On the stability of Riemannian manifolds

Abstract

M The identity map of a compact Riemannian manifold is always a harmonic map. Any harmonic map has its Jacobi operator determined by the second variational formula of the energy integral of the harmonic map. The Jacobi operator of the identity map of a compact manifold is a linear elliptic selfadjoint operator of second order on the vector fields of the manifold. So we consider the first eigenvalue of the Jacobi operator of the identity map. We call a Riemannian manifold stable if the first eigenvalue of the Jacobi operator of the identity map is non-negative and unstable otherwise. The stability of Riemannian manifolds has been studied by many people. Mostly they studied which Riemannian manifolds are stable or unstable. We consider the stability problem from the different point of view. We are interested in the problem how stability of a compact Riemannian manifold depends on its Riemannian metric. For example the three-dimensional sphere is unstable with its standard Riemannian metric but there also exists a Riemannian metric which makes the three-sphere stable. We formulate the problem as follows: