Relevant Degree Programs
Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - May 2019)
In this thesis we investigate some problems on the uniqueness of mean curvatureflow and the existence of minimal surfaces, by geometric and analyticmethods. A summary of the main results is as follows.(i) The special Lagrangian submanifolds form a very important class ofminimal submanifolds, which can be constructed via the method ofmean curvature flow. In the graphical setting, the potential functionfor the Lagrangian mean curvature ow satisfies a fully nonlinearparabolic equation [formula omitted]where the ⋋j's are the eigenvalues of the Hessian D²u.We prove a uniqueness result for unbounded solutions of (1) withoutany growth condition, via the method of viscosity solutions (, ):for any continuous u₀ in ℝn, there is a unique continuous viscositysolution to (1) in ℝn x [0;∞).(ii) Let N be a complete, homogeneously regular Riemannian manifoldof dimN ≥ 3 and let M be a compact submanifold of N. Let Ʃ bea compact Riemann surface with boundary. A branched immersionu : (Σ,∂Σ) → (N,M) is a minimal surface with free boundary in Mif u(Σ) has zero mean curvature and u(Σ) is orthogonal to M alongu(∂Σ)⊑ M.We study the free boundary problem for minimal immersions of compactbordered Riemann surfaces and prove that Σ if is not a disk, then there exists a free boundary minimalimmersion of Σ minimizing area in any given conjugacy class ofa map in C⁰(Σ,∂Σ;N,M) that is incompressible; the kernel of i* : π₁(M) → π₁(N) admits a generating set suchthat each member is freely homotopic to the boundary of an areaminimizing 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.
Master's Student Supervision (2010 - 2018)
Level set solutions are an important class of weak solutions to the mean curvature flow which allow the flow to be extended past singularities. Unfortunately, when singularities do develop it is possible for the Hausdorff dimension of the level set solution to increase. This behaviour is referred to as the fattening phenomenon. The purpose of this thesis is to discuss this phenomenon and to provide concrete examples, focusing especially on its relation to the uniqueness of smooth solutions. We first discuss the definition of level set solutions in arbitrary codimension, due to Ambrosio and Soner. We then prove some technical results about distance solutions, a type of set-theoretic subsolution to level set solutions. These include a new method of gluing together distance solutions. Next, we present several known results on the fattening phenomenon in the context of distance solutions. Finally, we provide a new example by proving that fattening occurs when immersed curves in ℝ³ develop self-intersections.