Spectral properties of bipolar minimal surfaces in S^4

$\mathbb{B}^3$ 和 $\mathbb{S}^3$ 中谱极值极小曲面的大拓扑渐近性

Large topology asymptotics for spectrally extremal minimal surfaces in $\mathbb{B}^3$ and $\mathbb{S}^3$

研究利用拉普拉斯与Steklov特征值的等变优化构造低维球和球面中给定拓扑的极小曲面，在大欧拉特征情形下建立尖锐面积估计与varifold极限，验证低面积极小曲面空间的相关猜想及Lawson曲面的特殊作用。

Spectral properties of a family of minimal tori of revolution in five-dimensional sphere

The normalized eigenvalues $Λ_i(M,g)$ of the Laplace-Beltrami operator can be considered as functionals on the space of all Riemannian metrics $g$ on a fixed surface $M$. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in $\mathbb{S}^3$ or $\mathbb{S}^4$. In the present paper a family of extremal metrics induced by minimal immersions in $\mathbb{S}^5$ is provided.

谱分解定理与极小超曲面的端

Spectral splitting theorem and ends of minimal hypersurfaces

本文通过构造加权最小测地线，证明了非负双曲曲率流形中有限指数极小超曲面具有有限端，推广了Li-Wang关于非负截面曲率流形的结果。

On the Existence of Minimal Tori in $S^3$ of Arbitrary Spectral Genus

Minimal tori that are linearly full in the 3-sphere possess a natural invariant g called their spectral genus, which was introduced by Hitchin. We show that for each g>0, there are countably many real g-dimensional families of minimally immersed tori with spectral genus g (two of these dimensions are just reparametrisations of the torus). Each family maps from a fixed torus of rectangular conformal type.

Euclidean minimal tori with planar ends and elliptic solitons

A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere $S^3=\R^3\cup \{\infty\}$. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type is necessarily the double of the spectral curve of an elliptic KP soliton. The simplest possible examples of minimal tori with planar ends are related to 1--gap Lamé potentials, the simplest non--trivial algebro geometric KdV potentials. If one allows for translational periods, …

谱选择与庞加莱十二面体空间的最小莫尔斯结构

Spectral Selection and Minimal Morse Structures on the Poincaré Dodecahedral Space

研究热方程在庞加莱十二面体空间的长期行为，引入谱选择性质P，证明在初始数据密集开集下解最终成为最小莫尔斯函数，通过特征值分析和扰动机制展示莫尔斯结构的失效与恢复。

Finite type minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$

We study minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ of finite type by relating them to harmonic maps $\mathbb{C} \to \mathbb{S}^2$ of finite type. We rephrase an iteration by Pinkall-Sterling in terms of polynomial Killing fields. We discuss spectral curves, spectral data and the geometry of the isospectral set. We consider polynomial Killing fields with zeroes and the corresponding singular spectral curves, bubbletons and simple factors. We investigate the differentiable structure on the isospectral set of any finite type minimal annulus. We apply the theory to a 2-parameter family of…

临界点、在谱Ricci界下的分裂定理以及稳定极小超曲面的拓扑

Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces

本文基于临界点理论和谱Ricci界，证明了流形上Δ+V的临界性准则及分裂定理，拓展了Cheeger-Gromoll定理，并研究了非负bi-Ricci或截面曲率流形中稳定和δ-稳定极小超曲面的拓扑性质。

欧几里得空间中稳定极小超曲面的格林核梯度估计及$\mathbb{R}^4$中的稳定伯恩斯坦定理

Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$

本文通过新积分不等式证明了稳定极小超曲面的性质，并给出$\mathbb{R}^4$中完整双侧稳定极小超曲面是平面的简单证明，核心方法基于非负曲率空间中稳定极小超曲面具有谱里奇曲率下界。

基于Montiel-Ros划分方法的自由边界极小曲面谱估计

Spectral estimates for free boundary minimal surfaces via Montiel-Ros partitioning methods

本研究推广Montiel-Ros方法至带边界紧流形，结合有限群作用与扭同态，得到自由边界极小曲面Morse指标增长率的拓扑定量界，并通过两族对称例子推导出去等变约束后的显式双侧线性界。

The Novikov-Veselov hierarchy of equations and integrable deformations of minimal Lagrangian tori in CP^2

We associate a periodic two-dimensional Schrodinger operator to every Lagrangian torus in CP^2 and define the spectral curve of a torus as the Floquet spectrum of this operator on the zero energy level. In this event minimal Lagrangian tori correspond to potential operators. We show that Novikov-Veselov hierarchy of equations induces integrable deformations of minimal Lagrangian torus in CP^2 preserving the spectral curve. We also show that the highest flows on the space of smooth periodic solutions of the Tzizeica equation are given by the Novikov-Veselov hierarchy.