Notices of the International Consortium of Chinese Mathematicians

Volume 7 (2019)

Number 1

Special Issue: Celebrating Shing-Tung Yau on his 70th birthday

Guest Editors: Shiu-Yuen Cheng, Lizhen Ji, Hao Xu, and Xiaokui Yang

Liouville properties

Pages: 16 – 26

DOI: https://dx.doi.org/10.4310/ICCM.2019.v7.n1.a10

Authors

Tobias H. Colding (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

William P. Minicozzi, II (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Abstract

The classical Liouville theorem states that a bounded harmonic function on all of $\mathbf{R}^n$ must be constant. In the early 1970s, S.-T. Yau vastly generalized this, showing that it holds for manifolds with nonnegative Ricci curvature. Moreover, he conjectured a stronger Liouville property that has generated many significant developments. We will first discuss this conjecture and some of the ideas that went into its proof.

We will also discuss two recent areas where this circle of ideas has played a major role. One is Kleiner’s new proof of Gromov’s classification of groups of polynomial growth and the developments this generated. Another is to understanding singularities of mean curvature flow in high codimension. We will see that some of the ideas discussed in this survey naturally lead to a new approach to studying and classifying singularities of mean curvature flow in higher codimension. This is a subject that has been notoriously difficult and where much less is known than for hypersurfaces.

Published 16 May 2019