Locally nearly spherical surfaces are almost-positively $c$-curved

Positively Curved Manifolds with Large Spherical Rank

Rigidity results are obtained for Riemannian d-manifolds with sec > 1 and spherical rank at least d − 2 > 0. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini– Study metric. This conjecture is verified in all odd dimensions, for metrics on dspheres when d 6= 6, for Riemannian manifolds satisfying the Rakić duality p...

Positively Curved Surfaces in the Three-sphere

In this talk I will discuss an example of the use of fully nonlinear parabolic flows to prove geometric results. I will emphasise the fact that there is a wide variety of geometric parabolic equations to choose from, and to get the best results it can be very important to choose the best flow. I will illustrate this in the setting of surfaces in a three-dimensional sphere. There are quite a few...

Positively curved cubic plane graphs are finite

Let G be an infinite plane graph such that G is locally finite and every face of G is bounded by a cycle. Then G is said to be positively curved if, for every vertex x of G, 1− d(x)/2 + Σx∈F 1 |F | > 0, where the summation is taken over all facial cycles F containing x. Note that if G is positively curved then the maximum degree of G is at most 5. As a discrete analogue of a result in Riemannia...

Positively Curved Surfaces with No Tangent Support Plane

We discuss a characterization of positively curved surfaces M with the property that at each point the tangent plane to M is not a support plane for the entire surface. A one parameter family of examples which have special relevance with respect to the characterization is also given. Each member of this family is a smooth embedded surface in R 3 that is topologically a disk, has everywhere posi...

A C Estimate for Nearly Umbilical Surfaces