A weighted renormalized curvature for manifolds with density

TL;DR

This paper introduces a new scalar invariant for manifolds with density, demonstrating its variational properties and stability of shrinking gradient Ricci solitons under a related functional.

Contribution

It defines a novel weighted renormalized curvature invariant and proves its variational nature, linking it to stability analysis of Ricci solitons.

Findings

The invariant is variational.

Shrinking gradient Ricci solitons are stable under the -functional.

Analogous to the conformal geometry volume coefficient v_3.

Abstract

We introduce a scalar invariant on manifolds with density which is analogous to the renormalized volume coefficient $v_3$ in conformal geometry. We show that this invariant is variational and that shrinking gradient Ricci solitons are stable with respect to the associated $\mathcal{W}$-functional.