On the conformal-biharmonic stability of the identity map of Einstein manifolds

TL;DR

This paper studies the stability of the identity map on Einstein manifolds under conformal-biharmonic and harmonic functionals, revealing a surprising equivalence except on the 4D sphere.

Contribution

It demonstrates that the conformal-biharmonic index matches the harmonic index on compact Einstein manifolds, except for the 4D sphere where stability differs.

Findings

Conformal-biharmonic index equals harmonic index on most Einstein manifolds.

The 4D Euclidean sphere is an exception with different stability properties.

The identity map's stability varies between energy and conformal-bienergy functionals.

Abstract

The identity map of an Einstein manifold is a critical point of both the classical energy functional and the conformal-bienergy functional. In this paper, we investigate the conformal-biharmonic stability of the identity map of compact Einstein manifolds of dimension at least four and with nonnegative scalar curvature, and we compare it with the harmonic stability, when the identity map is considered as a harmonic map. Somewhat surprisingly, we show that the conformal-biharmonic index coincides with the harmonic index, with a single notable exception: the four-dimensional Euclidean sphere. In this case, the identity map is unstable with respect to the energy functional, as shown independently by Mazet and Smith, whereas it is stable with respect to the conformal-bienergy functional.