The isoperimetric inequality for a minimal submanifold in Euclidean space

Relative Isoperimetric Inequality for Minimal Submanifolds in a Riemannian Manifold

Let Σ be a domain on an m-dimensional minimal submanifold in the outside of a convex set C in S or H. The modified volume M Σ is introduced by Choe and Gulliver 1992 and we prove a sharp modified relative isoperimetric inequality for the domain Σ, 1/2 mωmM Σ m−1 ≤ Volume ∂Σ−∂C , where ωm is the volume of the unit ball of R. For any domain Σ on a minimal surface in the outside convex set C in an...

Minimal Surfaces in Euclidean Space

The Contact Number of a Euclidean Submanifold

We introduce an invariant, called the contact number, associated with each Euclidean submanifold. We show that this invariant is, surprisingly, closely related to the notions of isotropic submanifolds and holomorphic curves. We are able to establish a simple criterion for a submanifold to have any given contact number. Moreover, we completely classify codimension-2 submanifolds with contact num...

On the Isoperimetric Problem in Euclidean Space with Density

We study the isoperimetric problem for Euclidean space endowedwith a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this con...

Tangent Bundle of the Hypersurfaces in a Euclidean Space

Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...