Sharp Nash inequalities on manifolds with boundary in the presence of symmetries

TL;DR

This paper determines the optimal constant for the Trace Nash inequality on symmetric compact Riemannian manifolds, highlighting how symmetries influence geometric inequalities.

Contribution

It introduces the best constant for the Trace Nash inequality considering symmetries, extending classical results to symmetric manifolds with infinite orbits.

Findings

Established the optimal constant A_{opt}(ar{M}) for the Trace Nash inequality.

Showed that symmetries improve the inequality bounds.

Applied results to manifolds with symmetry groups having infinite orbits.

Abstract

In this paper we establish the best constant $\widetilde A_{opt}(\bar{M})$ for the Trace Nash inequality on a $n-$dimensional compact Riemannian manifold in the presence of symmetries, which is an improvement over the classical case due to the symmetries which arise and reflect the geometry of manifold. This is particularly true when the data of the problem is invariant under the action of an arbitrary compact subgroup $G$ of the isometry group $Is(M,g)$, where all the orbits have infinite cardinal.