Liouville theorem for heat equation along ancient super Ricci flow via reduced geometry

Abstract

The aim of this article is to provide a Liouville theorem for heat equation along ancient super Ricci flow. We formulate such a Liouville theorem under a growth condition concerning Perelman's reduced distance.

Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry

“…This is a continuation of [22] on Liouville theorems for heat equation along ancient super Ricci flow. The aim of this paper is to generalize the target spaces, and formulate Liouville theorems for harmonic map heat flow.…”

“…We review some facts on Perelman's reduced geometry. The references are [8], [34], [35], [42], [43], [44], [22]. We mainly refer to [22,Section 3].…”

“…Guo-Philipowski-Thalmaier [13] have provided an approach to this problem from stochastic analytic viewpoint, and obtained a Liouville theorem under a growth condition for entropy (see [13,Theorem 2]). On the other hand, the authors [22] have done from Perelman's reduced geometric viewpoint (cf. [35]), and established a Liouville theorem under a growth condition concerning reduced distance.…”

“…[35]), and established a Liouville theorem under a growth condition concerning reduced distance. Now, let us recall the precise statement of the Liouville theorem in [22]. To do so, we fix some notations on a complete, time-dependent Riemannian manifold (M, g(τ )) τ ∈[0,∞) , which is not necessarily backward super Ricci flow.…”

“…The main result in [22] can be stated as follows (see [22,Theorem 2.2]): 22]). Let (M, g(τ )) τ ∈[0,∞) be an admissible, complete backward super Ricci flow.…”