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Critical metrics of the volume functional on manifolds with boundary


Authors: H. Baltazar and E. Ribeiro Jr.
Journal: Proc. Amer. Math. Soc. 145 (2017), 3513-3523
MSC (2010): Primary 53C25, 53C21; Secondary 53C24
DOI: https://doi.org/10.1090/proc/13619
Published electronically: April 6, 2017
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Abstract: The goal of this article is to study the space of smooth Riemannian structures on compact manifolds with boundary that satisfies a critical point equation associated with a boundary value problem. We provide an integral formula which enables us to show that if a critical metric of the volume functional on a connected $ n$-dimensional manifold $ M^n$ with boundary $ \partial M$ has parallel Ricci tensor, then $ M^n$ is isometric to a geodesic ball in a simply connected space form $ \mathbb{R}^{n}$, $ \mathbb{H}^{n}$ or $ \mathbb{S}^{n}$.


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Additional Information

H. Baltazar
Affiliation: Universidade Federal do Piauí - UFPI, Departamento de Matemática, Campus Petrônio Portella, 64049-550, Teresina/ PI, Brazil
Email: halyson@ufpi.edu.br

E. Ribeiro Jr.
Affiliation: Universidade Federal do Ceará - UFC, Departamento de Matemática, Campus do Pici, Av. Humberto Monte, Bloco 914, 60455-760, Fortaleza - CE, Brazil
Email: ernani@mat.ufc.br

DOI: https://doi.org/10.1090/proc/13619
Keywords: Volume functional, critical metrics, parallel Ricci curvature, harmonic Weyl tensor
Received by editor(s): November 5, 2015
Published electronically: April 6, 2017
Additional Notes: The first and second authors were partially supported by CNPq/Brazil
Communicated by: Lei Ni
Article copyright: © Copyright 2017 American Mathematical Society