Liouville properties
Name
1902.09366.pdf
Description
Submitted version
Size
244.55 KB
Format
Adobe PDF
Checksum (MD5)
5b19a693523570c894007a8c02fe9488
Author(s)
Colding, Tobias
Minicozzi, William
Date Issued
May 2019
Journal
Notices of the International Congress of Chinese Mathematicians
Publisher
International Press of Boston
Citation
Colding, Tobias Holck and William P. Minicozzi II. "Liouville properties." Notices of the International Congress of Chinese Mathematicians Volume 7 (2019) © 2019 The Author(s)
Version
Original manuscript
Abstract
The classical Liouville theorem states that a bounded harmonic function on allofRnmust be constant. In the early 1970s, S.T. Yau vastly generalized this, showing that itholds for manifolds with nonnegative Ricci curvature. Moreover, he conjectured a strongerLiouville property that has generated many significant developments. We will first discussthis 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 thedevelopments this generated. Another is to understanding singularities of mean curvatureflow in high codimension. We will see that some of the ideas discussed in this surveynaturally lead to a new approach to studying and classifying singularities of mean curvatureflow in higher codimension. This is a subject that has been notoriouslydifficult and wheremuch less is known than for hypersurfaces.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.4310/iccm.2019.v7.n1.a10
