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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. Colding–Minicozzi entropies in Cartan–Hadamard manifolds

   https://doi.org/10.1515/crelle-2025-0032  

   Bernstein, Jacob; Bhattacharya, Arunima (May 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We introduce a family of functionals defined on the set of submanifolds of Cartan–Hadamard manifolds which generalize the Colding–Minicozzi entropy of submanifolds of Euclidean space.We show these functionals are monotone under mean curvature flow under natural conditions.As a consequence, we obtain sharp lower bounds on these entropies for certain closed hypersurfaces and observe a novel rigidity phenomenon. 

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4. 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 $K\le <\#comment/>a\le <\#comment/>0$is isometric to a convex domain in a complete simply connected space of constant curvature $a$, provided that $K\equiv <\#comment/>a$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 $K\ge <\#comment/>a\ge <\#comment/>0$. 

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5. 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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