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1. Total mean curvatures of Riemannian hypersurfaces

   https://doi.org/10.1515/ans-2022-0029  

   Ghomi, Mohammad ; Spruck, Joel ( January 2023 , Advanced Nonlinear Studies)

   Abstract We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ \Gamma in a Cartan-Hadamard manifold M M . In particular, we show that the first mean curvature integral of a convex hypersurface γ \gamma nested inside Γ \Gamma cannot exceed that of Γ \Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ \Gamma in terms of the volume it bounds in M M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ \gamma is parallel to Γ \Gamma , or M M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds. 

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2. Rigidity of nonpositively curved manifolds with convex boundary

   https://doi.org/10.1090/proc/16475  

   Ghomi, Mohammad ; Spruck, Joel ( November 2023 , Proceedings of the American Mathematical Society)

   We show that a compact Riemannian$3$-manifold$M$with strictly convex simply connected boundary and sectional curvature$K\leq a\leq 0$is isometric to a convex domain in a complete simply connected space of constant curvature$a$, provided that$K\equiv a$on planes tangent to the boundary of$M$. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard$3$-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for$K\geq a\geq 0$.

    

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3. Capacity, quasi-local mass, and singular fill-ins

   https://doi.org/10.1515/crelle-2019-0040  

   Mantoulidis, Christos ; Miao, Pengzi ; Tam, Luen-Fai ( December 2019 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown–York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them. 

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4. The anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces

   https://doi.org/10.1002/cpa.22189  

   De_Philippis, Guido ; De_Rosa, Antonio ( July 2024 , Communications on Pure and Applied Mathematics)

   We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3–dimensional Riemannian manifolds. The constructed min‐max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension 3. 

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5. On far-outlying constant mean curvature spheres in asymptotically flat Riemannian 3-manifolds

   https://doi.org/10.1515/crelle-2019-0034  

   Chodosh, Otis ; Eichmair, Michael ( October 2020 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract We extend the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [S. Brendle and M. Eichmair,Isoperimetric and Weingarten surfaces in the Schwarzschild manifold,J. Differential Geom. 94 2013, 3, 387–407] to the “far-off-center” regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable constant mean curvature spheres that depend delicately on the behavior of scalar curvature at infinity. 

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