arXiv subjects

Frederick Wilhelm

Publications and source records attributed to Frederick Wilhelm.

24 records · Page 2Linked to original sources

arXiv2011

Infinity-harmonic maps and morphisms

We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $p\to \infty $. Infinity harmoncity appears in many familiar contexts. For example, metric projection onto the orbit of an isometric group action from a tubular neighborhood is infinity harmonic. Unfortunately, infinity-harmonicity is not preserved under composition. Those infinity harmonic maps that always preserve infinity harmonicity under pull back are called infinity harmonic morphisms. We show that infinity harmonic morphisms are precisely horizontally homothetic mas. Many example of infinity-harmonic maps are given, including some very important and well-known classes of maps between Riemannian manifolds.

math.DG↗

arXiv2010

Some Principles for Deforming Nonnegative Curvature

We describe a few abstract principles that are used in the deformation of the Gromoll-Meyer metric to positive curvature.

math.DG↗

arXiv2008

An exotic sphere with positive sectional curvature

We show that there is a metric on the Gromoll-Meyer sphere with positive sectional curvature.

math.DG↗

arXiv2006

Horizontally homothetic submersions and nonnegative curvature

We show that any horizontally homothetic submersion from a compact manifold of nonnegative sectional curvature is a Riemannian submersion.

math.DG↗

arXiv1999

Examples of Riemannian manifolds with positive curvature almost everywhere

We show that the unit tangent bundle of S^4 and a real cohomology CP^3 admit Riemannian metrics with positive sectional curvature almost everywhere. These are the only examples so far with positive curvature almost everywhere that are not also known to admit positive curvature.

math.DG↗

arXiv1995

The Radius Rigidity Theorem for Manifolds of Positive Curvature

Recall that the radius of a compact metric space $(X, dist)$ is given by $rad\ X = \min_{x\in X} \max_{y\in X} dist(x,y)$. In this paper we generalize Berger's $\frac{1}{4}$-pinched rigidity theorem and show that a closed, simply connected, Riemannian manifold with sectional curvature $\geq 1$ and radius $\geq \fracπ{2}$ is either homeomorphic to the sphere or isometric to a compact rank one symmetric space.

dg-ga↗