## 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 $$\mathbb{S}^3$$.

### MSC:

 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
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### References:

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