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Burkhard Wilking

Publications and source records attributed to Burkhard Wilking.

18 recordsLinked to original sources
arXiv2025

Positive curvature, torus symmetry, and matroids

We identify a link between regular matroids and torus representations all of whose isotropy groups have an odd number of components. Applying Seymour's 1980 classification of the former objects, we obtain a classification of the latter. In addition, we prove optimal upper bounds for the cogirth of regular matroids up to rank nine, and we apply this to prove the existence of fixed-point sets of circles with large dimension in a torus representation with this property up to rank nine. Finally, we apply these results to prove new obstructions to the existence of Riemannian metrics with positive sectional curvature and torus symmetry.

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arXiv2024

Smoothness of submetries

We prove that a Riemannian submersion between smooth, compact, non-negatively curved Riemannian manifolds has to be smooth, resolving a conjecture by Berestovskii--Guijarro. We show that without any curvature assumption, the smoothness of the base is implied by the smoothness of the total space. Results are proven in the much more general setting of submetries. These are metric generalizations of Riemannian submersions and isometric actions, which have recently appeared in different areas of geometry.

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arXiv2023

Actions on positively curved manifolds and boundary in the orbit space

We study isometric actions of compact Lie groups on complete orientable positively curved $n$-manifolds whose orbit spaces have non-empty boundary in the sense of Alexandrov geometry. In particular, we classify quotients of the unit sphere by actions of compact simple Lie groups with non-empty boundary. We deduce from this the list of representations of compact simple Lie groups that admit non-trivial reductions. As a tool of special interest, we introduce a new geometric invariant of a compact symmetric space, namely, the minimal number of points in a "spanning set" of the space.

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arXiv2021

Splitting of torus representations and applications in the Grove symmetry program

A 1930s conjecture of Hopf states that an even-dimensional compact Riemannian manifold with positive sectional curvature has positive Euler characteristic. We prove this conjecture under the additional assumption that the isometry group has rank at least five. The fundamental new tool used to achieve this is a reduction to, and structural results concerning, a representation theoretic problem involving torus representations all of whose isotropy groups are connected.

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arXiv2017

The Ricci flow under almost non-negative curvature conditions

We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $-1$ can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than $-C$. Here the time of existence and the constant $C$ only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kaehler case. We also get a local version of the main theorem. As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds).

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arXiv2015

On the Berger conjecture for manifolds all of whose geodesics are closed

A conjecture of Berger states that, for any simply connected Riemannian manifold all of whose geodesics are closed, all prime geodesics have the same length. We firstly show that the energy function on the free loop space of such a manifold is a perfect Morse-Bott function with respect to a suitable cohomology. Secondly we explain when the negative bundles along the critical manifolds are orientable. These two general results then lead to a solution of Berger's conjecture when the underlying manifold is a sphere of dimension at least four.

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arXiv2015

Revisiting homogeneous spaces with positive curvature

As was recently observed by M. Xu and J. Wolf, there is a gap in Berard Bergery's classification of odd dimensional positively curved homogeneous spaces. Since this classification has been used in other papers as well, we give a modern, complete and self contained proof (in odd as well as even dimensions), confirming that there are indeed no new examples.

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arXiv2013

Riemannian foliations of spheres

We show that a Riemannian foliation on a topological $n$-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.

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arXiv2013

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.

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arXiv2011

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.

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arXiv2011

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

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 $\Ad_{\SO(n,\C)}$. 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 $\CP^n$. Moreover, the methods can also be applied to prove Harnack inequalities. In addition to an earlier version the paper contains some remarks on negative results for Harnack inequalities.

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arXiv2009

Manifolds with 1/4-pinched flag curvature

We say that a nonnegatively curved manifold $(M,g)$ has quarter pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional curvature is bounded above by 4. We show that these manifolds have nonnegative complex sectional curvature. By combining with a theorem of Brendle and Schoen it follows that any positively curved manifold with strictly quarter pinched flag curvature must be a space form. This in turn generalizes a result of Andrews and Nguyen in dimension 4. For odd dimensional manifolds we obtain results for the case that the flag curvature is pinched with some constant below one quarter, one of which generalizes a recent work of Petersen and Tao.

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arXiv2003

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=π. 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.

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