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Creators/Authors contains: "Ghomi, Mohammad"

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  1. ; (, Transactions of the American Mathematical Society)
    Using fiber bundle theory and conformal mappings, we continuously select a point from the interior of Jordan domains in Riemannian surfaces. This selection can be made equivariant under isometries, and take on prescribed values such as the center of mass when the domains are convex. Analogous results for conformal transformations are obtained as well. It follows that the space of Jordan domains in surfaces of constant curvature admits an isometrically equivariant strong deformation retraction onto the space of round disks. Finally we develop a canonical procedure for selecting points from planar Jordan domains. 
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    Free, publicly-accessible full text available March 1, 2026
  2. (, Advanced Nonlinear Studies)
    Abstract We devise some differential forms after Chern to compute a family of formulas for comparing total mean curvatures of nested hypersurfaces in Riemannian manifolds. This yields a quicker proof of a recent result of the author with Joel Spruck, which had been obtained via Reilly’s identities. 
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  3. ; (, Proceedings of the American Mathematical Society)
    We show that a compact Riemannian 3 3 -manifold M M with strictly convex simply connected boundary and sectional curvature K ≤<#comment/> a ≤<#comment/> 0 K\leq a\leq 0 is isometric to a convex domain in a complete simply connected space of constant curvature a a , provided that K ≡<#comment/> a K\equiv a on planes tangent to the boundary of M M . This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard 3 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 ≥<#comment/> a ≥<#comment/> 0 K\geq a\geq 0 . 
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  4. ; (, Bulletin of the London Mathematical Society)
    Abstract We show that in Euclidean 3‐space any closed curve which contains the unit sphere within its convex hull has length , and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in dimensions, we include the estimate by Nazarov, which is sharp up to the constant . 
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  5. ; (, 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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  6. ; (, 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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  7. ; (, The Journal of Geometric Analysis)
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  8. ; (, Discrete & Computational Geometry)
    null (Ed.)
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  9. ; (, Calculus of Variations and Partial Differential Equations)
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  10. (, Notices of the American Mathematical Society)
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