 10. Given a minimal submanifold M of N , its index is de ned as the maximal dimension of a subspace of  c(M;TN) on which I is negative de nite. Remark 11. Roughly speaking, the index measures the number of linearly independent directions in the space of sections for which the second variation is negative. The lower the index, the closer M is to being locally volume minimizing. We say that, the lower the index, the more stable M is, i.e, the less likely it is to increase in volume when taking a variation. We may rewrite De nition 7 in terms of the index in De nition 10. De nition 12 (Stability 2). A minimal, orientable submanifold M in N is stable if its index is 0. Once again, a non-orientable submanifold is stable if its orientable double cover is stable. The minimal submanifold technique consists of two main types of theo- rems. The  rst, of independent interest, assumes some topological condition on a manifold, and tries to prove the existence of a minimal submanifold 62.2. The Submanifold has Non-empty Boundary with some stability, inside the ambient manifold. The second assumes we have a particular minimal submanifold and examines the second variation formula on the submanifold. For a stable, minimal submanifold the second variation must be non-negative and we can use this to gain some informa- tion about the curvature of the manifold. The idea is then to put the two theorems together. The rest of this chapter is devoted to examples of successful applications of the minimal submanifold technique. When using this technique there are two main cases that arise: the case where the submanifold has boundary and the case where it has no boundary. We deal with each of these separately. 2.2 The Submanifold has Non-empty Boundary The simplest example of this technique is Bonnet’s theorem. Here we are concerned with a stable, minimal 1-dimensional submanifold (stable geodesic) of a complete, connected Riemannian manifold N . We imme- diately have a theorem of the  rst type in this case: Proposition 13. Suppose N is a complete, connected Riemannian mani- fold. If p; q 2 N then there exists a length minimizing geodesic inside N whose boundary is fp; qg. Remark 14. This is usually stated in the following way; suppose N is a complete, connected Riemannian manifold, then for any p; q 2 N there is a length minimizing geodesic connecting p and q. Remark 15. This is a standard result in Riemannian geometry, and can normally be found as a corollary to the Hopf-Rinow theorem. This gives us a theorem of the  rst type. Our goal now is to get a theorem of the second type. We compute the second variation in the case that M =  is a unit speed geodesic  : [0; l]! N , and V is a normal variation of  which 72.2. The Submanifold has Non-empty Boundary is zero at the endpoints,  (0) and  (l). From Proposition 6 we have: d2 dt2 vol (ft (M)) jt=0 = Z M  krV k2  X i=1 RN (ei; V; V; ei) 2k V k2 ! dv = Z l 0 (hr _ V;r _ V i  RN ( _ ; V; V _ ) 2hV;r _ _ i) ds = Z l 0   @ @s hV;r _ V i  hr _ r _ V; V i   RN (V; _ ; _ ; V ) 2hV;H i  ds = [hV;r _ V i] l 0  Z l 0 (hr _ r _ V; V i  RN (V; _ ; _ ; V )) ds =  Z l 0 hr _ r _ V +R(V; _ ) _ ; V ids (2.1) where the last line follows since V is zero at 0 and l. We note that the expression in 2.1 contains a sectional curvature term so we see that the sectional curvature of N gives us information about the stability of geodesics in N . We may combine the existence result of Proposition 13 with an exami- nation of the second variation of a geodesic  under some positive curvature conditions on the ambient manifold N . This is the following theorem: Theorem 16 ([Bon55]). Let N be a complete, connected Riemannian man- ifold all of whose sectional curvatures are bounded below by  > 0. Then: 1. If  : [0; l]! N is a stable geodesic in N we have: length( )   p  2. In particular for all p; q 2M we have: d(p; q)   p  Moreover, M is compact with  nite fundamental group. Proof. Let  : [0; l] ! N be a stable unit speed geodesic of length l. We show l   p . Given e 2 TpN with kek = 1, he; _ i = 0, we may obtain a parallel normal vector  eld e along  by parallel transport. Let V=’e, where ’ : [0; l]! R is a smooth cut-o function with ’(0) = ’(l) = 0. Since 82.2. The Submanifold has Non-empty Boundary  is length minimizing it has non-negative second variation. Therefore by 2.1  Z l 0 hr _ r _ V +R (V; _ ) _ ; V ids  0: Now r _ V = r _ (’e) = ’0e + ’r _ e = ’0e since e is parallel along  . Re- peating this argument yields r _ r _ V = ’00e, and the inequality becomes:  Z l 0 h’00e+R (’e; _ ) _ ; ’eids =  Z l 0  ’00 +R (e; _ ; _ ; e)’  ’ds  0: (2.2) Now if we assume that all the sectional curvatures of M are bounded below by  > 0, then R (e; _ ; _ ; e)   and we have:  Z l 0  ’00 +  ’  ’ds  0: Substituting ’(s) = sin   s l  gives: Z l 0   2 l2    sin2   s l  ds  0:: Therefore  2 l2    0 and rearranging this gives l   p  . The second part of this statement follows easily from the  rst. Take p; q 2 N then by Proposition 13 there exists a length minimizing geodesic connecting p and q. Then from the above argument we have d(p; q) = length( )   p . From this bound it follows that for p 2 N the map expp : B(0;  p  )  TpN ! N is surjective. So N is equal to the image of a compact set under a continuous map and therefore it is compact. We now show that N has a  nite fundamental group. Let  : ~N ! N denote the universal covering space of N with the pullback metric ~g :=   g. Since N is complete and  is a local isometry one can show that ~N is also complete and the sectional curvatures ~K of ~N have the same bound, ~K   > 0. Thus ~N is also compact by the above argument. By the theory of covering spaces there is a one-to-one correspondence between  1(N) and   1(p) for any point p 2 N . If  1(N) were in nite then it would correspond to an in nite discrete set in ~N given by   1(p), contradicting the compactness of ~N . Thus  1(N) is  nite. In fact we can make slightly weaker assumptions about the curvature, as shown by Myers [Mye41]. 92.2. The Submanifold has Non-empty Boundary Theorem 17 ([Mye41]). Let N be a complete, connected Riemannian n- manifold whose Ricci tensor satis ed the following inequality for all V 2 TN ; Ric (V; V )  (n 1) kV k2 for some constant  > 0. Then if  : [0; l]! N is a stable geodesic in N we have: length( )   p  : Proof. As before suppose  : [0; l] :! N is a stable unit speed geodesic of length l. Let feigni=1 be a parallel orthonormal frame along  such that en = _ , and for each i = 1; : : : ; n  1 let Vi = ’ei where ’ = 0 at 0 and l. The same computation as 2.2 in the proof of Bonnet’s theorem (16) gives, for each i = 1; : : : ; n 1, the following inequality:  Z l 0  ’00 +R (ei; _ ; _ ; ei)’  ’ds  0: (2.3) Summing up 2.3 gives:  Z l 0  (n 1)’00 + Ric ( _ ; _ )’  ’ds: (2.4) Now Pn 1 i=1 R (ei; _ ; _ ; ei) = Ric ( _ ; _ )  (n 1) . Thus replacing this term in 2.4 yields:  Z l 0  (n 1)’00 + (n 1) ’  ’ds  0: Again substituting ’(s) = sin   s l  gives: Z l 0  (n 1)  2 l2  (n 1)  sin2   s l  ds  0 and following a similar argument to Bonnet’s theorem (16) yields l   p . Proposition 13 deals with 1-dimensional stable minimal submanifolds with boundary. The natural next step is to see if we can ask an analogous question in 2 dimensions. Our  rst formulation of this question is: given a simple closed curve  in Rn, can we  nd a 2-dimensional submanifold (surface) with  xed boundary  which minimizes the area among all such submanifolds? 102.3. The Submanifold Has No Boundary This is historically known as Plateau’s problem. It is named after Belgian physicist Joseph Plateau (1801 - 1883) who experimented with soap  lms to give a physical and intuitive understanding of the problem. Plateau’s experiments involved taking a wire, which represented a simple closed curve, and dipping it into soapy water. He would obtain a soapy  lm with the wire as its boundary. The surface obtained is a minimal surface and thus it can be seen as a solution to Plateau’s problem. In 1930 Rad o [Rad30] and Douglas [Dou30] independently arrived at a mathematical solution of Plateau’s problem. The precise formulation given by Douglas is as follows: Theorem 18 ([Dou30]). Given an arbitrary Jordan curve  in Rn there exists a minimal surface whose boundary is  . Remark 19. In the case where there exists a surface with  nite area and boundary  then the surface guaranteed by Theorem 18 is a surface of minimum area. However there exists a Jordan curve  in R2 such that no surface with boundary  has  nite area, see [Dou30], Part 27, page 320 for an example. Thus we may think of the condition that the surface be minimal as a generalization of idea that the surface minimizes area. Remark 20. Morrey [Mor48] gives a generalization of Theorem 18 to the case when the ambient manifold is a Riemannian manifold. Plateau’s problem in full generality is concerned with the question of existence of a minimal k-dimensional Riemannian submanifold M with pre- scribed boundary inside an n-dimensional Riemannian manifold N . The main result in this formulation can be found in Morrey [Mor65] and builds on the argument used by Reifenberg [Rei60] in the case N = Rn. 2.3 The Submanifold Has No Boundary We now give an expose of some successful applications of the minimal sub- manifold technique in the case where the minimal submanifold M has no boundary. The  rst step is an existence theory for stable minimal sub- manifolds with no boundary. In general we will make an assumption about the topology of the ambient manifold, which will guarantee the existence of a minimal submanifold with some stability. A simple example of this is showing the existence of a 1-dimensional, stable, minimal submanifold homeomorphic to S1 inside the ambient manifold N under the assumption  1(N) 6= 0. First we need a de nition. 112.3. The Submanifold Has No Boundary De nition 21. Given a manifold N and p 2 N the injectivity radius i(N; p) of N at p is de ned as: i(N; p) := supfr > 0 : expp is a di eomorphism on B(0; r)  TpNg: And the injectivity radius of N is de ned as: i(N) := inffi(N; p) : p 2 Ng: Remark 22. If d(p; q) < i(N) then there exists a unique length minimizing geodesic joining p and q. Then we have the following lemma: Lemma 23. Let N be a complete, compact Riemannian manifold. Then i(N) > 0 and if  0;  1 : S1 ! N are such that for all s 2 S1 we have d( 0(s);  1(s)) < i(M) then  0 and  1 are homotopic. Proof. We omit the proof. Now we have the following existence theorem: Proposition 24. Let N be a compact Riemmanian manifold with  1(N) 6= 0. Then there exists a 1-dimensional, stable, minimal submanifold  that is homeomorphic to S1, i.e a stable, closed geodesic. Proof. We shall prove the following: a non-trivial homotopy class of curves in  1(N) has a shortest curve and this curve is stable and minimal. Let C := [ ] be a non-trivial free homotopy class of closed curves in M . We will  rst show that there is a shortest curve in this class. Choose L > 0 large, so that there is at least one curve in C of length  L. We restrict to all such curves. De ne l as l := inf  2C length( ) > 0: Then we may take a sequence of curves  n : S1 = [0; 2 ) ! M in C, with length( n) a decreasing sequence, length( n)  L for all n, and length( n)! l. Now M is compact so we have that i(M) = r > 0 for some r. Since length( n)  L, for each curve  n there exists 0 = s0;n < s1;n < : : : sm 1;n < sm;n = 2 such that: length( nj[sj 1;n;sj;n])  r 2 : 122.3. The Submanifold Has No Boundary Note that m here is independent of n and in fact is any integer m  d2Lr e. Then for each j, d( n(sj 1;n);  n(sj;n))  l( nj[sj 1;n;sj;n])  r 2 and we replace the segment of  n joining  n(sj 1;n) and  n(sj;n) with the unique length min- imizing geodesic between these two points guaranteed by Remark 22. From Lemma 23 we know each of these segments are homotopic and thus the curves as whole are homotopic. We call this new curve  n then we have a se- quence f ng  C with length( n)  length( n) and length( n)! l. De ne pj;n :=  n(sj;n), then since M is compact we may, passing to a subsequence if necessary, assume that the points p0;n; : : : ; pm;n converge to points p0; : : : pm. Since d(pj 1;n; pj;n)  r2 for all n we know that d(pj 1; pj)  r 2 < r and we may join each pj 1, pj with the unique geodesic of shortest length between them to construct a new curve  . We claim that  2 C and length( ) = l. By Lemma 23  is homotopic to  n for some large n and thus  2 C. We show length( ) = l. The unique geodesic connecting pj 1;n and pj;n is con- tained in  n and since pj 1;n; pj;n converge to pj 1; pj we know that the length of the segment  nj[sj 1;n;sj;n] converges to the distance between pj 1 and pj . Since  contains the unique geodesic realizing the distance between these two points we may sum up the lengths of each segment to obtain: length( ) = mX j=1 length( j[pj 1;pj ]) = mX j=1 lim n!1 length( nj[pj 1;n;pj;n]) = lim n!1 mX j=1 length( nj[pj 1;n;pj;n]) = limn!1 length( n) = l: Thus  is our desired curve. We have shown there exists  2 C with the shortest length amongst all curves in C. So if we were to deform  via a homotopy we cannot obtain a shorter curve. Therefore  is locally length minimizing and thus it is minimal and stable. Thus we have a theorem of the  rst type. Now we look for a theorem of the second type. The idea is to construct a vector  eld on a closed geodesic, if one exists, that will shorten it. We assume our manifold is even dimensional and orientable with positive sectional curvatures. Proposition 25 ([Syn36]). Let Nn be a even dimensional, orientable man- ifold with positive sectional curvatures K > 0. Then any closed geodesic  is unstable, that is, can be shortened by a variation. Proof. Let  : [0; l] ! N be a closed, unit speed geodesic in N of length l, and let p0 2  . Consider the parallel transport Pl around  Pl : T (0)=p0N ! T (l)=p0 : 132.3. The Submanifold Has No Boundary We show there exists a non-zero vector V 2 Tp0N , orthogonal to _ , that is  xed by Pl, i.e. PlV = V . Since  is a geodesic, Pl leaves _ 2 Tp0N unchanged and thus leaves _ ? invariant. We restrict Pl to this space and consider Pl : _ ? ! _ ?. Parallel transport preserves the inner product so Pl is an orthogonal linear transformation and detPl =  1. Furthermore since N is orientable detPl = 1. Since dim _ ? = n  1 is odd it follows that Pl has 1 as an eigenvalue and the corresponding eigenvector V 2 _ ? is  xed by Pl. We may assume without loss of generality that kV k = 1. By parallel transporting V around  we obtain a well de ned normal vector  eld V (p) on  and we may de ne a variation fs(p) := expp(sV (p)) of  . The second variation for geodesics, as calculated in Section 2.2 is: d2 dt2 vol (ft ( )) jt=0 = Z l 0  kr _ V k 2  RN (V; _ ; _ ; V )  ds: (2.5) The vector  eld V (p) is parallel along  so kr _ V k2 = 0. Then 2.5 becomes: d2 dt2 vol (ft ( )) jt=0 =  Z l 0 RN (V; _ ; _ ; V ) ds: V and _ are orthonormal and therefore RN (V; _ ; _ ; V ) is equal to the sec- tional curvature of the plane spanned by these vectors. Since all sectional curvatures K > 0 are assumed to be positive we see the second variation is strictly negative. Hence  is unstable. Putting Proposition 24 and Proposition 25 together we obtain the fol- lowing theorem, due to Synge [Syn36]. Theorem 26 ([Syn36]). If N is even dimensional, compact and orientable with positive sectional curvature then it is simply connected. Proof. Suppose N has non-trivial fundamental group. Then since N is com- pact, Proposition 24 guarantees the existence of a stable, closed geodesic. But since N is even dimensional and orientable, by Proposition 25 this geodesic must be unstable, a contradiction. Therefore N is simply con- nected. Micallef and Moore [MM88] give a result that can be seen as a 2- dimensional version of Synge’s theorem. As with the 1-dimensional case, the  rst step is an existence theorem. In non-trivial work of Sacks and Uhlenbeck [SU81] they developed an analogous general existence theory for 142.3. The Submanifold Has No Boundary branched minimal surfaces; namely, assuming something about the topology of N they prove existence of a non-constant minimal branched two-sphere. Their result is as follows: Theorem 27 ([SU81]). Let N be a compact Riemannian manifold of di- mension  3 such that the universal covering space of N is not contractible. Then there exists a non-constant, branched, minimal two-sphere in N . The condition that the universal cover of N is not contractible is equiv- alent to the condition that  k(N) 6= 0 for some k  2. Micallef and Moore [MM88] extended the result of Sacks and Uhlenbeck [SU81] as follows: Theorem 28 ([MM88]). If N is a compact Riemannian manifold such that  k(N) 6= 0, where k  2, then there exists a non-constant minimal branched two-sphere in N of index  k  2. Remark 29. The index is introduced in De nition 10 and gives a measure of the stability of the submanifold. Notice that Proposition 24 states that if a manifold N has  1(N) 6= 0 then N has a minimal, stable 1-dimensional submanifold  that is homeo- morphic to S1. Theorem 28 can be seen as a 2-dimensional version of this. The topological condition  1(N) 6= 0 is replaced with the condition that N is compact and  k(N) 6= 0 for some k  2, and the existence of a stable, minimal 1-sphere is replaced by the existence of a non-constant, branched, minimal two-sphere of index  k  2. The second part of Synge’s theorem (26), Proposition 25, states that if N is an even dimensional, orientable manifold with positive sectional curva- tures K > 0 then a minimal 1-dimensional submanifold homeomorphic to S1 is unstable. Again, Micallef and Moore [MM88] have a theorem analogous to this. The assumption of positive sectional curvature is changed to an assumption of positive isotropic curvature, the condition that N is even di- mensional and orientable is changed to the condition that N has dimension n  4, and the result that a minimal 1-dimensional manifold homeomorphic to S1 is unstable is changed to the result that a branched minimal immer- sion f : S2 !M has index at least n2  3 2 . The result as stated by Micallef and Moore [MM88] is as follows: Theorem 30 ([MM88]). Let N be an n-dimensional Riemannian manifold with positive isotropic curvature. Then any branched minimal immersion f : S2 ! has index at least n2  3 2 . 152.3. The Submanifold Has No Boundary Micallef and Moore [MM88] then put these two theorems together to obtain a proof of their main theorem, and we sketch a proof of it here. Theorem 31 ([MM88]). Let N be a compact, simply connected n-dimensional Riemannian manifold which has positive isotropic curvature, where n  4. Then N is homeomorphic to a sphere. Sketch of Proof. Let k be the smallest such integer such that  k(N) 6= 0. Since N is assumed to be simply connected, k  2. By Theorem 28 there exists a non-constant minimal two-sphere in N whose index is m  k  2. On the other hand it follows from Theorem 30 that m  n2  3 2 and hence k > n2 . This is a purely topological condition and Micallef and Moore [MM88] exploit this to conclude that N must be homeomorphic to a sphere. They use a number of deep theorems in topology, and we give a rough sketch of their argument here: Since k > n2 one may use the Hurewicz isomorphism theorem and Poincar e duality to show that we must have k = n, i.e. the lowest non- vanishing homotopy group of N is  n(N). Then one can use Whitehead’s theorem to conclude that N must be a homotopy sphere, followed by the generalized Poincar e conjecture when n  4 to conclude that N is homeo- morphic to a sphere. The results of Sack and Uhlenbeck [SU81] and Micallef and Moore [MM88], Theorem 27 and Theorem 28 respectively, show the existence of a non- constant, minimal branched sphere in the ambient space N , after making some assumptions about the topology of N . Schoen and Yau [SY79] and Sacks and Uhlenbeck [SU82] proved an ex- istence theory for surfaces of an arbitrary genus g > 0 rather than spheres. The result as formulated by Schoen and Yau [SY79] is as follows: Theorem 32 ([SY79]). Suppose N is a compact Riemannian manifold, Mg is a Riemannian surface of genus g > 0 and f : Mg ! N is a continuous map such that the induced map on the fundamental group given by f] :  1(Mg)!  1(N) is injective. Then there is a branched minimal immersion h : Mg ! N so that h] = f] on  1(Mg) and the induced area of h is the least among all maps with the same action on  1(N). If N is 3-dimensional, it follows that h is an immersion. If  2(N) = 0, then h can be deformed from f continuously. In the case where the ambient manifold N is 3-dimensional we are guar- anteed that the resulting map has no branch points, that is, the image of 162.3. The Submanifold Has No Boundary h is an immersed submanifold. Thus we have a very strong theorem of the  rst type. Next, we examine the second variation and ask if there are theorems of the second type. In the same paper as the existence results, Schoen and Yau [SY79] give such a theorem; it examines the second variation formula and concludes that if the ambient manifold N has positive scalar curvature then any stable, minimal immersed 2-dimensional submanifold has genus 0. In the wording of Schoen and Yau [SY79]: Theorem 33 ([SY79]). Let N be a compact, oriented, 3-dimensional man- ifold with positive scalar curvature. Then N has no compact, immersed, stable, minimal 2-dimensional submanifolds of positive genus. Proof. This is just a restatement of Corollary 41. Thus if we consider a 3-dimensional manifold N we have a theorem of each type: a theorem of the  rst type that guarantees the existence of a stable minimal 2-submanifold under certain conditions, and a theorem of the second type that makes the assumption that N has positive scalar curvature and shows that only genus 0 stable minimal 2-submanifolds can occur. The idea then is to make some assumptions about the topology of our manifold N , use these to guarantee the existence of a stable, minimal 2-submanifold of a non-zero genus, then use Theorem 33 to conclude that N cannot have positive scalar curvature. The precise result, due to Schoen and Yau [SY79], is as follows: Theorem 34 ([SY79]). Let N3 be compact and oriented. Suppose either of the following two conditions hold for N : 1  1(N) contains a  nitely generated non-cyclic abelian subgroup 2  1(N) contains a subgroup isomorphic to the fundamental group of a surfaces of genus greater than zero. Then N admits no metric of positive curvature. In fact, any metric having non-negative scalar curvature is  at. Remark 35. The  rst condition is actually a special case of the second. If  1(N) contains a  nitely generated non-cyclic abelian group then it has an abelian subgroup of rank 2. Then one can show there is a map  : T 2 ! N , where T 2 is the usual 2-torus, such that  ] :  1(T 2) !  1(N) maps onto this subgroup injectively. 17Chapter 3 Inradius Bounds for Minimal Submanifolds 3.1 Introduction In this chapter we examine the minimal submanifold technique in the case of a stable, minimal surface M inside an ambient 3-dimensional manifold N , whose scalar curvature RN is bounded below by a constant  > 0. We note that condition that the scalar curvature is non-negative also appears naturally in the theory of general relativity. In general relativity we typically deal with a 4-dimensional manifold S with a Lorentzian metric g and an initial data set (N; g; ). The initial data set (N; g; ) consists of; a 3-dimensional \spacelike" submanifold N , the metric g, and the second fundamental form  of N inside S. Since N is a hypersurface of S we can think of  being a real valued (0; 2) tensor, rather then a vector valued (0; 2) tensor, and we may de ne the following quantities:  := 1 2  RN  k k 2 + tr( )2  J := div(  tr( )g): Then we typically impose the dominant energy condition;   kJk. This condition arises naturally in general relativity. Letting feig4i=1 be a local orthonormal frame with e4 normal to N we may rewrite tr( ) as: tr( ) = 3X i=1 h (ei; ei); e4i = 3X i=1 h(reiei) N ; e4i = hHN ; e4i: Then if N is minimal inside S we have tr( ) = hHN ; e4i = 0 and the dominant energy condition becomes: 1 2  RN  k k 2  kJk which implies: RN  2kJk+ k k 2  0: 183.2. Inradius Bounds Thus we have the condition RN  0. The paper of Schoen and Yau [SY83] proves the existence of black holes when enough matter is condensed in a small region. Proposition 1 of their paper shows that if the curvature of a manifold is positive in some region then that region is small in sense of a \2-dimensional diameter" or \ ll radius". This chapter provides an exposition of Proposition 1 in the language of Riemannian geometry. Speci cally; it gives an inradius bound for a stable, minimal surfaceM in a 3-dimensional manifoldN in terms of a positive lower bound on the scalar curvature RN of the ambient space. In the context of the result of Schoen and Yau [SY83], the ambient manifold N is the \spacelike" submanifold of the spacetime (S; g) and we concerned with stable, minimal 2-dimensional submanifolds M of N . 3.2 Inradius Bounds Recall the theorem of Bonnet-Myers, Theorem 17, gives an upper bound on the length of a minimizing geodesic in an n-dimensional manifold in terms of a positive lower bound on the Ricci curvature of the manifold. Our main theorem, Theorem 40, can be seen as a 2-dimensional version of this. We give an upper bound on the inradius of a stable, minimal, 2-dimensional sub- manifold (surface) in a ambient manifold in terms of a positive lower bound on the scalar curvature of the ambient manifold. However, we must assume that N is 3-dimensional. We refer the reader to Chapter 4 where we discuss possible extensions of this result to higher dimensions and the implications these would have in understanding manifolds of positive curvature. We give two inradius bounds, the  rst one is as follows: Theorem 36. Let M be a stable, minimal, orientable, complete 2-dimensional submanifold of a 3-dimensional Riemannian manifold (N; g). Suppose that there is a globally de ned unit length normal vector e3 on M . Suppose that on M we have RN   > 0. 1. If M has no boundary then: diam (M)  2 p  : 2. If M has non-empty boundary then: dM (p; @M) := inffdM (p; q) 2 R : q 2 @Mg  2 p  for all p 2M . 193.2. Inradius Bounds The proof uses the second variation formula, given in Proposition 6. This states that for normal variations V we have: d2 dt2 vol (ft (M)) jt=0 = Z M  kDV k2  X i=1 RN (ei; V; V; ei) k V k2 ! dv: The  rst thing we notice about this equation is the curvature term P i=1RN (ei; V; V; ei). When the submanifold is of one less dimension than the ambient space this is simply RicN (V; V ). The  rst step is to rewrite RicN (V; V ) in terms of the scalar curvatures of the manifolds, the mean curvature HM , and the second fundamental form  of M inside N . We derive such an expression now. We need only that M is of one less dimension than N . Proposition 37. For an n-dimensional submanifold M in an (n + 1)-dimensional manifold N we have for a normal vector V = ’en+1, where en+1 is unit normal vector on M , the following equality: RicN (V; V ) = 1 2 RN’ 2  1 2 RM’ 2 + 1 2 kHMk’ 2  1 2 k V k2: Proof. We check the equality at an arbitrary p 2 N . Let feig n+1 i=1 be a local orthonormal frame around p with en+1 normal to M . Then calculating ’2RN around p we have: ’2RN = ’ 2 n+1X i;j=1 RN (ei; ej ; ej ; ei) = ’2 n+1X i=1 RN (ei; en+1; en+1; ei) + ’ 2 n+1X i=1;j 6=n+1 RN (ei; ej ; ej ; ei) = RicN (V; V ) + ’ 2 n+1X i;j 6=n+1 RN (ei; ej ; ej ; ei) + ’ 2 n+1X j 6=n+1 RN (en+1; ej ; ej ; en+1) = 2 RicN (V; V ) + ’ 2 nX i;j=1 RN (ei; ej ; ej ; ei) : Expanding the second term via the Gauss equation we obtain: ’2 nX i;j=1 (RM (ei; ej ; ej ; ei) h (ei; ei) ; (ej ; ej)i+ h (ei; ej) ; (ei; ej)i) : 203.2. Inradius Bounds The  rst term of this equation is ’2RM , the second is  ’2kHMk2. Writing  (ei; ej) as  (ei; ej) = hreiej ; en+1ien+1 the third term becomes k V k2. Therefore we have: ’2RN = 2 RicN (V; V ) + ’ 2RM  ’ 2kHMk+ k V k2: Rearranging yields the result. Remark 38. In particular RicN (V; V ) = 12RN’ 2 12RM’ 2 12k V k2 if M is minimal in N . We now give a proof of our  rst diameter bound, Theorem 36. Recall Theorem 36 states that for a stable, minimal, 2-dimensional manifold M inside a 3-dimensional manifold N with RN   > 0 we have the following inradius bounds: 1 If M has no boundary then: diam (M)  2 p  : 2 If M has non-empty boundary then: dM (p; @M) := inffdM (p; q) 2 R : q 2 @Mg  2 p  for all p 2M . Proof of Theorem 36. First assume M has no boundary. Take a normal variation V := ’e3 where ’ is compactly supported on M and e3 is a globally de ned unit normal vector on M . Since M is stable and minimal from De nition 7 and Proposition 6 we have the following inequality: Z suppV  kDV k2  X i=1 RN (ei; V; V; ei) k V k2 ! dv  0: Since M is of one less dimension than N we have: X i=1 RN (ei; V; V; ei) = RicN (V; V ) = 1 2 RN’ 2  1 2 RM’ 2  1 2 k V k2 (3.1) where the last equality follows from Proposition 37 and the minimality of M . Let feig3i=1 be a local orthonormal frame that includes our globally 213.2. Inradius Bounds de ned e3. We evaluate kDV k2. kDV k2 = X i=1 h(reiV ) N ; (reiV ) N i = X i=1 hreiV; e3i 2 = X i=1 hei (’) e3 + ’reie3; e3i 2 = X i=1 ei (’) 2 : where the last equality follows since hreie3; e3i = 1 2eihe3; e3i = 1 2ei(1) = 0. A quick check shows that P i=1 ei (’) 2 = hgrad’; grad’i. Then since ’ = 0 on @ supp’, Green’s identities give: Z supp’ kDV k2 = Z supp’ hgrad’; grad’i =  Z supp’ ( M’)’: (3.2) Then substituting the expressions in 3.1 and 3.2 into the formula for the second variation we have: Z supp’     M’ 1 2 RN’+ 1 2 RM’  ’ 1 2 k V k2  dv  0 which implies: Z supp’    M’ 1 2 RN’+ 1 2 RM’  ’ dv  0: (3.3) Therefore the operator L de ned by: L’ :=   M’ 1 2 RN’+ 1 2 RM’ (3.4) has non-negative  rst eigenvalue on the space of ’ with compact support. Now take p; q 2 M with dM (p; q) = l > 0 and de ne B := BM (p; l) = fx 2M : dM (p; x) < lg,  B := fx 2M : dM (p; x)  lg and @B := fx 2M : dM (p; x) = lg. L is a self-adjoint operator and results in PDE theory (see [GT83], The- orem 8.38, page 214) guarantee the existence of an eigenfunction f for such an operator, with f > 0 on B and f = 0 on @B. Let   0 be the corre- sponding eigenvalue for f , then multiplying the equation Lf =  f by 2f 1 and rearranging gives: RM  2f  1 ( Mf) = RN + 2   > 0: Now consider B  S1 with the warped product metric ~g = g + f2d 2. Ap- pendix B gives a discussion of warped product metrics and derives a useful 223.2. Inradius Bounds computation of the scalar curvature of a warped space that we will use now. This is Proposition 66 and we use it to compute RB S1 as: RB S1 = RB  2f  1 ( Bf) = RM  2f  1 ( Mf) = RN + 2   where the second equality follows since B is just a subset of M . So B  S1 obeys the same curvature condition. For a curve  from p to the boundary @B we de ne the quantity Lf ( ) := 2 R  fds. Then let ds be the volume form of  and d~s be the volume form of the warped space   S1. Expanding d~s in local coordinates we have: d~s = p det ~gdsd = fdsd and if h is constant over S1 we have: Z   S1 hd~s = Z   S1 hfdsd = Z  hfds Z S1 d = 2 Z  hfds: (3.5) In particular if h = 1 on M we have: vol(  S1) = Z   S1 d~s = 2 Z  fds = Lf ( ): (3.6) Since  B is compact there exists a curve  minimizing Lf ( ) among all such curves. Without loss of generality we may assume there is no segment of  inside @B and  ends at some y 2 @B. We suppose  has unit speed parametrization as  : [0; L]!  B with  (0) = p and  (L) = y 2 @B. Note that length( ) = L  l. So  minimizes Lf amongst all curves from p to y and therefore from the computation in 3.6 we see that vol    S1  under the warped metric is also minimized amongst all such curves. Then if we take a normal variation vector  eld V along  and extend it to a  eld along   S1 such that V (s;  ) = V (s), the second variation of   S1 will be non-negative. We consider vector  elds V of the form V = ’e where e is a unit length, parallel, normal vector  eld on  and ’ is a smooth cut-o function on  with supp’  [0; L   ] for 0 <  < L. From the second variation formula for   S1 inside B S1 and a similar argument to that used in 3.1 and 3.2 we have following inequality: Z  ([0;L  ]) S1  hgrad’; grad’i  1 2 RB S1’ 2 + 1 2 R  S1’ 2  d~s  0: 233.2. Inradius Bounds Remark 39. This is the place we are using the fact that M is 2-dimensional.  is 1-dimensional and in order to get the above inequality we need   S1 to be a submanifold of M  S1 of one less dimension. Since RB S1   on B we can replace the RB S1 term to give: Z  ([0;L  ]) S1  hgrad’; grad’i  1 2  ’2 + 1 2 R  S1’ 2  d~s  0: By Proposition 66 we have R  S1 = R  2f  1 (  f) =  2f 1 (  f) where the last equality follows as R = 0, since  is 1-dimensional. This yields: Z  ([0;L  ]) S1  hgrad’; grad’i  1 2  ’2  f 1 (  f)’ 2  d~s  0: (3.7) Since all of these functions are constant over S1 we may use the equality in 3.5 to write this as an integral over  ([0; L  ]). Parameterizing the resulting integral by s and dividing through by 2 we can pull this back to an integral on [0; L  ], this gives: Z L  0   ’0  2 f  1 2  ’2f  f 00’2  ds  0: or Z L  0    ’0  2 f + 1 2  ’2f + f 00’2  ds  0: Since (’0)2 f > 0 we may change the coe cient of this term to 2 giving: Z L  0   2  ’0  2 f + 1 2  ’2f + f 00’2  ds  0: Making the substitution ’ = f 1 2 where  is another function with  = 0 at 0 and L  , we have: Z L  0   2   0  2 + 1 2   2  1 2  2  f 0  2 f 2 + 2  0f 1f 0 +  2f 00f 1  ds  0: (3.8) Expanding out dds  f 1f 0 2  gives: d ds  f 1f 0 2  =   2  f 0  2 f 2 + 2  0f 1f 0 +  2f 00f 1 243.2. Inradius Bounds and substituting this into the inequality in 3.8 yields: Z L  0   2   0  2 + 1 2   2 + 1 2  2  f 0  2 f 2 + d ds  f 1f 0 2   ds  0: The fundamental theorem of calculus gives: Z L  0 d ds  f 1f 0 2  = [f 1f 0 2]L  0 = 0: Since  2 (f 0)2 f 2  0 we may get rid of this term, which yields: Z L  0   2   0  2 + 1 2   2  ds  0: Integration by parts gives R L  0  00 ds =  R L  0 ( 0)2 ds, therefore we have: Z L  0  2 00 + 1 2   2  ds  0: This implies the operator L0 = 2 00 + 12  has non-positive eigenvalues. Choosing the eigenfunction  (s) = sin   s L   and computing its eigenvalue gives:   2 2 (L  )2 + 1 2    0 which implies: L   2 p  : Since 0 <  < L was arbitrary we have L  2 p . Then since l  L we have dM (p; q) = l  2 p . Finally since p; q 2 M were arbitrary it follows that diam (M)  2 p . Now assume M has non-empty boundary @M 6= ;. We use the notation  M = M [@M where M \@M = ;. We assume  M is topologically complete, that is; all closed and bounded sets are compact. Take p 2 M , then we show: dM (p; @M) := inffdM (p; q) 2 R : q 2 @Mg  2 p  : Suppose l := dM (p; @M) > 0. Then B := BM (p; l) = fx 2  M : dM (p; x) < lg is contained entirely within M and since  B := fx 2  M : dM (p; x)  lg is compact we may use the same argument as Theorem 36, starting from 3.4, and replacing B appropriately. Thus we have l  2 p and the bound is proven. 253.2. Inradius Bounds In fact the bound in Theorem 36 can be improved. This is our main theorem, Theorem 40. Theorem 40. Let M be a stable, minimal, orientable, complete 2-dimensional submanifold of a 3-dimensional Riemannian manifold (N; g). Suppose that there is a globally de ned unit length normal vector e3 on M . Suppose that on M we have RN   > 0. 1. If M has no boundary then: diam (M)  r 8 3  p  : 2. If M has non-empty boundary then: dM (p; @M) := inffdM (p; q) 2 R : q 2 @Mg  r 8 3  p  for all p 2M . Proof. We  rst assume that M has no boundary, @M = ;. We begin from 3.7 in Theorem 36, with the added condition 0 < 3 < L. Recall 3.7 states that for ’ with supp’  [0; L  ], we have the following stability inequality: Z  ([0;L  ]) S1  hgrad’; grad’i  1 2  ’2  f 1 (  f)’ 2  d~s  0: Since ’ = 0 on the boundary of  ([0; L  ]) we may use Green’s identity to give: Z   S1   (   S1’) 1 2  ’ f 1 (  f)’  ’d~s  0: (3.9) Now we compute    S1’. We let fE1 = @ @s ; E2 = f  1 @ @ g be an orthonor- mal frame on  ([0; L  ]) S1. From the connection identities in Proposition 64 we have:    S1’ = div  S1 grad’ = hrE1 grad’;E1i+ hrE2 grad’;E2i = div grad’+ h(grad’)(f)f  1E2; E2i =   ’+ hgrad’; grad fif  1: Therefore 3.9 becomes:  Z   S1  (  ’) + hgrad’; grad fif  1 + 1 2  ’+ f 1 (  f)’  ’d~s  0: 263.2. Inradius Bounds From the above inequality we see the operator L0 L0’ := (  ’) + hgrad’; grad fif  1 + 1 2  ’+ f 1 (  f)’ has non-positive eigenvalue on the space of functions with supp’  [0; L  ]. Let g be an eigenfunction of L0 with g > 0 on (0; L  ), g(0) = g(L  ) = 0 and eigenvalue  0  0. Then parameterizing  by s and using primes to denote dds we expand the inequality L0 (g) g  1 =  0  0, to obtain: g 1g00 + f 1f 00 + f 1g 1g0f 0 + 1 2   0: (3.10) Now let  be a function with supp  [ ; L  2 ], multiply both sides of 3:10 by  2 and integrate over [ ; L 2 ]. Integrating the  rst term by parts gives: Z L 2  g 1g00 2ds = [g 1g 2]L 2   Z L 2  g0   g 2g0 2 + 2  0g 1  ds = Z L 2  g 2  g0  2  2ds Z L 2  2  0g 1g0ds similarly, Z L 2  f 1f 00 2ds = Z L 2  f 2  f 0  2  2ds Z L 2  2  0f 1f 0ds: Thus after rearranging, we have: Z L 2  g 2  g0  2  2+f 2  f 0  2  2 + f 1g 1f 0g0 2 + 1 2   2ds  Z L 2  2  0  g 1g0 + f 1f 0  ds: (3.11) For notation’s sake we make the following de nitions: X := g 2(g0)2 + f 2(f 0)2 Y := f 1g 1f 0g0: Then (X + Y ) 2 gives the  rst three terms in the integrand of 3.11. We claim: 2  0  g 1g0 + f 1f 0   4 3   0  2 + (X + Y ) 2: (3.12) 273.2. Inradius Bounds We show this now. After expanding and rearranging the following inequality: 0  0 @ r 4 3  0  1 q 4 3   g 1g0 + f 1f 0  1 A 2 we obtain: 2  0  g 1g0 + f 1f 0   4 3 ( 0)2 + 3 4 (X + 2Y ) 2: Thus it su ces to show: 3 4 (X + 2Y )  (X + Y ), 2Y  X: This follows immediately from expanding and rearranging the following in- equality: (g 1g0  f 1f 0)2  0: Thus we have proven 3.12. Substituting this into 3.11 and cancelling we obtain: Z L 2  1 2   2ds  Z L 2  4 3   0  2 ds: Integration by parts gives R L 2  ( 0)2 ds =  R L 2   00 ds, making this sub- stitution and rearranging gives: Z L 2   1 2   + 4 3  00   ds  0: Choosing  (s) = sin   (s  ) L 3  and following the same argument as Theorem 36 gives: 1 2   4 3  2 (L 3 )2  0 which implies: L 3  r 8 3  p  : Since 0 < 3 < L was arbitrary we have L  q 8 3  p  . Then since l  L we have dM (p; q) = l  q 8 3  p  . Finally since p; q 2M were arbitrary it follows that diam (M)  q 8 3  p  . 283.2. Inradius Bounds Now assume that M has non-empty boundary @M 6= ;. Using a similar argument to Theorem 36 we may conclude that for any point p 2 M we have: dM (p; @M) := inffdM (p; q) 2 R : q 2 @Mg  r 8 3  p  : Corollary 41. A stable, minimal surface M without boundary in a 3- manifold N whose scalar curvature RN is bounded below by a positive con- stant  > 0 is homeomorphic to either S2 or RP 2 Proof. First we assume M is orientable. Since we have a bound on the diameter it follows that M is compact. Thus we may take a variation V = ’e3 whose support is all of M . From 3.3 we have Z M    M’ 1 2 RN’+ 1 2 RM’  ’ dv  0: Choosing ’  1 this becomes: Z M   1 2 RN + 1 2 RM  dv  0: Therefore, after rearranging, using the lower bound on the scalar curvature of N , and the Gauss-Bonnet theorem, we have: 0 < 1 2 Z M  dv < 1 2 Z M RN dv  1 2 Z M RM dv = Z M KM dv = 2  (M) : Now  (M) = 2  2g > 0 and thus we must have g = 0. Therefore M is homeomorphic to S2. Now suppose M is not orientable. Let  :  M !M be the orientable double cover of M with pullback metric  g :=   g. Then from the de nition of minimal and stable for non-orientable submanifolds we have that the inequality 3.3 applies to  M with scalar curvature  RM = RM   and  RN := RN   . Following the same argument as above we have  M  = S2 and thus M  = RP 2. Theorem 36 uses a second variation argument to draw geometric con- clusions about stable, minimal surfaces in 3-manifolds that have a lower positive bound on the scalar curvature. Viewing Theorem 36 (2) as a theo- rem of the second type in the context of the minimal submanifold technique when the submanifold has boundary, we may put it together with the theo- rem of the  rst type - the existence theorem for Plateau’s problem (Theorem 18) - to draw topological conclusions about 3-manifolds with positive scalar curvature: 293.2. Inradius Bounds Theorem 42 (Schoen and Yau). A compact, 3-dimensional, manifold N whose universal cover is contractible cannot have RN > 0. This is an unpublished result of Schoen and Yau. Note the result in Theorem 42 is for a 3-dimensional manifold. The following extension is still an open question in Riemannian geometry: can an n-dimensional, n  4, manifold N whose universal cover is contractible have RN > 0? 30Chapter 4 Conclusion The main theorem in this thesis, Theorem 40, states that if M is a stable, minimal, orientable, surface in a 3-dimensional Riemannian manifold N with scalar curvature RN bounded below by a positive constant  > 0, we have the following inradius bounds: 1. If M has no boundary then: diam (M)  r 8 3  p  : 2. If M has non-empty boundary then: dM (p; @M) := inffdM (p; q) 2 R : q 2 @Mg  r 8 3  p  for all p 2M . Theorem 40 is a version of Proposition 1 in Schoen and Yau [SY83] stated in the language of Riemannian geometry. While Theorem 40 gives an inradius bound for stable minimal surfaces in N , the result of Schoen and Yau [SY83] gives a bound on the \2-dimensional diameter" or \ ll radius" of N . We de ne this now: De nition 43. Let  be a smooth, simple closed curve in N which bounds a disk in N . Set Nr( ) to be: Nr( ) := fx 2 N : dN (x;  )  rg: The  ll radius ( llrad( )) of  is de ned as:  llrad( ) := ( supfr : dN ( ; @N) > r and  doesn’t bound a disk in Nr( )g @N 6= ; supfr :  doesn’t bound a disk in Nr( )g @N = ;: 31Chapter 4. Conclusion The  ll radius of N is de ned as:  llrad(N) := supf llrad( ) :  as above g: Then we may rewrite Theorem 40 in terms of the  ll radius: Theorem 44. Let N be a complete Riemannian 3-dimensional manifold with scalar curvature RN bounded below by RN   > 0. Then if  is a smooth, simple closed curve in N which bounds a disk in N we have:  llrad( )  r 8 3  p  : And in particular:  llrad(N) := supf llrad( ) :  as aboveg  r 8 3  p  : Proof. Let  be a smooth, simple closed curve in N that bounds a disk in N and let r :=  llrad( ). Since we know  bounds a disk in N we may use the generalization of Plateau’s problem, Theorem 18, given by Morrey [Mor48], to conclude that there exists stable minimal disk M with boundary  . From Theorem 40 we have the following bound for every point p 2M : dM (p; @M) := inffdM (p; q) 2 R : q 2 @Mg  r 8 3  p  : (4.1) Now  does not bound a disk in Nr( ) so therefore there is a point p 2 M \ (NnNr( )). Then from 4.1 we have dM (p; @M) = dM (p;  )  q 8 3  p  . On the other hand p 2 NnNr(M) and therefore dN (p;  ) > r. Then we have: r < dN (p;  )  dM (p;  )  r 8 3  p  : So  llrad( ) < q 8 3  p  , and therefore, taking the supremum, we may con- clude  llrad(N)  q 8 3  p  . The result as stated in Proposition 1 of Schoen and Yau [SY83] makes slightly weaker assumptions on the curvature. Their result is as follows: 32Chapter 4. Conclusion Theorem 45 ([SY83]). Let N be a complete 3-dimensional Riemannian manifold such that the operator   N+ 12RN has its  rst Dirichlet eigenvalue bounded below by  > 0. Then if  is a smooth simple closed curve in N which bounds a disk in N we have:  llrad( )  2 p 3  p  : And in particular:  llrad(N)  2 p 3  p  : Suppose we have that RN   . Then let  be the  rst Dirichlet eigenvalue of the operator   N + 12RN , with eigenfunction f . We have  f =   Nf + 12RNf . Multiplying by f and integrating yields:  Z N f2dv = Z N (  Nf)fdv + 1 2 Z N RNf 2dv = Z N hgrad f; grad fidv + 1 2 Z N RNf 2dv  Z N hgrad f; grad fidv + 1 2  Z N f2dv  1 2  Z N f2dv where the second equality follows from Green’s identity. This implies that   12 . Then taking  = 1 2 in Theorem 45 gives Theorem 44. Theorem 44 is interesting in its own right but is also important in a wider context. As we have seen in Theorem 42, it can be applied to give topological information about manifolds with positive scalar curvature. Note the result in Theorem 42 is for a 3-dimensional manifold. This leads naturally to the question: can an n-dimensional, n  4, manifold N whose universal cover is contractible have RN > 0? This is still an open question in Riemannain geometry, but would presumably follow if an extension of Theorem 44 to the case where N is n-dimensional could be proven. If it is true, proving a result like Theorem 44 in the general case of an n- dimensional ambient manifold N will require new techniques, since the proof in 3-dimensions makes essential use of the fact that M is of codimension 1 and the dimension of N is three. Perhaps a more approachable problem is to prove such a  ll radius bound under a stronger curvature assumption. To this e ect two conjectures have been made. The  rst conjecture involves an 33Chapter 4. Conclusion assumption involving two-positive Ricci curvature. A manifold N is said to have two-positive Ricci curvature greater or equal to  > 0 if the sum of the two smallest eigenvalues of the Ricci curvature is greater or equal to  > 0. The  rst conjecture is as follows: Conjecture 46. Let (N; g) be a complete Riemannian n-manifold with two- positive Ricci curvature bounded below by  , for a constant  > 0. If  is a smooth, simple closed curve in N which bounds a disk in N then:  llrad( )  C( ): The second conjecture involves positive isotropic curvature. It is as fol- lows: Conjecture 47. Let (N; g) be a complete Riemmanian n-manifold with positive isotropic curvature bounded below by  , for a constant  > 0. If  is a smooth, simple closed curve in N which bounds a disk in N then:  llrad( )  C( ): Remark 48. If a manifold N has either two-positive Ricci curvature bounded below by  > 0 or positive isotropic curvature bounded below by  > 0, then it immediately has scalar curvature RN bounded below by  > 0. If these conjectures are true then topological information can be ob- tained about the manifold N . A recent result of Ramachandran and Wolf- son [RW10] shows that if the universal cover ~N has bounded  ll radius then the fundamental group of N is \virtually free". Virtually free is a purely group theoretic notion, de ned as follows: De nition 49. A group G is said to be virtually free if it possesses a  nite index subgroup that is a free group. The result as stated by Ramachandran and Wolfson [RW10] is as follows: Theorem 50 ([RW10]). Let N be a closed Riemannian n-manifold. Suppose that the universal cover  : ~N ! N is given the pullback metric ~g :=   g. If ( ~N; ~g) has bounded  ll radius then the fundamental group of N is virtually free. 34Chapter 4. Conclusion Thus if Conjecture 46 or Conjecture 47 is true we may conclude that a manifold with two-postive Ricci curvature bounded below by  > 0 or with positive isotropic curvature bounded below by  > 0, has a virtually free fundamental group. Moreover recently Gadgil and Seshadri [GS09] have shown the following: Theorem 51 ([GS09]). Let N be a smooth, orientable, closed n-manifold such that  1(N) is a free group on k generators and  i(N) = 0 for 2  i  n2 . If n 6= 4 or k = 1 then N is homeomorphic to the connected sum of k copies of Sn 1  S1. If Conjecture 47 is true, Theorem 51 can be used along with Theorem 50 to say some more about the topology of manifolds with positive isotropic curvature. We give an outline of this argument now. Suppose Conjecture 47 is true. Let N be an n-dimensional manifold with isotropic curvature bounded below by  > 0. Then the universal cover ~N of N with the pullback metric ~g =   g also has isotropic curvature bounded below by  > 0. Then if Conjecture 47 is true ~N has bounded  ll radius and therefore by Theorem 50 the fundamental group of N is virtually free. If the fundamental group of N is virtually free it has a subgroup that is a free group with  nite index k, and we may  nd a  nite cover  N of N with fundamental group  1(  N) isomorphic to this free group. If  N is given the pull-back metric from N then it also has isotropic curvature bounded below by  > 0. Then from the work done by Micallef and Moore [MM88] we may use the argument given in Theorem 31 to conclude that  i(  N) = 0 for 2  i  n2 . Then  N almost satis es the hypothesis of Theorem 51. If we have that either n 6= 4 or that k = 1, (i.e.  1(  N) = Z), then we may conclude that  N is homeomorphic to the connected sum of k copies of Sn 1  S1. Thus from the assumption of positive isotropic curvature we would have a fairly strong topological result: N has a  nite cover  N that is homeomorphic to the connected sum of k copies of Sn 1  S1. 35Bibliography [Bon55] P. Bonnet, Sur quelques propri et es des lignes g eod esiques, C. R. Math. Acad. Sci. Paris 40 (1855), 1311{1313. [Dou30] J. Douglas, Solution of the problem of Plateau, Proc. Natl. Acad. Sci. USA 16 (1930), 242{248. [GS09] S. Gadgil and H. Seshadri, On the topology of manifolds with posi- tive isotropic curvature, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1807{1811. [GT83] D. Gilbarg and N. S. Trudinger, Elliptic partial di erential equa- tions of second order, second ed., Grundlehren der mathematischen Wissenschaften ;224, Springer Verlag, Berlin; New York, 1983. [MM88] M. J. Micallef and J. D. Moore, Minimal two-spheres and the topol- ogy of manifolds with positive curvature on totally isotropic two- planes, Ann. of Math. 127 (1988), no. 1, 199{227 (English). [Mor48] C. B. Morrey, The problem of Plateau on a Riemannian manifold, Ann. of Math. (2) 49 (1948), 807{851. [Mor65] , The higher-dimensional Plateau problem on a Riemannian manifold, Proc. Nat. Acad. Sci. USA 54 (1965), no. 4, 1029. [Mye41] S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401{404. [O’N83] B. O’Neill, Semi-riemannian geometry: with applications to rela- tivity, Pure and applied mathematics ;103, Academic Press, New York, 1983. [Rad30] T. Rad o, On Plateau’s problem, Ann. of Math. (2) 31 (1930), no. 3, 457{469. [Rei60] E. R. Reifenberg, Solution of the Plateau problem for m- dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1{92. 36Chapter 4. Bibliography [RW10] M. Ramachandran and J. Wolfson, Fill radius and the fundamental group, J. Topol. Anal. 2 (2010), no. 1, 99{107. [SU81] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1{24. [SU82] , Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc 271 (1982), 639{652. [SY79] R. Schoen and S. T. Yau, Existence of incompressible minimal sur- faces and the topology of three-dimensional manifolds with non- negative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127{142. [SY83] , The existence of a black hole due to condensation of mat- ter, Comm. Math. Phys. 90 (1983), no. 4, 575{579. [Syn36] J. L. Synge, On the connectivity of spaces of positive curvature, Q. J. Math. os-7 (1936), no. 1, 316{320 (English). 37Appendix A The First and Second Variation Formulae A stable, minimal submanifold M in N is a manifold which locally minimizes volume inside the ambient manifold. To obtain a concrete de nition for this we consider a deformation or variation of M inside N indexed by a variable t 2 (  ;  ), which leaves M unchanged when t = 0. We may write this as a map f : (  ;  )  M ! N , with f(0;M) = f0(M) = M . Then we can consider t ! vol(ft(M)) as a function from R to R. If f0(M) = M locally minimizes volume we know from single variable calculus that d dt vol(ft(M))jt=0 = 0 and d2 dt2 vol(ft(M))jt=0  0. We shall derive formulae for these expressions soon but  rst we need a lemma from linear algebra. Lemma 52. If A (t) is a smooth function of t into the space of invertible matrices and A (t0) = id, then d dt detA (t) jt=t0 = trA 0 (t0) = X i=1 A0 (t0)ii Proof. Let ! be the standard n-form on Rn, then if feig is the standard basis we have: d dt detA (t) jt=t0 = d dt ! (A (t) e1; : : : ; A (t) en) jt=t0 = X i=1 !  A (t0) e1; : : : ; A 0 (t0) ei; : : : ; A (t0) en  = X i=1 !  e1; : : : ; A 0 (t0) ei; : : : ; en  = X i=1 A0 (t0)ii ! (e1; : : : ; en) = X i=1 A0 (t0)ii where the third inequality follows since A (t0) = id. 38Appendix A. The First and Second Variation Formulae Lemma 53. If B (t) is a smooth function of t into the space of invertible matrices then d dt detB (t) = detB (t) X i;j=1 B 1 (t)ij B 0 (t)ji where B 1 (t) := B (t) 1. Proof. We check this at a point t = t0. De ne A (t) := B 1 (t0)B (t), then d dt detA (t) jt=t0 = d dt det  B 1 (t0)B (t)  jt=t0 = d dt  detB 1 (t0)  (detB (t)) jt=t0 = detB 1 (t0) d dt detB (t) jt=t0 = 1 detB (t0) d dt detB (t) jt=t0 Now A (t) satis es the conditions of Lemma 52, therefore: 1 detB (t0) d dt detB (t) jt=t0 = d dt detA (t) jt=t0 = X i=1 A0 (t0)ii = X i=1  B 1 (t0)B 0 (t0)  ii = X i;j=1 B 1 (t0)ij B 0 (t0)ji multiplying through by detB (t0) we have our result. Given a normal variation vector  eld V 2  (M;TN), we now derive forumulae for ddt vol(ft(M))jt=0 and d2 dt2 vol(ft(M))jt=0, known as the  rst and second variation formulae. Here M may have possible boundary @M and if so we require that the boundary is  xed in the variation. This is equivalent to saying V = 0 on @M . We are measuring the volume of M immersed in N , so we give it the pullback metric f t g, thus the volume element of the metric induced on M by ft is given by dvt = p det g (t)dx. The volume of ft (M) is then: vol (ft (M)) = Z M p det g (t)dx Then the  rst variation is as follows: Proposition 54. The  rst variation of a compact, orientable submanifold M (with possible boundary @M) of a Riemannian manifold (N; g) with nor- mal variation vector  eld V (with V = 0 on @M), is given by: d dt vol(ft(M))jt=0 =  Z M hV;HM idv 39Appendix A. The First and Second Variation Formulae where HM is the mean curvature of M in N and dv is the Riemannian volume form associated with the pullback metric f 0 g. Proof. We wish to calculate ddt vol (ft (M)) jt=0, we have from Lemma 53: d dt p det g (t) = 1 2 (det g (t)) 1 2 d dt det g (t) = 1 2 p det g (t) X i;j=1 g 1 (t)ij g 0 (t)ji Then letting g = det g (t) and suppressing the dependence on t we may write this in a more compact form: d dt p g = 1 2 p g X i;j=1 gij _gij where we have used the fact that g 1 (t) is symmetric and renamed i; j. Now if we denote fi = df  @ @xi  , then the induced metric on ft (M) is given by gij = hfi; fji. Then we have: _gij = @ @t hfi; fji = hrtfi; fji+ hfi;rtfji So at t = 0 we have: d dt p gjt=0 = p g X i;j=1 gijhrtfi; fji = p g X i;j=1 gijhrfiV; fji = p g divM V where the second equality follows since [fi; V ] = df [@i; @t] = df (0) = 0. Then since the tangential component of V is zero we may write V = V N and we have: divM V = divM V N = X i=1 hreiV N ; eii =  X i=1 hV N ;reieii =  X i=1 hV; (reiei) N i =  hV; X i=1 (reiei) N i =  hV;HM i where HM is the mean curvature vector of M inside N . Therefore d dt vol (ft (M)) jt=0 = Z M divM V Npgdx =  Z M hV;HM idv From examining the integral in Proposition 54 we can see that the con- dition that ddt vol(f(M))jt=0 = 0 for any normal variation vector  eld V is equivalent to saying that HM is zero. This motivates the next de nition. 40Appendix A. The First and Second Variation Formulae De nition 55. A submanifold M inside N is minimal if HM = 0. Remark 56. HM is de ned for non-compact and non-orientable manifolds so this is a generalization of the condition that the  rst variation is zero. Given a minimal submanifold M of N and a variation f : (  ;  ) M ! N with normal variation vector  eld V we now derive an expression for the second variation d 2 dt2 vol(ft(M))jt=0. Again we assume that M is compact and orientable. Proposition 57. Assuming M is minimal, i.e. HM = 0, the second varia- tion of a compact, orientable submanifold M (with possible boundary @M) in N with normal variation vector  eld V (with V = 0 on @M), is given by: d2 dt2 vol (ft (M)) jt=0 = Z M  krV k2  X i=1 RN (ei; V; V; ei) 2k V k2 ! dv = Z M  kDV k2  X i=1 RN (ei; V; V; ei) k V k2 ! dv where D is the normal connection on TN jM !M , i.e. DXY = (rXY ) N . Proof. We want to compute d 2 dt2 vol (ft (M)) jt=0 = R M d2 dt2 p det g(t)jt=0dx, thus we begin by computing d 2 dt2 p g where g = det g(t). d2 dt2 p g = d dt  d dt p g  = d dt 0 @ 1 2 p g X i;j=1 gij _gij 1 A = 1 2 0 @ 1 2 p g X i;j=1 gij _gij 1 A X i;j=1 gij _gij + 1 2 p g X i;j=1 _gij _gij + 1 2 p g X i;j=1 gij gij = 1 4 p g 0 @ X i;j=1 gij _gij 1 A 2 + 1 2 p g X i;j=1 _gij _gij + 1 2 p g X i;j=1 gij gij If G is the matrix gij then  GG 1  = id, so ddt  GG 1  = 0 therefore, by the product rule: dG dt G 1 +G dG 1 dt = 0 so dG 1 dt =  G 1 dG dt G 1 41Appendix A. The First and Second Variation Formulae and therefore _gij =  P k;l=1 g ik _gklglj . Thus we have: d2 dt2 p g = 1 4 p g 0 @ X i;j=1 gij _gij 1 A 2  1 2 p g X i;j;k;l=1 gik _gklg lj _gij + 1 2 p g X i;j=1 gij gij (A.1) We evaluate A.1 at t = 0. Examining the  rst term we have: 1 4 0 @ X i;j=1 gij _gij 1 A 2 = 1 4 0 @2 X i;j=1 gijhriV; fji 1 A 2 = 1 4 (2 divM V ) 2 = (divM V ) 2 =  divM V N + div V T  2 =   hV;HM i+ div V T  2 = 0 where the last equality follows since HM = 0 and V T = 0. We now examine the third term in A.1,  rst we compute  gij at t = 0  gij =hrtrtfi; fji+ hrtfi;rtfji+ hrtfi;rtfji+ hfi;rtrtfji =hrtriV; fji+ 2hriV;rjV i+ hrtrjV; fii =hRN (V; fi)V; fji+ hrirtV; fji+ 2hriV;rjV i+ hRN (V; fj)V; fii + hrjrtV; fii Therefore, if feig is a local orthonormal frame in M we may write this as: X i;j=1 gij gij = 2 X i;j=1 gijhriV;rjV i  2 X i;j=1 gijRN (fi; V; V; fj) + 2 divM (rtV ) = 2krV k2  2 X i=1 RN (ei; V; V; ei) + 2 divM (rtV ) where r is the pullback connection on TN jM ! M and rV is the total covariant derivative of V on M . Finally we compute the second term in A.1. Writing it in terms of feig X i;j;k;l=1 gik _gklg lj _gij = X i;j;l;k=1 gik (hrkV; fli+ hfk;rlV i) g lj (hriV; fji+ hfi;rjV i) = X i;j=1  hreiV; eji+ hei;rejV i  2 = X i;j=1 ( hV;reieji  hreiej ; V i) 2 = 4 X i;j=1 hV;reieji 2 = 4 X i;j=1 hV; (reiej) N i2 = 4k V k2 42Appendix A. The First and Second Variation Formulae Thus making the substitutions in A.1 at t = 0 we may compute d2 dt2 vol (ft (M)) jt=0 = R M d2 dt2 p gjt=0dx as: d2 dt2 vol (ft (M)) jt=0 = Z M  krV k2  X i=1 RN (ei; V; V; ei) + divM (rtV ) 2k V k2 ! dv where  V is the V component of the second fundamental form  . Now V is zero on the boundary, thus using the divergence theorem to compute the third term of this we have: Z M divM (rtV ) = Z @M hrtV; vidv@M = 0: where v is the outward pointing unit normal  eld to the boundary. Making this substitution gives the  rst equation: d2 dt2 vol (ft (M)) jt=0 = Z M  krV k2  X i=1 RN (ei; V; V; ei) 2k V k2 ! dv (A.2) Looking at the  rst term and splitting up reiV as reiV = (reiV ) N + (reiV ) T we have krV k2 = X i=1 h(reiV ) N ; (reiV ) N i+ X i=1 h(reiV ) T ; (reiV ) T i = X i=1 kDeiV k 2 + X i=1 X j=1 hreiV; eji 2 = kDV k2 + X i;j=1 hV;reieji 2 = kDV k2 + k V k2 Therefore substituting this into the A.2 we obtain: d2 dt2 vol (ft (M)) jt=0 = Z M  kDV k2  X i=1 RN (ei; V; V; ei) k V k2 ! dv This motivates the following de nition: 43Appendix A. The First and Second Variation Formulae De nition 58. An orientable submanifold M inside N is stable if it is minimal and the integral in Proposition 57 is non-negative for all compactly supported normal variation vector  elds V 2  c(M;TN). A non-orientable submanifold is stable if its orientable double cover is stable. Remark 59. Again, the above de nition generalizes the condition d2 dt2 vol(ft(M))jt=0  0 to non-compact and non-orientable manifolds. 44Appendix B Warped Product Metrics Given two Riemannian manifolds one can construct the product manifold and endow it with a metric in a natural way. This idea is generalized with the notion of a warped product metric, where a smooth function f > 0 is taken on the  rst manifold and used to warp the overall structure. This idea is very important in general relativity. In this section we provide a quick exposition of some of the curvature calculations for a warped product. De nition 60. Suppose (B; gB) and (F; gF ) are Riemannian manifolds and let f > 0 be a smooth function on B. The warped product M = B  f F is the product manifold B  F with the metric tensor: g = p 1gB + (f  p1) 2 p 2gF where p1 : B  F ! B, p2 : B  F ! F are the usual projections. Remark 61. The usual Riemannian product manifold B  F can be ob- tained from the above de nition by taking f = 1 on B. Let M = B  f F be such a warped product, then TM splits up canoni- cally as TM = TB TF . We make the identi cations TB = TB f0g and TF = f0g  TF and we say (X; 0) is the lift of X 2 TB, similarly for TF . We now give an expose of some of the curvature computations for a warped space B fF . We begin with calculation of the gradient of a function h that is constant in F . Proposition 62. If h : B ! R is a function on B, then the gradient gradB fF of the lift h  p1 : B  F ! R is the lift to B  f F of the gradient gradB of h. Proof. Take V = (0; V ) 2 TB  TF = TM . Then hgrad(h  p1); V i = V (h  p1) = dp1(V )h = 0 since dp1(V ) = 0. Therefore gradB fF (h  p1) 2 TB  f0g  TM , we must now check that it is the lift of gradB h. This amounts to showing that for X = (X; 0) 2 TB  TF = TM we have 45Appendix B. Warped Product Metrics hdp1 gradB fF (h  p1); dp1Xi = hgradB h; dp1Xi. We show this now: hdp1 gradB fF (h  p1); dp1Xi = hgradB fF (h  p1); Xi = X(h  p1) = dp1(X)h = hgradB h; dp1Xi This concludes the proof. Remark 63. So if our function h is constant in F there should be no confusion if we drop the subscripts and simply write gradh for the gradient of h. We now give calculations for the Levi-Civita connection on B  f F , writing it in terms of the connections on B and F . Proposition 64. We write r for the Levi-Civita connection on M = B f F , and rB, rF for the Levi-Civita connections on B and F respectively. For X;Y 2 TB  f0g and V;W 2 f0g  TF we have: 1. rXY is the lift of rBXY on B. 2. rXV = rVX = X(f)f 1V 3. dp1 (rVW ) =  (hV;W if 1) grad f 4. dp2 (rVW ) = rFVW Proof. The proof can be found in [O’N83], page 206. We may use Proposition 64 to compute the curvature of the warped space as follows: Proposition 65. Let M = B  f F be a warped product with Levi-Civita connection r and Riemannian curvature tensor R. We write RB and RF for the curvature tensors of B and F respectively. If X;Y; Z 2 TB  f0g and U; V;W 2 f0g  TF then: 1. R (X;Y )Z is the lift of RB (X;Y )Z on B 2. R (V;X)Y =  f 1Hf (X;Y )V where Hf is the Hessian of f 3. R (X;Y )V = R (V;W )X = 0 4. R (X;V )W = f 1hV;W irX grad f 5. R (V;W )U = RF (V;W )U  f 2k grad fk2 (hV;UiW  hW;UiV ) 46Appendix B. Warped Product Metrics Proof. The proof can be found in [O’N83], page 210. Proposition 66. Let M be a Riemmanian n-manifold and f > 0 a smooth function on M . Consider the warped product M  f S1, where S1 is given the usual metric. Then RM fS1 = RM  2 ( Mf) f  1 Proof. Take p 2 M  S1 and let feig n+1 i=1 be a local orthonormal frame for TpM  S1 with ei 2 TM for i = 1; : : : n, en+1 2 TS1 and reiej jp = 0 for all i; j. We use the notation R for the curvature tensor of M  f S1 and we compute RicM fS1 (ei; ei), when i  n RicM fS1 (ei; ei) = n+1X k=1 R (ek; ei; ei; ek) = nX k=1 RM (ek; ei; ei; ek) +R (en+1; ei; ei; en+1) = RicM (ei; ei) f  1Hf (ei; ei) and when i = n+ 1 RicM fS1 (en+1; en+1) = nX k=1 R (ek; en+1; en+1; ek) =  f  1 nX k=1 hrekgradf; eki where we have used Proposition 65. Now hrekgradf; eki = ekhgradf; eki  hgradf;rekeki = ekek (f)  rekek (f) = H f (ek; ek) Therefore computing RM fS1 RM fS1 = n+1X i=1 RicM fS1 (ei; ei) = nX i=1 RicM fS1 (ei; ei) + RicM fS1 (en+1; en+1) = nX i=1  RicM (ei; ei) f  1Hf (ei; ei)   f 1 nX k=1 Hf (ek; ek) = RM  2f  1 nX i=1 Hf (ei; ei) 47Appendix B. Warped Product Metrics Now Hf (ei; ei) = eiei (f) reiei (f) = eiei (f) since reiei = 0 at p. There- fore: RM fS1 = RM  2f  1 nX i=1 eiei (f) = RM  2 ( Mf) f  1 48

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