Optimal bounds for ancient caloric functions
Differential Geometry
2021-02-09 v3 Analysis of PDEs
Classical Analysis and ODEs
Group Theory
Geometric Topology
Abstract
For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions on any space where Yau's 1974 conjecture about polynomial growth harmonic functions holds.
Cite
@article{arxiv.1902.01736,
title = {Optimal bounds for ancient caloric functions},
author = {Tobias Holck Colding and William P. Minicozzi},
journal= {arXiv preprint arXiv:1902.01736},
year = {2021}
}
Comments
A stronger sharp dimension bound is added which is an equality on Euclidean space. To appear in Duke Math. Journal