More Like this

1. 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. 

   more » « less
2. The Neumann problem on the domain in 𝕊 ³ bounded by the Clifford torus

   https://doi.org/10.1515/ans-2022-0072  

   Case, Jeffrey S; Chen, Eric; Wang, Yi; Yang, Paul; Yung, Po-Lam (January 2023, Advanced Nonlinear Studies)

   Abstract In this study, the solution of the Neumann problem associated with the CR Yamabe operator on a subset $\mathrm{\Omega }$\Omegaof the CR manifold ${\mathbb{S}}^3${{\mathbb{S}}}^{3}bounded by the Clifford torus $\mathrm{\Sigma }$\Sigmais discussed. The Yamabe-type problem of finding a contact form on $\mathrm{\Omega }$\Omegawhich has zero Tanaka-Webster scalar curvature and for which $\mathrm{\Sigma }$\Sigmahas a constant $p$p-mean curvature is also discussed. 

   more » « less
3. 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. 

   more » « less
4. 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. 

   more » « less
5. Uniqueness of Blowups for Forced Mean Curvature Flow

   https://doi.org/10.1093/imrn/rnaf023  

   Hirsch, Sven; Zhu, Jonathan J (February 2025, International Mathematics Research Notices)

   Abstract We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. The corresponding results for mean curvature flow in Euclidean space were proven by Schulze and Colding–Minicozzi, respectively. We adapt their methods to handle the presence of the forcing term, which vanishes in the blow-up limit but complicates the analysis along the rescaled flow. Our results naturally include the case of mean curvature flows in Riemannian manifolds. 

   more » « less