Mohammad Ghomi

The problem of optimal smoothing for convex functions

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 process, and it is shown that it can be done successfully provided that the original function is strictly convex over the boundary of the smooth regions.

Optimal Smoothing for Convex Polytopes

It is proved that given a convex polytope P in R, 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.

The convex hull property and topology of hypersurfaces with nonnegative curvature

Abstract 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 nonsmooth category is obtained as well. We show that our boundary conditions determine the topology of the surface up to at most two choices. The proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure, a noncollapsing convergence theorem, and a gluing procedure.

Skew loops and quadric surfaces

Abstract. A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in R3 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.

Gauss map, topology, and convexity of hypersurfaces with nonvanishing curvature

Abstract 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 their Gauss map; particularly, in relation to a conjecture of Meeks on minimal surfaces with convex boundary. It is also proved, as another application, that a compact hypersurface with nonvanishing curvature is convex, if its boundary lies in a hyperplane.

Topology of Riemannian submanifolds with prescribed boundary

We prove that a smooth compact submanifold of codimension 2 immersed in R, 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. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash’s isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman.

principles for hypersurfaces with prescribed principle curvatures and directions

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. Further we construct regular homotopies (resp. isotopies) which control the principal curvatures and directions of hypersurfaces in a variety of ways. These results, which we prove by holonomic approximation, establish some h-principles in the sense of Gromov, and generalize theorems of Gluck and Pan on embedding and knotting of positively curved surfaces in 3-space.

Tangent bundle embeddings of manifolds in Euclidean space

For a given

Nonexistence of skew loops on ellipsoids

n

A Riemannian four vertex theorem for surfaces with boundary

-dimensional manifold