Uniqueness of solutions of the Yamabe problem on manifolds with boundary. (October 2019)

• Record Type: Journal Article
• Authors: Cárdenas, Elkin; Sierra, Willy
• Is Part Of: Nonlinear analysis. Volume 187(2019)
• Page Start: 125
• Page End: 133
• Publication Date: 2019-10
• Subjects: The Yamabe problem -- Flat scalar curvature -- Constant mean curvature -- Uniqueness of solutions -- Riemannian metrics; 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.2019.04.010
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Given a compact manifold with boundary M of dimension m ≥ 3 and a nondegenerate Riemannian metric g ∗ having null scalar curvature, constant mean curvature, and unitary volume on the boundary, we show that the set of Riemannian metrics with null scalar curvature, constant mean curvature, and unitary volume on the boundary, near to g ∗ is an embedded submanifold of the manifold of all Riemannian metrics on M . Additionally, such submanifold is strongly transversal to the conformal classes. We also prove, using recent results of compactness, that conformal classes of metrics closed to g ∗ contain only one of these metrics. Highlights: We study results related to the Yamabe problem on compact manifolds with boundary M . We prove a result of global uniqueness for solutions of the Yamabe problem on M . A certain set of metrics is a submanifold of the set of Riemannian metrics on M .