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

Seungsu Hwang, Jeongwook Chang and Gabjin Yun

Vol. 261 (2013), No. 2, 395–415

DOI: 10.2140/pjm.2013.261.395

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 s_g is a nonnegative constant. Moreover, if the Ricci curvature does not vanish, then s_g∕(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.

Keywords

critical point equation, fourth-order elliptic operator, eigenvalue, Einstein metric, Laplace operator, scalar curvature, total scalar curvature

Mathematical Subject Classification 2010

Primary: 53C21

Milestones

Received: 2 December 2011

Revised: 26 September 2012

Accepted: 22 October 2012

Published: 20 March 2013