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

TL;DR

This paper establishes a Bernstein type theorem for graphical self-shrinkers in four-dimensional space, showing that under certain Jacobian conditions, such self-shrinkers must be affine linear, extending classical minimal surface results.

Contribution

It proves a Bernstein type result for codimension two graphical self-shrinkers in , linking Jacobian conditions to linearity of the self-shrinker graphs.

Findings

Self-shrinkers satisfying certain Jacobian conditions are affine linear.

Derived structure equations for graphical self-shrinkers in .

Identified positive functions related to Jacobian conditions on self-shrinkers.

Abstract

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, and the existence of some positive functions on self-shrinkers related to these Jacobian conditions.