Nonexistence of minimizers for the second conformal eigenvalue near the round sphere in low dimensions

TL;DR

This paper proves that in dimensions 3 to 10, there are no minimizers for the second conformal eigenvalue near the round sphere, contrasting with higher dimensions where minimizers do exist, and also provides bounds for Yamabe solutions.

Contribution

It demonstrates the nonexistence of minimizers in low dimensions and extends bounds on Yamabe solutions, revealing dimension-dependent behavior of conformal eigenvalues.

Findings

No minimizers for the second conformal eigenvalue in dimensions 3-10 near the sphere.

Existence of minimizers in dimensions 11 and higher for the same problem.

Lower bounds on the energy of sign-changing Yamabe solutions in dimensions 3-5.

Abstract

We consider the problem of minimizing the second conformal eigenvalue of the conformal Laplacian in a conformal class of metrics with renormalized volume. We prove, in dimensions $n\in\left\{3,\dotsc,10\right\}$, that a minimizer for this problem does not exist for metrics sufficiently close to the round metric on the sphere. This is in striking contrast with the situation in dimensions $n \ge 11$, where Ammann and Humbert obtained the existence of minimizers for the second conformal eigenvalue on any smooth closed non-locally conformally flat manifold. As a byproduct of our techniques, we also obtain a lower bound on the energy of sign-changing solutions of the \nobreak Yamabe equation in dimensions 3, 4 and 5, which extends a result obtained by Weth in the case of the round sphere.