×

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.

MSC:

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

Citations:

Zbl 1175.49043
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. · Zbl 0984.53001
[2] R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1 – 49. · Zbl 0191.52002
[3] Shiu Yuen Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (1975), no. 3, 289 – 297. · Zbl 0329.53035
[4] Justin Corvino, On the existence and stability of the Penrose compactification, Ann. Henri Poincaré 8 (2007), no. 3, 597 – 620. · Zbl 1120.83008
[5] Arthur E. Fischer and Jerrold E. Marsden, Deformations of the scalar curvature, Duke Math. J. 42 (1975), no. 3, 519 – 547. · Zbl 0336.53032
[6] Doris Fischer-Colbrie and Richard Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199 – 211. · Zbl 0439.53060
[7] Richard S. Hamilton, A compactness property for solutions of the Ricci flow, Amer. J. Math. 117 (1995), no. 3, 545 – 572. · Zbl 0840.53029
[8] Osamu Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34 (1982), no. 4, 665 – 675. · Zbl 0486.53034
[9] Osamu Kobayashi and Morio Obata, Conformally-flatness and static space-time, Manifolds and Lie groups (Notre Dame, Ind., 1980) Progr. Math., vol. 14, Birkhäuser, Boston, Mass., 1981, pp. 197 – 206. · Zbl 0485.58020
[10] Jacques Lafontaine, Conformal geometry from the Riemannian viewpoint, Conformal geometry (Bonn, 1985/1986) Aspects Math., E12, Friedr. Vieweg, Braunschweig, 1988, pp. 65 – 92. · Zbl 0661.53008
[11] Pengzi Miao and Luen-Fai Tam, On the volume functional of compact manifolds with boundary with constant scalar curvature, Calc. Var. Partial Differential Equations 36 (2009), no. 2, 141 – 171. · Zbl 1175.49043
[12] Richard Schoen and Shing Tung Yau, Proof of the positive mass theorem. II, Comm. Math. Phys. 79 (1981), no. 2, 231 – 260. · Zbl 0494.53028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.