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

Abstract

For the dual operator $s_g^{\prime \ast }$ of the linearization $s_g^{\prime }$ of the scalar curvature function, it is well-known that if $\ker s_g^{\prime \ast }\ne 0$, 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 $\ker s_g^{\prime \ast }$. 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 $\ker s_g^{\prime \ast }\ne 0$, and if the first eigenvalue of the Laplace operator is large compared to its scalar curvature, then $\nu$ is positive and $\ker s_g^{\prime \ast }=0$. Furthermore, we calculated the lower bound on $\nu$ in the case of $\ker s_g^{\prime \ast }=0$. 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.

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