Critical metrics of the volume functional on compact three-manifolds with smooth boundary. (English) Zbl 1372.58010

A Miao-Tam critical metric is a 3-tuple \((M^n,\,g,\,f)\) where \((M^{n},\,g)\) is a compact Riemannian manifold of dimension \(n\geq 3\) with a smooth boundary \(\partial M\) and \(f: M^{n}\to \mathbb{R}\) is a smooth function such that \(f^{-1}(0)=\partial M\) satisfying the overdetermined-elliptic system \(\mathfrak{L}_{g}^{*}(f)=g\) where \(\mathfrak{L}_{g}^{*}(f)=-(\Delta f)g+\mathrm{Hess } f-\mathrm{Ric}\) is the formal \(L^{2}\)-adjoint of the linearization of the scalar curvature operator \(\mathfrak{L}_{g}\). Such a function \(f\) is called a potential function. In [Calc. Var. Partial Differ. Equ. 36, No. 2, 141–171 (2009; Zbl 1175.49043)] P. Miao and L.-F. Tam showed that these critical metrics arise as critical points of the volume functional on \(M^n\) when restricted to the class of metrics \(g\) with prescribed constant scalar curvature such that \(g_{|_{T \partial M}}=h\) for a prescribed Riemannian metric \(h\) on the boundary. In this paper, the authors obtain an estimate to the area of the boundary \(\partial M\) of Miao-Tam critical metrics on compact three-manifolds. In addition, the authors obtain a Böchner type formula which enables to show that a Miao-Tam critical metric on a compact three-manifold with positive scalar curvature must be isometric to a geodesic ball in \(\mathbb{S}^3\).


58E11 Critical metrics
58D17 Manifolds of metrics (especially Riemannian)
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C65 Integral geometry


Zbl 1175.49043
Full Text: DOI arXiv


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