Counterexamples to Escobar's conjecture

Abstract

Escobar (J Funct Anal 165(1):101-116, 1999) conjectured that for every $n\ge 3$, an $n$-dimensional compact Riemannian manifold with nonnegative Ricci curvature and all boundary principal curvatures bounded below by $\kappa >0$ must satisfy ${\sigma }_1\ge \kappa$. We disprove this conjecture for every $n\ge 3$ by constructing conformal deformations of the Euclidean unit ball. We first establish a perturbative criterion, then construct explicit polynomial conformal factors satisfying this criterion. For every sufficiently small $t>0$, the resulting metrics $g_t=e^{2t\Phi }{g}_{{\mathbb{R}}^n}$ have positive Ricci curvature, every boundary principal curvature is strictly larger than $1$, and ${\sigma }_1({\mathbb{B}}^n,g_t)<1$. The proof requires several computations, some of which were carried out in Mathematica. The Mathematica code is attached to this submission.