On energy gap phenomena of the Whitney spheres in $\mathbb{C}^n$ or $\mathbb{CP}^n$

New characterizations of the Whitney spheres and the contact Whitney spheres

In this paper, based on the classical K. Yano's formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction $J\vec{H}$ and the norm of the covariant differentiation of the second fundamental form $h$, where $J$ is the almost complex structure and $\vec{H}$ is the mean curvature vector field. Second and analogously, for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure $(φ,ξ,η,g)$, we also establish an optimal integral inequality involving the Ricc…

A note on the evolution of the Whitney sphere along mean curvature flow

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in $\mathbb{C}^n$ satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.

Lagrangian mean curvature flow of Whitney spheres

It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian subspace. In particular this result applies to the Whitney spheres.

On energy gap phenomena of the Whitney sphere and related problems

In this paper, we study Lagrangian submanifolds satisfying ${\rm \nabla^*} T=0$ introduced by Zhang \cite{Zh} in the complex space forms $N(4c)(c=0\ or \ 1)$, where $T ={\rm \nabla^*}\tilde{h}$ and $\tilde{h}$ is the Lagrangian trace-free second fundamental form. We obtain some Simons' type integral inequalities and rigidity theorems for such Lagrangian submanifolds. Moreover we study Lagrangian submanifolds in $\mathbb{C}^n$ satisfying $\nabla^*\nabla^*T=0$ and introduce a flow method related to them.

An energy gap phenomenon for the Whitney sphere

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.

Heisenberg群中平均曲率泛函的意外现象

Unexpected phenomena for mean curvature functionals in the Heisenberg group

本文发现欧几里得空间中球面优化平均曲率变分问题的范式在子黎曼Heisenberg群中失效，通过建立一般黎曼流形上的变分公式并借助黎曼逼近方案，在第一个Heisenberg群中研究了面积约束下总平均曲率的最优化问题，引入了一族新的旋转不变临界曲面——Pansu-Minkowski球面，并证明了其中最优成员唯一最小化Minkowski商，同时分析了其稳定性。