Convex hull property for ancient harmonic map heat flows

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.

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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