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1. 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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2. Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

   https://doi.org/10.1515/crelle-2017-0019  

   Isenberg, James ; Wu, Haotian ( September 2019 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the “vanishing” time T : (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter {\gamma>\frac{1}{2}} , there is a solution with the highest curvature blowing up at the rate {(T-t)^{{-(\gamma+\frac{1}{2})}}} . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the “Grim Reaper” solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ. 

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3. 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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4. Existence of hypersurfaces with prescribed mean curvature I – generic min-max

   Zhou, Xin ; Zhu, Jonathan J. ( April 2020 , Cambridge journal of mathematics)

   We prove that, for a generic set of smooth prescription functions h on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature h. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous min-max theory, developed for constant mean curvature hypersurfaces, can be extended to construct min-max prescribed mean curvature hypersurfaces for certain classes of prescription function, including a generic set of smooth functions, and all nonzero analytic functions. In particular we do not need to assume that h has a sign. 

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5. 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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