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

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.

Final version. To appear International Mathematics Research Notices

References in corpus (1)

• On the self-shrinking systems in arbitrary codimension spaces