Jingyi Chen


Research Classification

Research Interests

Differential Geometry, Partial Differential Equations

Relevant Thesis-Based Degree Programs



Master's students
Doctoral students
Postdoctoral Fellows

Geometric Analysis 

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Graduate Student Supervision

Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

Index estimates and compactness for constant mean curvature surfaces and Steklov eigenvalues (2021)

In this work, we focus on three problems. First, we give a relationship between the number of eigenvalues of the Jacobi operator below a certain threshold and the topology of closed constant mean curvature (CMC) surfaces in three-dimensional Riemannian manifolds. We then obtain that the (weak) Morse index of CMC surfaces in an arbitrary 3-manifold is bounded below by a linear function of the genus when the constant mean curvature is greater than a certain nonnegative value. In particular, this implies that stable CMC surfaces are topological spheres. Corresponding results for CMC surfaces with free boundary in 3-manifolds with boundary are obtained as well. Second, we consider the space of embedded free boundary CMC surfaces with bounded topology, bounded area, and bounded boundary length in a 3-manifold N with boundary. We show that this space is almost compact in the sense that any sequence of surfaces in this space has a convergent subsequence that converges to a free boundary CMC surface, graphically and smoothly except on a finite set of singularities. If in addition Ric_N>0 and the boundary of N is convex, then the convergence is at most 2-sheeted. In particular, it is 1-sheeted if the limiting surface is not a minimal surface. Third, we consider the maximization of Steklov eigenvalues in higher dimensions. We show that for compact manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimension 3 and higher.

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Geometric properties of the space of Lagrangian self-shrinking tori in R^4 (2020)

No abstract available.

Uniqueness of Lagrangian mean curvature flow and minimal immersions with free boundary (2013)

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 ([4], [13]):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.

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Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

The fattening phenomenon for level set solutions of the mean curvature flow (2017)

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.

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