Variational characterizations of the total scalar curvature and eigenvalues of the Laplacian

TL;DR

This paper introduces variational characterizations of the kernel of the dual operator of scalar curvature linearization, using a new elliptic operator and geometric invariant, linking eigenvalues of the Laplacian to scalar curvature properties.

Contribution

It develops a novel variational framework involving a fourth-order elliptic operator and geometric invariant to analyze the kernel of the scalar curvature linearization.

Findings

The invariant $ u$ vanishes iff the kernel is non-trivial.

Large first Laplacian eigenvalue implies $ u$ is positive and kernel is trivial.

Lower bounds on $ u$ are established when the kernel is trivial.

Abstract

For the dual operator $s_g'^*$ of the linearization $s_g'$ of the scalar curvature function, it is well-known that if $\ker s_g'^*\neq 0$, then $s_g$ is a non-negative constant. In particular, if the Ricci curvature is not flat, then $ {s_g}/(n-1)$ is an eigenvalue of the Laplacian of the metric $g$. In this work, some variational characterizations were performed for the space $\ker s_g'^*$. To accomplish this task, we introduce a fourth-order elliptic differential operator $\mathcal A$ and a related geometric invariant $\nu$. We prove that $\nu$ vanishes if and only if $\ker s_g'^* \ne 0$, and if the first eigenvalue of the Laplace operator is large compared to its scalar curvature, then $\nu$ is positive and $\ker s_g'^*= 0$. Furthermore, we calculated the lower bound on $\nu$ in the case of $\ker s_g'^* = 0$. We also show that if there exists a function which is $\mathcal A$-superharmonic and the Ricci curvature has a lower bound, then the first non-zero eigenvalue of the Laplace operator has an upper bound.