On the geometric dependence of Riemannian Sobolev best constants

TL;DR

This paper investigates how the second Riemannian L^p-Sobolev best constant varies with the geometry of the manifold, establishing continuity in the C^2-topology for 1 <= p <= 2 and deriving related existence and compactness results.

Contribution

It proves the continuity of the Sobolev best constant with respect to the Riemannian metric in the C^2-topology for p between 1 and 2, and shows the topology's sharpness at p=2.

Findings

B_0(p,g) depends continuously on g in C^2-topology for 1 <= p <= 2

Continuity is sharp at p=2

Existence and C^0-compactness results for extremal functions

Abstract

We concerns here with the continuity on the geometry of the second Riemannian L^p-Sobolev best constant B_0(p,g) associated to the AB program. Precisely, for 1 <= p <= 2, we prove that B_0(p,g) depends continuously on g in the C^2-topology. Moreover, this topology is sharp for p = 2. From this discussion, we deduce some existence and C^0-compactness results on extremal functions.