The Second Variational Formula For the Functional $\int v^{(6)}(g)dV_g$

TL;DR

This paper derives the second variational formula for a curvature-related functional on manifolds, showing Einstein manifolds with positive scalar curvature are local maxima within their conformal class, generalizing previous results.

Contribution

It computes the second variational formula for the Graham-Juhl functional and extends known results to non-locally conformally flat Einstein manifolds.

Findings

Einstein manifolds with positive scalar curvature are strict local maxima within their conformal class

The second variational formula for the functional is explicitly derived

Generalizes previous results to broader class of manifolds

Abstract

In this note, we compute the second variational formula for the functional $\int_M v^{(6)}(g)dv_g$, which was introduced by Graham-Juhl and the first variational formula was obtained by Chang-Fang. We also prove that Einstein manifolds (with dimension $\ge 7$) with positive scalar curvature is a strict local maximum within its conformal class, unless the manifold is isometric to round sphere with the standard metric up to a multiple of constant. Note that when $(M,g)$ is locally conformally flat, this functional reduces to the well-studied $\int_M \sigma_3(g)dv_g$. Hence, our result generalize a previous result of Jeff Viaclovsky without the locally conformally flat restraint.