Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold $M$ . Under the assumption that the sectional curvature $K^M$ is strictly positive, we prove the existence of a smooth immersion $f:{\mathbb {S}}^2 \rightarrow M$ minimizing the $L^2$ integral of the second fundamental form. Assuming instead that $K^M \le 2$ and that there is some point $\overline{x} \in M$ with scalar curvature $R^M(\overline{x}) > 6$ , we obtain a smooth minimizer $f:{\mathbb {S}}^2 \rightarrow M$ for the functional $\int \frac{1}{4}|H|^2+1$ , where $H$ is the mean curvature.     

On tangential stabilization in curvature driven flows of planar curves

We discuss the role of tangential stabilization in a curvature driven flow of planar curves. The governing system of nonlinear parabolic equations includes a nontrivial tangential velocity functional yielding a uniform redistribution of grid points along the evolving family of curves preventing numerically computed curves from forming various instabilities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Motion by curvature of planar curves with end points moving freely on a line

This paper deals with the motion by curvature of planar curves having end points moving freely along a line with fixed contact angles to this line. We first prove the existence and uniqueness of self-similar shrinking solution. Then we show that the curve shrinks to a point in a self-similar manner, if initially the curve is a graph.     

Combinatorial curvature for planar graphs

Regarding an infinite planar graph G as a discrete analogue of a noncompact simply connected Riemannian surface, we introduce the combinatorial curvature of G corresponding to the sectional curvature of a manifold. We show this curvature has the property that its negative values are bounded above by a universal negative constant. We also prove that G is hyperbolic if its curvature is negative. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 220–229, 2001     

Submersions and curves of constant geodesic curvature

Considering Riemannian submersions, we find necessary and sufficient conditions for when sub‐Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvature. We describe a canonical extension of the sub‐Riemannian metric and study geometric properties of the obtained Riemannian manifold. This work contains several examples illustrating the results.     

Generalized planar curves and quaternionic geometry

Motivated by the analogies between the projective and the almost quaternionic geometries, we first study the generalized planar curves and mappings. We follow, recover, and extend the classical approach, see e.g., (Sov. Math. 27(1) 63–70 (1983), Rediconti del circolo matematico di Palermo, Serie II, Suppl. 54 75–81) (1998), Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of ℍ-planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaternionic geometries.     

Approximation by planar elastic curves

We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good initial guess for the approximating curve. A gradient-driven optimization is then used to find the approximating elastic curve.     

Existence and uniqueness for Legendre curves

We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.     

Boundary regularity for minimizing currents with prescribed mean curvature

We prove complete boundary regularity for energy minimizing integer multiplicity rectifiablen currents in ⁿ⁺¹ of prescribed mean curvatureH with boundaryB= represented by an oriented smooth submanifold of dimensionn – 1 in sun+1. We also give applications to the Plateau problem for surfaces with prescribed mean curvature.This article was processed by the author using the LaT_EX style filepljour1 from Springer-Verlag.