Einstein and conformally flat critical metrics of the volume functional. (English) Zbl 1222.53041

Let \(\Omega\) be a connected, compact \(n\)-dimensional \((n\geq 3)\) manifold with smooth boundary \(\Sigma\), \(\gamma\) a metric on \(\Sigma\) and \(R\) a constant. Denote by \({\mathcal M}^R_\gamma\) the space of metrics on \(\Omega\) which have constant scalar curvature \(R\) and induce on \(\Sigma\) the metric \(\gamma\). It is known that, if \(g\) is an element of \({\mathcal M}^R_\gamma\) such that the first Dirichlet eigenvalue of \((n-1)\Delta_g+ R\) on \(\Omega\) is positive, then \({\mathcal M}^R_\gamma\) has a manifold structure near \(g\). A metric \(g\) is a critical point of the volume functional in \({\mathcal M}^R_\gamma\) if and only if there exists a function \(\lambda\) on \(\Omega\) such that \(\lambda= 0\) on \(\Sigma\) and
\[ -(\Delta_g\lambda) g+ \nabla^2_g\lambda- \lambda\text{\,Ric}(g)= g\tag{1} \]
holds on \(\Omega\), where \(\Delta_g\), \(\nabla^2_g\) are the Laplacian and the Hessian operators with respect to \(g\), resp., and \(\text{Ric}(g)\) is the Ricci curvature.
A critical metric is a metric \(g\) on \(\Omega\) which satisfies (1) for some function \(\lambda\) which vanishes on the boundary of \(\Omega\).
In order to characterize critical metrics, in [Calc. Var. Partial Differ. Equ. 36, No. 2, 141–171 (2009; Zbl 1175.49043)] the authors obtained a partial description of critical metrics defined on a bounded domain in a simply connected space-form.
This paper consists in a detailed study of critical metrics, under the hypothesis that they are Einstein or conformally flat metrics. The first main result is the following:
Theorem 1. Let \((\Omega^n, g)\) be a connected compact Einstein manifold with smooth boundary. If \(g\) is a critical metric, then \((\Omega^n, g)\) is isometric to a geodesic ball in a simply connected space-form, i.e., \(\mathbb{R}^n\), \(\mathbb{S}^n\) or \(\mathbb{H}^n\).
To discuss conformally flat metrics, firstly the authors construct explicit examples of critical metrics which are in the form of warped products. These examples include the spatial Schwarzschild metric and the ADS-Schwarzschild metrics restricted to suitable domains. Then, they prove that any conformally flat non-Einstein critical metric either is one of the constructed warped products or it is covered by such a warped product metric. In particular, if \((\Omega^n, g)\) is conformally flat, simply connected and its boundary is isometric to a standard round sphere, \(g\) being a critical metric, then \((\Omega^n, g)\) is isometric to a geodesic ball in \(\mathbb{R}^n\), \(\mathbb{S}^n\) or \(\mathbb{H}^n\).
The paper includes an appendix on estimates of graphical representations of hypersurfaces with bounded second fundamental form, which are needed for the classification of conformally flat critical metrics.


53C20 Global Riemannian geometry, including pinching
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)


Zbl 1175.49043
Full Text: DOI arXiv


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