Publications

We show that in each dimension 4n+3, n>1, there exist infinite sequences of closed smooth simply connected manifolds M of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension 7, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, inconjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such M the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold M × R also has infinitely many path components.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.

Related project(s):
24Minimal surfaces in metric spaces

We derive various pinching results for small Dirac eigenvalues using the classification of spinc and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spinc manifolds which involves a general study on convergence of Riemannian manifolds with a principal S1-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S1-bundle on spinc manifolds with Killing spinors.

Related project(s):
5Index theory on Lorentzian manifolds

The stable converse soul question (SCSQ) asks whether, given a real vector bundle \(E\) over a compact manifold, some stabilization \(E\times\mathbb{R}^k\) admits a metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all real vector bundles over any simply connected homogeneous manifold with positive curvature, except possibly for the Berger space \(B^{13}\). Along the way, we show that the same is true for all simply connected homogeneous spaces of dimension at most seven, for arbitrary products of simply connected compact rank one symmetric spaces of dimensions multiples of four, and for certain products of spheres. Moreover, we observe that the SCSQ is "stable under tangential homotopy equivalence": if it has an affirmative answer for all vector bundles over a certain manifold \(M\), then the same is true for any manifold tangentially homotopy equivalent to~\(M\). Our main tool is topological K-theory. Over \(B^{13}\), there is essentially one stable class of real vector bundles for which our method fails.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive  sectional curvature.  In the present article we classify closed, simply-connected  cohomogeneity-one Alexandrov spaces in dimensions $5$, $6$ and  $7$. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of  closed, simply-connected  cohomogeneity-one Alexandrov spaces in dimensions at most $7$.

Related project(s):
11Topological and equivariant rigidity in the presence of lower curvature bounds

We prove that sufficiently collapsed, closed and irreducible three-dimensional Alexandrov spaces are modeled on one of the eight three-dimensional Thurston geometries. This extends a result of Shioya and Yamaguchi, originally formulated for Riemannian manifolds, to the Alexandrov setting.

Related project(s):
11Topological and equivariant rigidity in the presence of lower curvature bounds

The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to  the universal covering as a result of their local feature.  As a consequence, we lift a class of spectral $\zeta$-invariants using lifted defect formulae which express discrepancies of  $\zeta$-regularised traces in terms of Wodzicki residues.  We derive Atiyah's $L^2$-index theorem as an instance of the $\mathbb{Z}_2$-graded generalisation of the canonical lift of spectral $\zeta$-invariants and we show that certain  lifted spectral $\zeta$-invariants for geometric operators are integrals of Pontryagin and Chern forms.

Related project(s):
4Secondary invariants for foliations

We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods.

Related project(s):
19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

We study magnetic Schrödinger operators on graphs. We extend the notion of sparseness of graphs by including a magnetic quantity called the frustration index. This notion of magnetic sparse turn out to be equivalent to the fact that the form domain is an \(\ell^2\) space. As a consequence, we get criteria of discreteness for the spectrum and eigenvalue asymptotics.

Related project(s):
19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

In this paper we give an algebraic construction of the (active) reflected Dirichlet form. We prove that it is the maximal Silverstein extension whenever the given form does not possess a killing part and we prove that Dirichlet forms need not have a maximal Silverstein extension if a killing is present. For regular Dirichlet forms we provide an alternative construction of the reflected process on a compactification (minus one point) of the underlying space.

Related project(s):
19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

Related project(s):
19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

Given a three-dimensional Riemannian manifold containing a ball with an explicit lower bound on its Ricci curvature and positive lower bound on its volume, we use Ricci flow to perturb the Riemannian metric on the interior to a nearby Riemannian metric still with such lower bounds on its Ricci curvature and volume, but additionally with uniform bounds on its full curvature tensor and all its derivatives. The new manifold is near to the old one not just in the Gromov-Hausdorff sense, but also in the sense that the distance function is uniformly close to what it was before, and additionally  we have Hölder/Lipschitz equivalence of the old and new manifolds.

One consequence is that we obtain a local bi-Hölder correspondence between Ricci limit spaces in three dimensions and smooth manifolds.

This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson-Cheeger-Colding-Tian, describing how Ricci limit spaces in three dimensions must be homeomorphic to manifolds, and we obtain this in the most general, locally non-collapsed case.

The proofs build on results and ideas  from recent papers of Hochard and the current authors.

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${\mathcal B} \to {\mathrm{D}}_{\mathcal B}(1 + {\mathrm{D}}_{\mathcal B}^2)^{-\frac{1}{2}}$ depends on Lipschitz smoothness and ellipticity of ${\mathcal B}$ and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.

Related project(s):
5Index theory on Lorentzian manifolds

 Let M(n,D) be the space of closed n-dimensional Riemannian manifolds (M,g) with diam(M)≤D and |secM|≤1. In this paper we consider sequences (Mi,gi) in M(n,D) converging in the Gromov–Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient vol(BMir(x))/injMi(x) can be uniformly bounded from below by a positive constant C(n, r, Y) for all points x∈Mi. On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient vol(BMir(x))/injMi(x) uniformly from below for all x∈Mi. As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in M(n,D) with C≤vol(M)/inj(M).

Related project(s):
5Index theory on Lorentzian manifolds

We investigate the problem of calculating the Fredholm index of a geometric Dirac operator subject to local (e.g. Dirichlet and Neumann) and non-local (APS) boundary conditions posed on the strata of a manifold with corners. The boundary strata of the manifold with corners can intersect in higher codimension. To calculate the index we introduce a glueing construction and a corresponding Lie groupoid. We describe the Dirac operator subject to mixed boundary conditions via an equivariant family of Dirac operators on the fibers of the Lie groupoid. Using a heat kernel method with rescaling we derive a general index formula of the Atiyah-Singer type.

Related project(s):
3Geometric operators on a class of manifolds with bounded geometry

We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus at least 3 without punctures is Sp_2g(2), thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz's results on C-linear representations of mapping class groups to projective representations over any field.

Related project(s):
8Parabolics and invariants

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal \(L^q\)-regularity space for all times and is instantaneously smooth in space and time, where the maximal \(L^q\)-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.

Related project(s):
30Nonlinear evolution equations on singular manifolds

We construct a codimension 3completely non-holonomic subbundle on the Gromoll–Meyer exotic 7-sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method can be applied to construct a codimension 3 completely non-holonomic subbundle on the standard 7-sphere (or more general on a (4n +3)-dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides a new and simple proof for the standard sphere S7.

Related project(s):
6Spectral Analysis of Sub-Riemannian Structures

In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.

Related project(s):
4Secondary invariants for foliations11Topological and equivariant rigidity in the presence of lower curvature bounds

We decompose locally in time maximal \(L^{q}\)-regular solutions of abstract quasilinear parabolic equations as a sum of a smooth term and an arbitrary small−with respect to the maximal \(L^{q}\)-regularity space norm−remainder. In view of this observation, we next consider the porous medium equation and the Swift-Hohenberg equation on manifolds with conical singularities. We write locally in time each solution as a sum of three terms, namely a term that near the singularity is expressed as a linear combination of complex powers and logarithmic integer powers of the singular variable, a term that decays to zero close to the singularity faster than each of the non-constant summands of the previous term and a remainder that can be chosen arbitrary small with respect e.g. to the \(C^{0}\)-norm. The powers in the first term are time independent and determined explicitly by the local geometry around the singularity, e.g. by the spectrum of the boundary Laplacian in the situation of straight conical tips. The case of the above two problems on closed manifolds is also considered and local space asymptotics for the solutions are provided.

Related project(s):
30Nonlinear evolution equations on singular manifolds