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- Title
A Bernstein Type Result for Graphical Self-Shrinkers in ${\boldsymbol{\mathbb{R}}}^{\bf 4}$.
- Authors
Zhou, Hengyu
- 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.
- Subjects
MINIMAL surfaces; PLATEAU'S problem; MAXIMA &; minima; JACOBIAN matrices; MATHEMATICS theorems
- Publication
IMRN: International Mathematics Research Notices, 2018, Vol 2018, Issue 21, p6798
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnx089