Members & Former Members

In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds in Rn+m. We first provide, following Cabré \cite{Cabre2008}, an ABP proof of the relative isoperimetric inequality proved in Choe-Ghomi-Ritoré \cite{CGR07}, by generalizing ideas of restricted normal cones given in \cite{CGR06}. Then we prove a relative isoperimetric inequalities for minimal submanifolds in Rn+m, which is optimal when the codimension m≤2. In other words we obtain a relative version of isoperimetric inequalities for minimal submanifolds proved recently by Brendle \cite{Brendle2019}. When the codimension m≤2, our result gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6. As another application we prove an optimal logarithmic Sobolev inequality for free boundary submanifolds in the Euclidean space following a trick of Brendle in \cite{Brendle2019b}.

Related project(s):
22Willmore functional and Lagrangian surfaces

Abstract: For an immersed Lagrangian submanifold, let \check{A} be the Lagrangian trace-free second fundamental form. In this note we consider the equation \nabla^*T=0 on Lagrangian surfaces immersed in \mathbb{C}^2, where T=-2\nabla^*(\check{A}\lrcornerω), and we prove a gap theorem for the Whitney sphere as a solution to this equation.

Related project(s):
22Willmore functional and Lagrangian surfaces

In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.

Related project(s):
22Willmore functional and Lagrangian surfaces

In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. This solves completely a long-standing open problem. In the proof one of crucial ingredients is a new Minkowski type formula. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.

Related project(s):
22Willmore functional and Lagrangian surfaces

In this note, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the free boundary hypersurfaces and capillary hypersurfaces. Then we present some interesting minimal Lagrangian submanifolds examples satisfying this boundary condition and we prove a Lagrangian version of Nitsche (or Hopf) type theorem. Some problems are proposed at the end of this note.

Related project(s):
22Willmore functional and Lagrangian surfaces

In this paper we prove that any smooth surfaces can be locally isometrically embedded into as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.

Related project(s):
22Willmore functional and Lagrangian surfaces