Mohammad Ghomi

Mohammad Ghomi
Verified
Mohammad verified their affiliation via an institutional email.
Verified
Mohammad verified their affiliation via an institutional email.
Georgia Institute of Technology | GT · School of Mathematics

PhD
Differential Geometry

About

78
Publications
7,797
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
775
Citations
Introduction
Mohammad Ghomi works in the School of Math at Georgia Tech. He is interested in geometry of curves and surfaces in Euclidean space, Riemannian submanifolds, and all aspects of mathematics related to these objects, specially notions of convexity. His papers since 2003 are available on the arXiv. See his website for all his papers arranged by subject area: http://people.math.gatech.edu/~ghomi/Papers/index.html
Additional affiliations
August 2005 - present
Georgia Institute of Technology
Position
  • Professor

Publications

Publications (78)
Preprint
Motivated by Nirenberg's problem on isometric rigidity of tight surfaces, we study closed asymptotic curves $\Gamma$ on negatively curved surfaces $M$ in Euclidean $3$-space. In particular, using C\u{a}lug\u{a}reanu's theorem, we obtain a formula for the linking number $Lk(\Gamma,n)$ of $\Gamma$ with the normal $n$ of $M$. It follows that when $Lk(...
Article
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 we...
Preprint
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.
Preprint
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 isotop...
Article
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...
Article
Full-text available
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.
Preprint
We show that in Cartan-Hadamard manifolds $M^n$, $n\geq 3$, closed infinitesimally convex hypersurfaces $\Gamma$ bound convex flat regions, if curvature of $M^n$ vanishes on tangent planes of $\Gamma$. This encompasses Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Carta...
Preprint
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 we...
Article
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...
Preprint
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.
Preprint
We show that a compact Riemannian 3 3 -manifold M M with strictly convex simply connected boundary and sectional curvature K ≤ a ≤ 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 ≡ a K\equiv a on planes tangent to the boundary of M M . This yields a characterization of...
Article
Full-text available
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 in a Cartan-Hadamard manifold M . In particular, we show that the first mean curvature integral of a convex hypersurface γ nested inside Γ cannot e...
Preprint
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 conve...
Preprint
We show that in Euclidean 3-space any closed curve $\gamma$ which contains the unit sphere within its convex hull has length $L\geq4\pi$, and characterize the case of equality. This result generalizes the authors' recent solution to a conjecture of Zalgaller. Furthermore, for the analogous problem in $n$ dimensions, we include the estimate $L\geq C...
Preprint
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...
Preprint
We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces, via Reilly's identities. As applications we derive a number of geometric inequalities for a convex hypersurface $\Gamma$ in a Cartan-Hadamard manifold $M$. In particular we show that the first mean curvature integral of a convex hypersurface $\gamma$ nested...
Article
We obtain an explicit formula for comparing total curvature of level sets of functions on Riemannian manifolds and develop some applications of this result to the isoperimetric problem in spaces of nonpositive curvature.
Article
We show that in Euclidean 3-space any closed curve γ which lies outside the unit sphere and contains the sphere within its convex hull has length ≥ 4 ⁢ π {\geq 4\pi} . Equality holds only when γ is composed of four semicircles of length π, arranged in the shape of a baseball seam, as conjectured by V. A. Zalgaller in 1996.
Preprint
Full-text available
We show that in Euclidean 3-space any closed curve which lies outside the unit sphere and contains the sphere within its convex hull has length at least $4\pi$. Equality holds only when the curve is composed of $4$ semicircles of length $\pi$, arranged in the shape of a baseball seam, as conjectured by V. A. Zalgaller in 1996.
Article
Full-text available
A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph E with respect to which K is not unfoldable. Thus Durer's conjecture do...
Article
Full-text available
We prove that any properly oriented $\mathcal{C}^{2,1}$ isometric immersion of a positively curved Riemannian surface $M$ into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in $M$. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable conditions on t...
Preprint
Full-text available
We prove that the total positive Gauss-Kronecker curvature of any closed hypersurface embedded in a complete simply connected manifold of nonpositive curvature $M^n$, $n\geq 2$, is bounded below by the volume of the unit sphere in Euclidean space $\mathbf{R}^n$. This yields the optimal isoperimetric inequality for bounded regions of finite perimete...
Article
Full-text available
We prove that in Euclidean space Rn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {R}^{n+1}$$\end{document} any compact immersed nonnegatively curved hypersu...
Article
Full-text available
The width $w$ of a curve $\gamma$ in Euclidean space $R^n$ is the infimum of the distances between all pairs of parallel hyperplanes which bound $\gamma$, while its inradius $r$ is the supremum of the radii of all spheres which are contained in the convex hull of $\gamma$ and are disjoint from $\gamma$. We use a mixture of topological and integral...
Preprint
Full-text available
We prove that any properly oriented $C^{2,1}$ isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in M. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable conditions on their parabolic...
Preprint
A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable. The proof is based on a resu...
Preprint
We prove that in Euclidean space $R^{n+1}$ any compact immersed nonnegatively curved hypersurface $M$ with free boundary on the sphere $S^n$ is an embedded convex topological disk. In particular, when the $m^{th}$ mean curvature of $M$ is constant, for any $1\leq m\leq n$, $M$ is a spherical cap or an equatorial disk.
Article
Full-text available
We prove the existence of a center, or continuous selection of a point, in the relative interior of $C^1$ embedded $k$-disks in Riemannian $n$-manifolds. If $k\le 3$ the center can be made equivariant with respect to the isometries of the manifold, and under mild assumptions the same holds for $k=4=n$. By contrast, for every $n\ge k\ge 6$ there are...
Preprint
We prove the existence of a center, or continuous selection of a point, in the relative interior of $C^1$ embedded $k$-disks in Riemannian $n$-manifolds. If $k\le 3$ the center can be made equivariant with respect to the isometries of the manifold, and under mild assumptions the same holds for $k=4=n$. By contrast, for every $n\ge k\ge 6$ there are...
Article
Full-text available
We show that the torsion of any simple closed curve $\Gamma$ in Euclidean 3-space changes sign at least $4$ times provided that it is star-shaped and locally convex with respect to a point $o$ in the interior of its convex hull. The latter condition means that through each point $p$ of $\Gamma$ there passes a plane $H$, not containing $o$, such tha...
Preprint
We show that the torsion of any simple closed curve $\Gamma$ in Euclidean 3-space changes sign at least $4$ times provided that it is star-shaped and locally convex with respect to a point $o$ in the interior of its convex hull. The latter condition means that through each point $p$ of $\Gamma$ there passes a plane $H$, not containing $o$, such tha...
Preprint
The width $w$ of a curve $\gamma$ in Euclidean space $R^n$ is the infimum of the distances between all pairs of parallel hyperplanes which bound $\gamma$, while its inradius $r$ is the supremum of the radii of all spheres which are contained in the convex hull of $\gamma$ and are disjoint from $\gamma$. We use a mixture of topological and integral...
Article
Full-text available
We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4 vertex theorem of Sedykh for co...
Article
Full-text available
The total diameter of a closed planar curve $C\subset R^2$ is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of $C$. Furthermore, when $C$ is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when $C...
Article
Full-text available
We show that every convex polyhedron admits a simple edge unfolding after an affine transformation. In particular there exists no combinatorial obstruction to a positive resolution of Durer's unfoldability problem, which answers a question of Croft, Falconer, and Guy. Among other techniques, the proof employs a topological characterization for embe...
Article
Full-text available
We show that Carathéodory’s conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R 2 → R whose gradient decays uniformly faster than 1/r. The divergence theorem then yields a pair of integral equations for the normal curvatures of these graphs,...
Article
Full-text available
We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. We also show that 2(P'+I)+V >3, where P' is the number of pair...
Article
Full-text available
We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than 1/r. The divergence theorem then yields a pair of integral equations for the normal curvatures of these graphs,...
Article
Full-text available
We prove the existence of C 1 isometric embeddings, and C ∞ approx-imate isometric embeddings, of Riemannian manifolds into Euclidean space with prescribed values in a neighborhood of a point.
Article
Full-text available
We prove that every metric of constant curvature on a compact sur-face M with boundary ∂M induces at least four vertices, i.e., local extrema of geodesic curvature, on a connected component of ∂M , if, and only if, M is simply connected. Indeed, when M is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant cur...
Article
Full-text available
Given any finite subset X of the sphere S^n, n>1, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space R^{n+1} whose Gauss map misses X. In particular, this answers a question of M. Gromov. Comment: Minor revisions over the last draft
Article
Full-text available
We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. Furt...
Article
Full-text available
We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is $\C^1$. In the real a...
Article
Full-text available
We prove that a smooth compact submanifold of codimension 2 immersed in Rn, n ≥ 3, bounds at most finitely many topologically distinct, compact, nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well...
Article
Full-text available
We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in a canonical way. Further, the total cur...
Article
Full-text available
We find a quartic example of a smooth embedded negatively curved surface in R 3 homeomorphic to a doubly punctured torus. This constitutes an explicit solution to Hadamard's problem on constructing complete surfaces with negative curvature and Euler characteristic in R 3 . Further we show that our solu-tion has the optimal degree of algebraic compl...
Article
Full-text available
We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an n-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generalized vector field problem, existence of non-singular bilinear...
Article
Full-text available
We prove that any closed orientable surface may be smoothly embedded in Euclidean 3-space so that when it is illuminated by parallel rays from any direction the shade cast on the surface is connected.
Article
Full-text available
We prove that the area of a hypersurface Σ which traps a given volume outside a convex domain C in Euclidean space R n is bigger than or equal to the area of a hemisphere which traps the same volume on one side of a hyperplane. Further, when C has smooth boundary ∂C, we show that equality holds if and only if Σ is a hemisphere which meets ∂C ortho...
Article
Full-text available
We prove that C¥{\mathcal{C}}^\infty curves of constant curvature satisfy, in the sense of Gromov, the relative C1{\mathcal{C}}^1-dense h-principle in the space of immersed curves in Euclidean space R n≥ 3. In particular, in the isotopy class of any given C1{\mathcal{C}}^1 knot f there exists a C¥{\mathcal{C}}^\infty knot f͂ of constant curvature...
Article
Full-text available
We prove that any compact orientable hypersurface with bound-ary immersed (resp. embedded) in Euclidean space is regularly homotopic (resp. isotopic) to a hypersurface with principal directions which may have any prescribed homotopy type, and principal curvatures each of which may be prescribed to within an arbitrary small error of any constant. Fu...
Article
We prove that any compact orientable hypersurface with boundary immersed (resp. embedded) in Euclidean space is regularly homotopic (resp. isotopic) to a hypersurface with principal directions which may have any prescribed homotopy type, and principal curvatures each of which may be prescribed to within an arbitrary small error of any constant. Fur...
Article
Full-text available
For a given n-dimensional manifold Mn we study the prob- lem of nding the smallest integer N(Mn) such that Mn admits a smooth embedding in the Euclidean space RN without intersecting tangent spaces.
Article
Full-text available
We prove that if Σ is a compact hypersurface in Euclidean space R<sup>n</sup>, its boundary lies on the boundary of a convex body C, and meets C orthogonally from the outside, then the total positive curvature of Σ is bigger than or equal to half the area of the sphere S<sup>n-1</sup>. Also, we obtain necessary and sufficient conditions for the equ...
Article
Full-text available
We prove that every C, closed curve without parallel tangent lines immersed in
Article
We prove that every C 1 C^1 closed curve immersed on an ellipsoid has a pair of parallel tangent lines. This establishes the optimal regularity for a phenomenon first observed by B. Segre. Our proof uses an approximation argument with the aid of an estimate for the size of loops in the tangential spherical image of a spherical curve.
Article
Full-text available
We study the geometry and topology of immersed surfaces in Euclidean 3-space whose Gauss map satisfies a certain two-piece-property, and solve the ``shadow problem" formulated by H. Wente.
Article
Full-text available
We prove that in Euclidean space R n+1 , every metrically complete, positively curved immersed hypersurface M , with compact boundary ∂M, lies outside the convex hull of ∂M provided that ∂M is embedded on the boundary of a convex body and n > 2. For n = 2, on the other hand, we construct examples which contradict this property.
Article
Full-text available
It is proved that given a convex polytope P in R n , together with a collection of compact convex subsets in the interior of each facet of P , there exists a smooth convex body arbitrarily close to P which coincides with each facet precisely along the prescribed sets, and has positive curvature elsewhere.
Article
Full-text available
We show that the length of any periodic billiard trajectory in any convex body K Ì Rn K \subset \mathbf{R}^n is always at least 4 times the inradius of K; the equality holds precisely when the width of K is twice its inradius, e.g., K is centrally symmetric, in which case we prove that the shortest periodic trajectories are all bouncing ball (2...
Article
Solomon (Projecting codimension-two cycles to zero on hyperplanes in RN+1, Topology (2004) X-ref: doi:10.1016/S0040-9383(03)00043-0) has studied the problem of existence of a simple closed curve in R3 whose projections into planes in three linearly independent directions vanish in the sense of currents. He discovered some nonsmooth examples of such...
Article
Full-text available
We prove that, in Euclidean space, any nonnegatively curved, compact, smoothly immersed hypersurface lies outside the convex hull of its boundary, provided the boundary satisfies certain required conditions. This gives a convex hull property, dual to the classical one for surfaces with nonpositive curvature. A version of this result in the nonsmoot...
Preprint
We obtain bounds on the least dimension of an affine space that can contain an $n$-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points. This problem is closely related to the generalized vector field problem, non-singular bilinear maps, and the immersion problem for real projective spaces.
Article
Full-text available
A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in R 3 with a point of positive curvature and no skew loops are the quadrics. In particular: Ellipsoids are the only closed surfaces without skew loops. Our efforts also yield results about skew loops on cylinders and positively curved surfaces.
Article
Full-text available
A procedure is described for smoothing a convex function which not only preserves its convexity, but also, under suitable conditions, leaves the function unchanged over nearly all the regions where it is already smooth. The method is based on a convolution followed by a gluing. Controlling the Hessian of the resulting function is the key to this pr...
Preprint
Full-text available
A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in euclidean 3-space with a point of positive curvature and no skew loops are the quadrics. In particular, ellipsoids are the only closed surfaces without skew loops. We also prove results about skew loops on cylinders and positively curved surfaces.
Article
Full-text available
It is proved that, for n⩾2, every immersion of a compact connected n-manifold into a sphere of the same dimension is an embedding, if it is one-to-one on each boundary component of the manifold. Some applications of this result are discussed for studying geometry and topology of hypersurfaces with non-vanishing curvature in Euclidean space, via the...
Article
Full-text available
We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The pro...
Preprint
We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The pro...
Article
Full-text available
We construct smooth closed hypersurfaces of positive curvature with prescribed submanifolds and tangent planes. Further, we develop some applications to boundary value problems via Monge-Ampére equations, smoothing of convex polytopes, and an extension of Hadamard's ovaloid theorem to hypersurfaces with boundary.
Article
Full-text available
By a classical problem in differential geometry I mean one which involves smooth curves or surfaces in three dimensional Euclidean space. We list here a number of such problems. For other problems in differential geometry or geometric analysis see [40]. Some problems and many references may also be found in [6]. A large collection of problems in di...
Article
Full-text available
We find quartic examples of smooth embedded negatively curved surfaces in R 3 homeomorphic to singly and doubly punctured tori, and a triply punctured sphere. These constitute explicit solutions to Hadamard's problem on constructing complete surfaces with negative curvature and Euler characteristic in R 3 . Further we show that our solutions have t...
Article
Full-text available
We prove that a smooth compact immersed submanifold of codimen-sion 2 in R n , n ≥ 3, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. Analogous results for noncompact fillings are obtained as well. On the other hand, we show that these topological finite-ness theorems may not hold if the prescribed bo...

Network

Cited By