Related papers: Veronese minimizes normal curvatures

We calculate the index and nullity of the three orientable focal manifolds of isoparametric hypersurfaces in spheres with three distinct principal curvatures. It turns out that the index is equal to the dimension of the ambient Euclidean…

We give an optimal bound on normal curvatures of immersed n-torus in a Euclidean ball of large dimension.

In this note, we give a characterization of immersed submanifolds of simply-connected space forms for which the quotient of the extrinsic diameter by the focal radius achieves the minimum possible value of $2$. They are essentially round…

This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4\pi must be smoothly embedded. Related results are proved for varifolds and for soap film surfaces.

It is verified that the number of vertices in a $d$-dimensional cubical pseudomanifold is at least $2^{d+1}$. Using Adin's cubical $h$-vector, the generalized lower bound conjecture is established for all cubical 4-spheres, as well as for…

We identify as topological spheres those complete submanifolds lying with any codimension in hyperbolic space whose Ricci curvature satisfies a lower bound contingent solely upon the length of the mean curvature vector of the immersion.

There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a…

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…

A classical fact is that normal bundles of rational normal curves are well-balanced. We generalize this by proving that all Veronese normal bundles are slope semistable. We also determine the line bundle decomposition of the restriction of…

Let $C$ be a smooth (irreducible) curve of degree $d$ in $\mathbb{P}^{2}$. Let $\mathbb{P}^{2} \hookrightarrow \mathbb{P}^{5}$ be the Veronese embedding and let $\mathcal{I}_{C}$ denote the homogeneous ideal of $C$ on $\mathbb{P}^{5}$. In…

For a Riemannian manifold $M^{n+1}$ and a compact domain $\Omega \subset M^{n+1}$ bounded by a hypersurface $\partial \Omega$ with normal curvature bounded below, estimates are obtained in terms of the distance from $O$ to $\partial \Omega$…

Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth…

We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we…

Let M be a complete simply connected Riemannian manifold, with sectional curvature K bounded above by -1. Under some assumptions on the geometry of the boundary of M, which are satisfied for instance if M is a symmetric space, or has…

We prove that if a topological sphere smoothly embedded into $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ is contained in an open ball of radius $2$, then the region it bounds must contain a unit ball. This result…

We present an analytically explicit study of optimal discrete quantization on spherical geometries equipped with the geodesic metric, focusing on highly symmetric configurations on the unit sphere $\mathbb S^2$. Three discrete uniform…

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant…

We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4\pi. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also…

In this article, we consider compact Riemannian 3-manifolds with boundary. We prove that if the $L^2$-norm of the curvature is small and if the $H^{1/2}$-norm of the difference of the fundamental forms of the boundary is small, then the…

Let M be a compact Riemannian manifold without boundary and let E be a Riemannian vector bundle over M. If $\Sigma$ denotes the sphere subbundle of E, we look for embeddings of $\Sigma$ into E admitting a prescribed mean curvatures of…