We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all −∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f . We show, for instance, that if an X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurfac...

In this paper we will show that on any complete noncompact Riemannian manifold with a finite volume there exist geodesic loops of arbitrarily small length.

Introduction We consider a Riemannian manifold M n with a positive density function Ψ (x) used to weight volume and hypersurface area. In terms of the underlying Riemannian volume dV 0 and area dA 0 , the new, weighted volume and area are given by

We show that if L is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of L are all in the interval (−δ, δ) for a fixed δ ∈ [0, 1) and no complimentary region of L is an interval bundle over a surface, then each boundary leaf of L has a nontrivial fundamental group. We also prove existence of a fixed constant δ0 > 0 such that i...

In this paper, we prove that every closed nil 3-manifold is diffeomorphic to a cusp cross-section of a finite volume complex hyperbolic 2-orbifold.

Article history: Received 25 May 2012 Received in revised form 14 September 2012 Accepted 21 November 2012 Available online 19 December 2012

We prove convergence of discrete duality finite volume (DDFV) schemes on distorted meshes for a class of simplified macroscopic bidomain models of the electrical activity in the heart. Both time-implicit and linearised time-implicit schemes are treated. A short description is given of the 3D DDFV meshes and of some of the associated discrete calculus tools. Several numerical tests are presented.

Poroelasticity theory models the dynamics of porous, fluid-saturated media. It was pioneered by Maurice Biot in the 1930s through 1960s, and has applications in several fields, including geophysics and modeling of in vivo bone. A wide variety of methods have been used to model poroelasticity, including finite difference, finite element, pseudospectral, and discontinuous Galerkin methods. In thi...

We review the spatial discretization procedure for three high-order, unstructured grid methods recently developed for use within ocean circulation modeling. They are based respectively on spectral finite element and high-order finite volume formulations. We discuss and contrast the three methods in the context of the shallow water equations.

A new volume deformation technique using FEM and voxel-texture (voxture) maps has been developed. In contrast to the traditional deformation techniques, the new algorithm is aimed at volume transformations, deformations and volume metamorphosis.