Field electron emission from nanometer-scale objects deviates the predictions of classical theory as both electrostatic potential curves within tunneling region and image planar one. This impels inclusion additional correction terms in barrier. At apex a tip-like rotationally symmetric surface, these are proportional to (single) local emitter curvature. The present paper generalizes this relati...

Chen conjecture states that every Euclidean biharmonic submanifold is minimal. In this paper we consider the Chen conjecture for Lk-operators. The new conjecture (Lk-conjecture) is formulated as follows: If Lkx = 0 then Hk+1 = 0 where x : M → R is an isometric immersion of a Riemannian manifold M into the Euclidean space R, Hk+1 is the (k+1)-th mean curvature of M , and Lk is the linearized ope...

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in H satisfying f(κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is uni...

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in H n+1 satisfying f(κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is...

This paper describes an approach for 3D triangular meshes segmentation. The method that we adopt for the 3D mesh segmentation is based on the unbiased hierarchical queue which has been used for 2D images. Our method uses the hierarchical transformation on a connected faces structure. The approach that we propose is based on the principal curvature to calculate the faces curvature. The results s...

It is fair to say that Riemannian geometry started with Gauss’s famous ”Disquisitiones generales” from 1827 in which one finds a rigorous discussion of what we now call the Gauss curvature of a surface. Much has been written about the importance and influence of this paper, see in particular the article [Do] by P.Dombrowski for a careful discussion of its contents and influence during that time...

Associated to oriented surfaces immersed in R 3 , here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature K, given by the product of the principal curvatures k1, k2 of the immersion, is positive. The leaves of the foliations are the lines of M-mean curvature, along which the normal curvature of the immersion is given by a...

The crest lines, salient subsets of the extrema of the principal curvatures over their corresponding curvature lines, are powerful shape descriptors which are widely used for shape matching, interrogation, and visualization purposes. In this paper, we develop fast, accurate, and reliable methods for detecting the crest lines on surfaces approximated by dense triangle meshes. The methods exploit...

where H is the mean curvature of the immersion and K its Gauss curvature. This functional models the elastic energy of thin shells. Willmore studied in [7] the functional W0, by now called Willmore functional. First we note that Wγ(Γ) ≥ 0 holds for every γ ∈ [0, 1]. Let κ1, κ2 ∈ R denote the principal curvatures of the surface. Then H − γK = 1 4 (κ1 + κ2) 2 − γκ1 · κ2 = 1−γ 4 (κ1 + κ2) 2 + γ 4 ...

We consider the evolution of compact surfaces by fully nonlinear, parabolic curvature ows for which the normal speed is given by a smooth, degree one homogeneous function of the principal curvatures of the evolving surface. Under no further restrictions on the speed function, we prove that initial surfaces on which the speed is positive become weakly convex at a singularity of the flow. This ge...