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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. Minkowski Inequality in Cartan–Hadamard Manifolds

   https://doi.org/10.1093/imrn/rnad114  

   Ghomi, Mohammad ; Spruck, Joel ( June 2023 , International Mathematics Research Notices)

   Using harmonic mean curvature flow, we establish a sharp Minkowski-type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard $3$-manifolds. This inequality also improves the known estimates for total mean curvature in hyperbolic $3$-space. As an application, we obtain a Bonnesen-style isoperimetric inequality for surfaces with convex distance function in nonpositively curved $3$-spaces, via monotonicity results for total mean curvature. This connection between the Minkowski and isoperimetric inequalities is extended to Cartan–Hadamard manifolds of any dimension. 

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3. Kasner-like regions near crushing singularities *

   https://doi.org/10.1088/1361-6382/abd3e1  

   Lott, John ( December 2020 , Classical and Quantum Gravity)

   Abstract We consider vacuum spacetimes with a crushing singularity. Under some scale-invariant curvature bounds, we relate the existence of Kasner-like regions to the asymptotics of spatial volume densities. 

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4. Mean Curvature Flow in Null Hypersurfaces and the Detection of MOTS

   https://doi.org/10.1007/s00220-022-04326-9  

   Roesch, Henri ; Scheuer, Julian ( February 2022 , Communications in Mathematical Physics)

   We study the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a global foliation of the past of an outermost MOTS, provided the null hypersurface admits an un-trapped foliation asymptotically.

    

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5. Uniqueness of convex ancient solutions to mean curvature flow in R^3.

   https://doi.org/10.1007/s00222-019-00859-4  

   Brendle, Simon ; Choi, Kyeongsu ( July 2019 , Inventiones mathematicae)

   A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension 3 which have positive sectional curvature and are κ-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in R^3, and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in R^3 which is strictly convex and noncollapsed. 

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