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

Abstract

We study harmonic map heat flow along ancient super Ricci flow, and derive several Liouville theorems with controlled growth from Perelman’s reduced geometric viewpoint. For non-positively curved target spaces, our growth condition is sharp. For positively curved target spaces, our Liouville theorem is new even in the static case (i.e., for harmonic maps); moreover, we point out that the growth condition can be improved, and almost sharp in the static case. This fills the gap between the Liouville theorem of Choi and the example constructed by Schoen-Uhlenbeck. 1. Background 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. 1.1. Ancient super Ricci flow. A smooth manifold (M, g(t))t∈I with a time-dependent Riemannian metric is called Ricci flow when ∂tg = −2Ric, which has been introduced by Hamilton [14]. A supersolution to this equation is called super Ricci flow. Namely, (M, g(t))t∈I is called super Ricci flow if ∂tg ≥ −2Ric, which has been introduced by McCann-Topping [33] from the viewpoint of optimal transport theory. Recently, the super Ricci flow has begun to be investigated from various perspectives, especially metric measure geometry (see e.g., [3], [4], [16], [19], [20], [21], [25], [26], [27], [28], [29], [30], [39]). A Ricci flow (M, g(t))t∈I is said to be ancient when I = (−∞, 0], which is one of the crucial concepts in singular analysis of Ricci flow. In the present paper, we will focus on ancient super Ricci flow. 1.2. Liouville theorems for ancient solutions to heat equation. The celebrated Yau’s Liouville theorem states that on a complete manifold of non-negative Ricci curvature, any positive harmonic functions must be constant. One of the natural research directions is to generalize his Liouville theorem for ancient solutions to heat equation ∂tu = ∆u. Souplet-Zhang [38] have proven the following parabolic analogue (see [38, Theorem 1.2]): Theorem 1.1 ([38]). Let (M, g) be a complete Riemannian manifold of non-negative Ricci curvature. Then we have the following: Date: April 8, 2021. 2010 Mathematics Subject Classification. Primary 53C44; Secondly 53C43.