Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group

关于整体图曲面的Bernstein型定理综述

A survey on Bernstein-type theorems for entire graphical surfaces

本文综述欧几里得与Lorentz-Minkowski空间中图曲面的Bernstein型定理，涵盖极小图、Heinz型平均曲率估计、常平均曲率图及Calabi-Bernstein定理。

A Bernstein type result for graphical self-shrinkers in $\mathbb{R}^4$

Self-shrinkers are important geometric objects in the study of mean curvature flows, while the Bernstein Theorem is one of the most profound results in minimal surface theory. We prove a Bernstein type result for graphical self-shrinker surfaces with codimension two in $\mathbb{R}^4$. Namely, under certain natural conditions on the Jacobian of any smooth map from $\mathbb{R}^2$ to $\mathbb{R}^2$, we show that the self-shrinker which is the graph of this map must be affine linear. The proof relies on the derivation of structure equations of graphical self-shrinkers in terms of the parallel form…

Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips

We prove a monotonicity result at specific points for the Horizontal Perimeter for a class of surfaces in the Heisenberg group.

平均曲率流的平移解的伯恩斯坦问题

A Bernstein problem for translating solutions to the mean curvature flow

该研究针对平均曲率流的平移解的伯恩斯坦问题，在N≥8时构造出非凸的平均凸整体图平移子，完善了相关刚性理论。

Complete translating solitons to the mean curvature flow in $\mathbb{R}^3$ with nonnegative mean curvature

We prove that any complete immersed two-sided mean convex translating soliton $Σ\subset \mathbb{R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in $\mathbb{R}^3$ is the axisymmetric "bowl soliton". We also show that if the mean curvature of $Σ$ tends to zero at infinity, then $Σ$ can be represented as an entire graph and so is the "bowl soliton". Finally we classify all locally strictly convex graphical translating solitons defined over strip regions.