Extremals of Log Sobolev inequality on non-compact manifolds and Ricci soliton structures

TL;DR

This paper proves the existence and decay properties of extremals for the Log Sobolev inequality on certain non-compact manifolds and shows that shrinking Ricci solitons support gradient structures under these conditions.

Contribution

It establishes the existence of extremals for the Log Sobolev functional on non-compact manifolds with Ricci bounds and links these to Ricci soliton structures, introducing new geometric estimates.

Findings

Existence of extremals for Log Sobolev functional under Ricci curvature conditions.

Exponential decay of extremals when Ricci curvature is bounded above.

Non-trivial shrinking Ricci solitons support gradient Ricci soliton structures.

Abstract

In this paper we establish the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity. When Ricci curvature is also bounded from above we get exponential decay at infinity of the extremals. As a consequence of these analytical results we establish, under the same assumptions, that non-trivial shrinking Ricci solitons support a gradient Ricci soliton structure. On the way, we prove two results of independent interest: the existence of a distance-like function with uniformly controlled gradient and Hessian on complete non-compact manifolds with bounded Ricci curvature and strictly positive injectivity radius and a general growth estimate for the norm of the soliton vector field on manifolds with bounded Ricci curvature.