# Liouville properties

## TL;DR

This paper surveys the classical and generalized Liouville theorems, discusses Yau's conjecture, and explores recent applications in geometric group theory and mean curvature flow singularities, proposing new approaches for high codimension cases.

## Contribution

It provides a comprehensive overview of Liouville properties, discusses Yau's conjecture, and introduces new ideas for analyzing singularities in high codimension mean curvature flow.

## Key findings

- Yau's conjecture remains influential in geometric analysis.
- Recent methods have advanced understanding of polynomial growth groups.
- New approaches are proposed for classifying high codimension mean curvature flow singularities.

## Abstract

The classical Liouville theorem states that a bounded harmonic function on all of $\RR^n$ 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.