Mathematical Research Letters

Volume 29 (2022)

Number 5

Convex hull property for ancient harmonic map heat flows

Pages: 1571 – 1594



Chiung-Jue Anna Sung (Department of Mathematics, National Tsing Hua University, Hsin-Chu, Taiwan)


For an ancient solution u to the harmonic map heat flow from a complete manifold $M$ into a Cartan–Hadamard manifold $N$ with curvature bounded between two negative constants, we show that the image of $u$ is contained in the convex hull of its intersection with the ideal boundary of $N$ together with at most $k$ interior points in $N$, where $k$ is the dimension of the space of bounded ancient solutions to the heat equation on $M$. In the case $M$ has nonnegative Ricci curvature and $u$ is of polynomial growth, its image is contained in an ideal polyhedron with estimable number of vertices in terms of the growth order.

The author was partially supported by a grant from the National Science and Technology Council of Taiwan.

Received 21 July 2020

Accepted 28 March 2021

Published 21 April 2023