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Uniqueness of Lagrangian mean curvature flow and minimal immersions with free boundary

University of British Columbia

In this thesis we investigate some problems on the uniqueness of mean curvature flow and the existence of minimal surfaces, by geometric and analytic methods. A summary of the main results is as follows. (i) The special Lagrangian submanifolds form a very important class of minimal submanifolds, which can be constructed via the method of mean curvature flow. In the graphical setting, the potential function for the Lagrangian mean curvature ow satisfies a fully nonlinear parabolic equation [formula omitted] where the ⋋j's are the eigenvalues of the Hessian D²u. We prove a uniqueness result for unbounded solutions of (1) without any growth condition, via the method of viscosity solutions ([4], [13]): for any continuous u₀ in ℝn, there is a unique continuous viscosity solution to (1) in ℝn x [0;∞). (ii) Let N be a complete, homogeneously regular Riemannian manifold of dimN ≥ 3 and let M be a compact submanifold of N. Let Ʃ be a compact Riemann surface with boundary. A branched immersion u : (Σ,∂Σ) → (N,M) is a minimal surface with free boundary in M if u(Σ) has zero mean curvature and u(Σ) is orthogonal to M along u(∂Σ)⊑ M. We study the free boundary problem for minimal immersions of compact bordered Riemann surfaces and prove that Σ if is not a disk, then there exists a free boundary minimal immersion of Σ minimizing area in any given conjugacy class of a map in C⁰(Σ,∂Σ;N,M) that is incompressible; the kernel of i* : π₁(M) → π₁(N) admits a generating set such that each member is freely homotopic to the boundary of an area minimizing disk that solves the free boundary problem. (iii) Under certain nonnegativity assumptions on the curvature of a 3-manifold N and convexity assumptions on the boundary M=∂N, we investigate controlling topology for free boundary minimal surfaces of low index: • We derive bounds on the genus, number of boundary components; • We prove a rigidity result; • We give area estimates in term of the scalar curvature of N.

Text

2013-05-28

Attribution-NoDerivs 3.0 Unported

http://hdl.handle.net/2429/44511

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nd/3.0/