Umbilic foliations and curvature

Umbilic and Tangential Singularities on Configurations of Principal Curvature Lines

In this paper is studied the relationship between quadratic tangencies of principal lines with the boundary of a surface and the Darbouxian umbilics of a smooth boundaryless surface which approximates it through the process of thickening and smoothing defined here.

Foliations by constant mean curvature tubes

Let Γ be a nondegenerate geodesic in a compact Riemannian manifold M . We prove the existence of a partial foliation of a neighbourhood of Γ by CMC surfaces which are small perturbations of the geodesic tubes about Γ. There are gaps in this foliation, which correspond to a bifurcation phenomenon. Conversely, we also prove, under certain restrictions, that the existence of a partial CMC foliatio...

Constant mean curvature foliations in cosmological spacetimes

Foliations by constant mean curvature hypersurfaces provide a possibility of defining a preferred time coordinate in general relativity. In the following various conjectures are made about the existence of foliations of this kind in spacetimes satisfying the strong energy condition and possessing compact Cauchy hypersurfaces. Recent progress on proving these conjectures under supplementary assu...

Constant curvature foliations in asymptotically hyperbolic spaces

Let (M, g) be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on ∂M and Weingarten foliations in some neighbourhood of infinity inM . We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations whe...

Foliations on Constant Curvature Surfaces and Nonlinear Partial Differential Equations