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

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 ≠ 0, then sg is a nonnegative constant. Moreover, if the Ricci curvature does not vanish, then sg/(n-1) is an eigenvalue of the Laplacian of the metric g. In this work, we give some variational characterizations for the space ker s'*g . To accomplish this, we introduce a fourth-order elliptic differential operator A and a related geometric invariant ν. We prove that ν vanishes if and only if ker s'*g ≠ 0, and if the first eigenvalue of the Laplace operator is large compared to its scalar curvature, then ν is positive and ker s'*g = 0. We calculate a lower bound for ν in the case of ker s'*g = 0. We also show that if there exists a function which is A-superharmonic and the Ricci curvature has a lower bound, then the first nonzero eigenvalue of the Laplace operator has an upper bound. © 2013 Mathematical Sciences Publishers.