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arXiv · 2610.08759

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

Santiago Cordero-Misteli
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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 $3\leq n\leq 14$. We show that this dimensional threshold $n_*=14$ is sharp by constructing in $n\geq 15$ 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, $n_*^{LCF}=18$, and the metrics with umbilic boundary, $n_*^{umb} =20$. In all three settings we also prove that the Weyl--umbilicity vanishing conjecture holds for $4\leq n\leq n_*$ and construct counterexamples in $n>n_*$ 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.

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BibTeXRIS

Santiago Cordero-Misteli. 2026-10-06. Compactness and Noncompactness for the Positive Yamabe--Escobar Problem with Minimal Boundary. https://arxiv.org/abs/2610.08759

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