Topology Of Hypersurfaces Research Articles

The convex hull property and topology of hypersurfaces with nonnegative curvature

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.

Topology of Complex Polynomials and Jacobian Conjecture

The purpose of this paper is to report on some new results on the Topology of Complex Polynomials f : C 2→ C . These results lead to an open problem closely related to the Jacobian Conjecture. Works by S.A. Broughton [Invent. Math. 92 (1988) 217], A. Dimca [Singularities and Topology of Hypersurfaces, Springer, 1992] and LeDung Trang, C. Weber [C. R. Acad. Sci. Paris 320 (1995) 581] can provide more insight into the problem and J. Milnor [Singular Points of Complex Hypersurfaces, Princeton Univ. Press, 1988] is a good reference for general information on the subject. The main result proved here is similar to one proved by R. Ephraim [Proc. Sympos. Pure Math., Vol. 40, 1983, p. 353, Corollary 3.14], but our proof is short and clearer due to the use of Ha–Lê Theorem. Also, we bring new light to the points at infinity.

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

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.