Liouville properties

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.