Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

Explicit formulas, symmetry and symmetry breaking for Willmore surfaces of revolution

In this paper we prove explicit formulas for all Willmore surfaces of revolution and demonstrate their use in the discussion of the associated Dirichlet boundary value problems. It is shown by an explicit example that symmetric Dirichlet boundary conditions do in general not entail the symmetry of the surface. In addition we prove a symmetry result for a subclass of Willmore surfaces satisfying symmetric Dirichlet boundary data.

Generalized Weierstrass formulae, soliton equations and Willmore surfaces. I. Tori of revolution and the mKdV equation

A new approach is proposed for study structure and properties of the total squared mean curvature $W$ of surfaces in ${\bf R}^3$. It is based on the generalized Weierstrass formulae for inducing surfaces. The quantity $W$ (Willmore functional) is shown to be invariant under the modified Novikov--Veselov hierarchy of integrable flows. The $1+1$--dimensional case and, in particular, Willmore tori of revolution, are studied in details. The Willmore conjecture is proved for the mKDV--invariant Willmore tori.

旋转环面的Willmore流

The Willmore Flow of Tori of Revolution

研究旋转环面Willmore流的长时间行为，证明能量低于8π时收敛至Clifford环面，并应用于共形约束极小化。

基于 Simon 的 Li-Yau 不等式的带边界 Willmore 流的全局存在性

Global existence for the Willmore flow with boundary via Simon's Li-Yau inequality

本文利用修改后的 Simon 的 Li-Yau 不等式和几何测度论工具，证明了初始能量低于特定阈值的带 Dirichlet 边界数据的圆柱旋转曲面 Willmore 流的全局存在性，并改进了平稳问题中 Willmore 极小化子的存在性结果。

Almost Monotonicity Formula for H-minimal Legendrian Surfaces in the Heisenberg Group

We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called {\it contact stationary legendrian immersions}) in the Heisenberg Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. This last result happens to be in particular a central ingredient in the analysis part of the proof of the Willmore conjecture given by the author in arXiv:2007.05467

叶状超曲面的Willmore型变分问题

Willmore-type variational problem for foliated hypersurfaces

本文研究带叶状结构超曲面的Willmore型变分问题，建立Reilly型泛函，计算第一、第二变分并证明部分共形不变性，推导临界超曲面的Euler-Lagrange方程，并构造旋转临界超曲面且证明其局部极小性。