Compactness and blow up results for doubly perturbed Yamabe problems on manifolds with umbilic boundary. (April 2023)

• Record Type: Journal Article
• Authors: Ghimenti, Marco G.; Micheletti, Anna Maria
• Is Part Of: Nonlinear analysis. Volume 229(2023)
• Publication Date: 2023-04
• Subjects: 35J65 -- 53C21; Umbilic boundary -- Yamabe problem -- Compactness -- Blow up analysis; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113206
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Given a compact Riemannian manifold with umbilic boundary, the Yamabe boundary problem studies if there exist conformal scalar-flat metrics such that ∂ M has constant mean curvature. In this paper we address to the stability of this problem with respect of perturbation of mean curvature of the boundary and scalar curvature of the manifold. In particular we prove that the Yamabe boundary problem is stable under perturbation of the mean curvature and the scalar curvature from below, while it is not stable if one of the two curvatures is perturbed from above.