Burkhard Wilking

Torus actions on manifolds of positive sectional curvature

or equivMently as the largest number d such tha t a d-dimensional torus acts effectively and isometrically on M. Grove and Searle [13] showed tha t symrank((M,g))~< [ 89 provided tha t M is a compact manifold of positive sectional curvature. They also studied the case of equali ty and showed tha t this can only occur if the under lying manifold is diffeomorphic to C P n, S ~, or to a lens space. Our main new tool is the following basic result.

A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry

We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S^3 which can be realized as an extremal set with respect to an inner metric on S^3 which has nonnegative curvature in the Alexandrov sense.

On fundamental groups of manifolds of nonnegative curvature

Abstract We will characterize the fundamental groups of compact manifolds of (almost) nonnegative Ricci curvature and also the fundamental groups of manifolds that admit bounded curvature collapses to manifolds of nonnegative sectional curvature. Actually it turns out that the known necessary conditions on these groups are sufficient as well. Furthermore, we reduce the Milnor problem—are the fundamental groups of open manifolds of nonnegative Ricci curvature finitely generated?—to manifolds with abelian fundamental groups. Moreover, we prove for each positive integer n that there are only finitely many non-cyclic, finite, simple groups acting effectively on some complete n -manifold of nonnegative Ricci curvature. Finally, sharping a result of Cheeger and Gromoll [6], we show for a compact Riemannian manifold (M,g 0 ) of nonnegative Ricci curvature that there is a continuous family of metrics (g λ ),λ∈[0,1] such that the universal covering spaces of (M,g λ ) are mutually isometric and (M,g 1 ) is finitely covered by a Riemannian product N×T d , where T d is a torus and N is simply connected.

How to produce a Ricci flow via Cheeger–Gromoll exhaustion

We prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature. We do not require upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger-Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these closed manifolds, we obtain a sequence of closed solutions of the Ricci flow with nonnegative complex sectional curvature which subconverge to a solution of the Ricci flow on the open manifold. Furthermore, we find an optimal volume growth condition which guarantees long time existence, and we give an analysis of the long time behaviour of the Ricci flow. Finally, we construct an explicit example of an immortal nonnegatively curved solution of the Ricci flow with unbounded curvature for all time.

A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities

Abstract We consider a subset S of the complex Lie algebra 𝔰𝔬(n, ℂ) and the cone C(S) of curvature operators which are nonnegative on S. We show that C(S) defines a Ricci flow invariant curvature condition if S is invariant under AdSO(n, ℂ). The analogue for Kähler curvature operators holds as well. Although the proof is very simple and short, it recovers all previously known invariant nonnegativity conditions. As an application we reprove that a compact Kähler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thus biholomorphic to ℂℙn. Moreover, the methods can also be applied to prove Harnack inequalities.

Spherical rank rigidity and Blaschke manifolds

Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=\pi. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and positive spherical rank, then M is isometric to a compact, rank one symmetric space (CROSS) i.e., isometric to a sphere, complex projective space, quaternionic projective space or to the Cayley plane. The notion of spherical rank is analogous to the notions of Euclidean rank and hyperbolic rank studied by several people (see references). The main theorem is proved in two steps: first we show that M is a so called Blaschke manifold with extremal injectivity radius (equal to diameter). Then we prove that such M is isometric to a CROSS.

Riemannian foliations of spheres

We show that a Riemannian foliation on a topological

Manifolds with positive curvature operators are space forms

n

Positively curved cohomogeneity one manifolds and 3-Sasakian geometry

-sphere has leaf dimension 1 or 3 unless n=15 and the Riemannian foliation is given by the fibers of a Riemannian submersion to an 8-dimensional sphere. This allows us to classify Riemannian foliations on round spheres up to metric congruence.