On 2-surfaces in R^4 and R^n

Foliations and Chern-Heinz inequalities

We extend the Chern-Heinz inequalities about mean curvature and scalar curvature of graphs of $C^{2}$-functions to leaves of transversally oriented codimension one $C^{2}$-foliations of Riemannian manifolds. That extends partially Salavessa's work on mean curvature of graphs and generalize results of Barbosa-Kenmotsu-Oshikiri \cite{barbosa-kenmotsu-Oshikiri} and Barbosa-Gomes-Silveira \cite{barbosa-gomes-silveira} about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces. These Chern-Heinz inequalities for foliations can be applied to prove Haymann-Makai-Osserm…

A note on two-dimensional minimal surface graphs in R^n and a theorem of Bernstein-Liouville type

Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of two-dimensional minimal surface graphs in R^n. This leads us to a theorem of Bernstein-Liouville type.

Gaussian curvature of minimal graphs in $M\times \mathbb{R}$

In this paper, we consider minimal graphs in the three-dimensional Riemannian manifold $M\times\mathbb{R}$. We mainly estimate the Gaussian curvature of such surfaces. We consider the minimal disks and minimal graphs bounded by two Jordan curves in parallel planes. The key to the proofs is the Weierstrass representation of those surfaces via $\wp-$harmonic mappings. We also prove some Schwarz lemma type results and some Heinz type results for harmonic mappings between geodesic disks in Riemannian surfaces.

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。

$H_R=H_L$ 曲面方程解的临界点

Critical points of the solution to the $H_R=H_L$ surface equation

本文研究 $H_R=H_L$ 曲面方程解的临界点，给出临界点的必要几何条件、Dirichlet 问题的新唯一性结果及内半径的 Heinz 型界，并推广到 $\mathcal{C}^2$ 类函数。

关于极小图曲率猜想的一个尖锐形式

On a sharp form of curvature conjecture for minimal graphs

针对单位圆盘上极小图的曲率问题，作者基于高斯曲率严格不等式的证明，改进了Hopf常数估计，并证明对任意单位法向量存在极小图，其原点正上方曲率接近相关Scherk型曲面曲率，强化了经典结果。