Compactness and Noncompactness for the Positive Yamabe--Escobar Problem with Minimal Boundary

Abstract

We study the Yamabe--Escobar equation prescribing positive constant scalar curvature and minimal boundary. The compactness conjecture asks whether being conformally equivalent to the round hemisphere is the only obstruction for its set of positive solutions to be compact. In this paper we show that this is true for dimensions $3\le n\le 14$. We show that this dimensional threshold $n_{\ast }=14$ is sharp by constructing in $n\ge 15$ nonumbilic and non-locally-conformally-flat examples of metrics in the hemisphere for which a noncompact sequence of solutions exists. We do the same for the special classes of locally conformally flat metrics, $n_{\ast }^{LCF}=18$, and the metrics with umbilic boundary, $n_{\ast }^{umb}=20$. In all three settings we also prove that the Weyl--umbilicity vanishing conjecture holds for $4\le n\le n_{\ast }$ and construct counterexamples in $n>n_{\ast }$ by perturbing the noncompactness examples. To the best of the author's knowledge, these are the first locally conformally flat noncompactness constructions for the positive-scalar-curvature, minimal-boundary case of the Yamabe--Escobar equation, outside the conformal class of the round hemisphere.

arXiv commentv1: 103 pages; v2: Numerical supplement added (missing in v1 due to submission difficulties). Introduction and bibliography revised, terminology and typos corrected, and some proof details expanded, 107 pages