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Mohammad Ghomi

Publications and source records attributed to Mohammad Ghomi.

At least 19 recordsLinked to original sources
arXiv2026

Shades and Convexity of Hypersurfaces

Let $f\colon M^n\to\mathbf{R}^{n+1}$, $n\geq2$, be a smooth immersion of a closed oriented manifold, with Gauss map $ν\colon M\to\mathbf{S}^n$. For each unit vector $u$, the shade $S_u\subset M$ is the set of points where $ν$ makes an acute angle with $u$. We show that $f$ is a convex embedding if and only if every shade is contractible. More generally, if every shade is $k$-connected, then $f$ is a convex embedding provided that $n\leq 2k+1$, or $n=2k+2$ and $H_{k+1}(M;\mathbf{Z})=0$. On the other hand, for every $n\geq 2k+3$ there is a nonconvex embedding $f\colon\mathbf{S}^n\to\mathbf{R}^{n+1}$ all of whose shades are $k$-connected. These results give sharp answers to a question of H. Wente in all dimensions.

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arXiv2026

A Converse to the Archimedes Sphere Theorem

We show that a convex surface in Euclidean 3-space is a sphere if the area cut off from it by a slab of some fixed width depends only on the direction of the slab. This result generalizes theorems of Blaschke and Stamm, which assumed the slab-area property for every width. It holds also for any smooth closed embedded surface, and we obtain analogous results in higher dimensions. The proof is based on an averaging identity, which relates slab areas to the Newtonian potential of the surface, the electrostatic characterization of spheres by Mendez and Reichel, and analytic microlocal properties of the Radon transform, which are used to show that the slab-area property forces real analyticity, via theorems of Kashiwara and Boman.

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arXiv2026

Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds

We show that if the curvature of a Cartan-Hadamard $n$-manifold is constant near a convex hypersurface $Γ$, then the total Gauss-Kronecker curvature $\mathcal{G}(Γ)$ is not less than that of any convex hypersurface nested inside $Γ$. This extends Borbély's monotonicity theorem in hyperbolic space. It follows that $\mathcal{G}(Γ)$ is bounded below by the volume of the unit sphere in Euclidean space $\mathbf{R}^n$. If $n=3$ or $Γ$ is $\mathcal{C}^{1,1}$, it suffices that the curvature be constant on $Γ$.

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arXiv2026

A counterexample to Nirenberg's conjecture on closed asymptotic curves

We construct a smooth embedded annulus in Euclidean 3-space with negative Gauss curvature in its interior whose boundary consists of strictly convex curves in planes tangent to the annulus, and which contains closed asymptotic curves. This gives a counterexample to a 1962 conjecture of Nirenberg. The construction is based on Lelieuvre's formula, which yields a closed curve that is then realized as an asymptotic curve on a ruled annulus. The second asymptotic curve is obtained from a periodic solution of a Riccati equation.

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arXiv2026

The isoperimetric inequality and CMC hypersurfaces in Cartan-Hadamard manifolds

We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $n$-manifolds of nonpositive sectional curvature, $3\leq n\leq9$, which establishes the Cartan-Hadamard conjecture in these dimensions. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, together with estimates for Jacobi fields along geodesic chords of the boundary. The weights in these integrals depend on the length of the chord and its angles with the boundary, and are chosen dimension by dimension, with a Green function pole. The inequality persists for boundaries of isoperimetric regions trapped in geodesic balls, whose mean curvature is constant only on the free part and which may have singularities. The isoperimetric-profile argument of Kleiner and Ghomi-Spruck completes the proof.

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arXiv2026

The Cartan-Hadamard conjecture in dimension five

We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, in the spirit of Banchoff-Pohl, together with an estimate for Jacobi fields along geodesic chords. The inequality persists for boundaries of isoperimetric regions in geodesic balls, whose mean curvature is constant only on the free part. An isoperimetric-profile argument, after Kleiner, completes the proof. Our method also gives a new proof in dimension $3$.

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arXiv2026

Mean curvatures and symmetry of convex hypersurfaces

Let $M^n$ be a $C^2$ closed convex hypersurface in Euclidean space, and $σ_m$ be its $m$th mean curvature. We show that $M$ is symmetric with respect to a hyperplane orthogonal to a given direction $e$, if $σ_m(p)\leqσ_m(q)$ whenever $p-q$ is parallel to $e$. For convex hypersurfaces, this settles a conjecture of Li, extends the mean-curvature theorem of Li-Yan-Yao to all $σ_m$, and strengthens some earlier results of Li-Nirenberg by removing nondegeneracy assumptions. The proof is based on the theory of mixed volumes, specifically the rigidity of quermassintegrals under Steiner symmetrization.

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arXiv2026

Deformations of curves with constant curvature

We prove that curves of constant curvature satisfy the parametric $C^1$-dense relative $h$-principle in the space of immersed curves with nonvanishing curvature in Euclidean space $R^{n\geq 3}$. It follows that two knots of constant curvature in $R^3$ are isotopic, resp. homotopic, through curves of constant curvature if and only if they are isotopic, resp. homotopic, and their self-linking numbers, resp. self-linking numbers mod $2$, are equal. The proofs are based on convex integration techniques, utilizing parametric versions of classical theorems by Carathéodory and Steinitz in convex geometry.

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arXiv2026

Total curvature and isoperimetric inequalities in pinched Cartan-Hadamard manifolds

We establish a sharp lower bound for the total Gauss-Kronecker curvature of convex hypersurfaces in Cartan-Hadamard manifolds with pinched negative curvature. The bound holds in all dimensions when the diameter is small relative to the curvature scale, and in dimensions 4 and 5 without any restriction on the diameter, provided that the pinching is sufficiently tight. The proofs are based on the Chern-Gauss-Bonnet theorem and weighted Hsiung-Minkowski inequalities. As an application, we obtain the isoperimetric inequality of the Cartan-Hadamard conjecture in dimension 5 under sufficiently pinched curvature.

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arXiv2026

A quantitative Schur comparison theorem for curves in CAT(k) spaces

We obtain a quantitative form of Schur's comparison theorem for curves with finite total curvature in CAT(k) spaces. This sharpens and extends the classical arm and bow lemmas in Euclidean space, as well as their Riemannian analogues. The proof is based on a comparison formula for curves in model planes, expressed in terms of curvature measures and a notion of moment arm borrowed from mechanics. Another ingredient is a refinement of Reshetnyak's theorem that controls the curvature of the majorizing curve.

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arXiv2026

Total absolute curvature and rigidity of surfaces in Cartan-Hadamard manifolds

We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in 1985. Our proof is based on an isometric embedding construction via holonomy, and uses Pogorelov's theory of surfaces with bounded extrinsic curvature. Along the way, we obtain a regularity result for convex hulls and a Schur-type comparison theorem for curves in Cartan-Hadamard manifolds.

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arXiv2026

A local isoperimetric inequality for balls with nonpositive curvature

We show that small perturbations of the metric of a ball in Euclidean n-space to metrics with nonpositive curvature do not reduce the isoperimetric ratio. Furthermore, the isoperimetric ratio is preserved only if the perturbation corresponds to a homothety of the ball. These results establish a sharp local version of the Cartan-Hadamard conjecture.

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arXiv2025

Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

We show that in Cartan-Hadamard manifolds $M^n$, $n\geq 3$, closed infinitesimally convex hypersurfaces $Γ$ bound convex flat regions, if curvature of $M^n$ vanishes on tangent planes of $Γ$. This encompasses Chern-Lashof-Sacksteder characterization of compact convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in $M^3$ with minimal total absolute curvature bound Euclidean convex bodies, as stated by Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT($k$) spaces, and other techniques from Alexandrov geometry outlined by Petrunin.

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arXiv2025

$h$-Principles for smooth curves and knots with prescribed curvature

We show that smooth curves with prescribed curvature satisfy a $C^1$-dense $h$-principle in the space of immersed curves in Euclidean space. More precisely, every $C^{α\geq 2}$ curve with nonvanishing curvature in $R^{n\geq 3}$ can be $C^1$-approximated by $C^α$ curves of any larger curvature, prescribed as a function of arclength. It follows that there exist $C^\infty$ knots of prescribed curvature in every isotopy class of closed curves embedded in $R^3$.

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arXiv2025

$h$-Principles for curves and knots of constant torsion

We prove that curves of constant torsion satisfy the $C^1$-dense h-principle in the space of immersed curves in Euclidean space. In particular, there exists a knot of constant torsion in each isotopy class. Our methods, which involve convex integration and degree theory, quickly establish these results for curves of constant curvature as well.

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arXiv2025

Topology of closed asymptotic curves on negatively curved surfaces

Motivated by Nirenberg's problem on isometric rigidity of tight surfaces, we study closed asymptotic curves $Γ$ on negatively curved surfaces $M$ in Euclidean $3$-space. In particular, using Călugăreanu's theorem, we obtain a formula for the linking number $Lk(Γ,n)$ of $Γ$ with the normal $n$ of $M$. It follows that when $Lk(Γ, n)=0$, $Γ$ cannot have any locally star-shaped planar projections with vanishing crossing number, which extends observations of Kovaleva, Panov and Arnold. These results hold also for curves with nonvanishing torsion and their binormal vector field. Furthermore we construct an example where $n$ is injective but $Lk(Γ, n)\neq 0$, and discuss various restrictions on $Γ$ when $n$ is injective.

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