A rigidity theorem for asymptotically flat static manifolds and its applications (Q6567173)

The paper focuses on asymptotically flat manifolds \((M^n,g)\) with boundary, \(3\leq n\leq 7\), endowed with a static potential \(V\), i.e., a function such that\N\[\N\nabla^2_g V=V\operatorname{Ric}_g\,,\qquad \Delta_g V=0\,,\qquad \lim_{|x|\to+\infty}V(x)=1\,.\N\]\NThe first result is an improvement of the rigidity statement of the Minkowski inequality for an outer-minimizing hypersurface \(\Sigma\subset\partial M\), proved in [\textit{S. McCormick}, Proc. Am. Math. Soc. 146, No. 9, 4039--4046 (2018; Zbl 1501.53082)]. Namely, the authors prove that if either \(V{|_{\Sigma}}\) or the mean curvature of \(\Sigma\) is constant, then equality holds in the Minkowski inequality if and only if \((M,g)\) is isometric to the exterior of a coordinate sphere in the Schwarzschild manifold with mass \(m\).\N\NThe authors also prove a different Minkowski inequality in the case where \(\partial M=\Sigma\) is connected and outer-minimizing, provided \(V{|_\Sigma}=V_0>0\) is constant:\N\[\NV_0\leq\frac{1}{(n-1)w_{n-1}}\left(\frac{|\Sigma|}{w_{n-1}}\right)^{\frac{2-n}{n-1}}\int_\Sigma H d\sigma\,.\N\]\NAgain, equality is achieved only for rotationally symmetric regions of the Schwarzschild manifold.\N\NThe authors then provide some applications, among which are a uniqueness result for static extensions for a class of Bartnik data, improving on a result in [\textit{P. Miao}, Classical Quantum Gravity 22, No. 11, L53--L59 (2005; Zbl 1071.83007)], and a uniqueness result for equipotential photon surfaces with small Einstein-Hilbert energy, that should be compared with the ones in [\textit{C. Cederbaum} and \textit{G. J. Galloway}, J. Math. Phys. 62, No. 3, Article ID 032504, 22 p. (2021; Zbl 1470.83033); \textit{S. Raulot}, Classical Quantum Gravity 38, No. 8, Article ID 085015, 22 p. (2021; Zbl 1482.58016)].