Abstract
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.