A Bernstein Type Result for Graphical Self-Shrinkers in ${\boldsymbol{\mathbb{R}}}^{\bf 4}$. (15th May 2017)

• Record Type: Journal Article
• Authors: Zhou, Hengyu
• Is Part Of: International mathematics research notices. Volume 2018:Number 21(2018)
• Page Start: 6798
• Page End: 6815
• Publication Date: 2017-05-15
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx089
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: Self-shrinkers are important geometric objects in the evolution 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 co-dimension 2 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 through 0. 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.