Publications

Let (N,h) be a time- and space-oriented Lorentzian spin manifold, and let M be a compact spacelike hypersurface of N with induced Riemannian metric g and second fundamental form K. If (N,h) satisfies the dominant energy condition in a strict sense, then the Dirac-Witten operator of M⊂ N is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial data pairs on $M$ satisfying the dominant energy condition in a strict sense. The central tool will be a Lorentzian analogue of Hitchin's α-invariant. In case that the dominant energy condition only holds in a weak sense, the Dirac-Witten operator may be non-invertible, and we will study the kernel of this operator in this case.

We will show that the kernel may only be non-trivial if π1(M) is virtually solvable of derived length at most 2. This allows to extend the index theoretical methods to spaces of initial data pairs, satisfying the dominant energy condition in the weak sense.

We will show further that a spinor φ is in the kernel of the Dirac--Witten operator on (M,g,K) if and only if (M,g,K,φ) admits an extension to a Lorentzian manifold (N,h) with parallel spinor ψ such that M is a Cauchy hypersurface of (N,h), such that g and K are the induced metric and second fundamental form of M, respectively, and φ is the restriction of ψ to M.

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Related project(s):
35Geometric operators on singular domains

We provide a direct connection between the Zmax (or essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank 2.

We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the Zmax-JSJ decomposition of such groups.

Related project(s):
60Property (T)

We initiate the study of torsion-free algebraically hyperbolic groups; these groups generalise, and are intricately related to, groups with no Baumslag-Solitar subgroups. Indeed, for groups of cohomological dimension 2 we prove that algebraic hyperbolicity is equivalent to containing no Baumslag-Solitar subgroups. This links algebraically hyperbolic groups to two famous questions of Gromov; recent work has shown these questions to have negative answers in general, but they remain open for groups of cohomological dimension 2.

We also prove that algebraically hyperbolic groups are CSA, and so have canonical abelian JSJ-decompositions. In the two-generated case we give a precise description of the form of these decompositions.

Related project(s):
60Property (T)

We prove that twisted ℓ2-Betti numbers of locally indicable groups are equal to the usual ℓ2-Betti numbers rescaled by the dimension of the twisting representation; this answers a question of Lück for this class of groups. It also leads to two formulae: given a fibration E with base space B having locally indicable fundamental group, and with a simply-connected fibre F, the first formula bounds ℓ2-Betti numbers b(2)i(E) of E in terms of ℓ2-Betti numbers of B and usual Betti numbers of F; the second formula computes b(2)i(E) exactly in terms of the same data, provided that F is a high-dimensional sphere.

We also present an inequality between twisted Alexander and Thurston norms for free-by-cyclic groups and 3-manifolds. The technical tools we use come from the theory of generalised agrarian invariants, whose study we initiate in this paper.

Related project(s):
60Property (T)

We construct a new type of expanders, from measure-preserving affine actions with spectral gap on origami surfaces, in each genus g⩾1. These actions are the first examples of actions with spectral gap on surfaces of genus g>1. We prove that the new expanders are coarsely distinct from the classical expanders obtained via the Laplacian as Cayley graphs of finite quotients of a group. In genus g=1, this implies that the Margulis expander, and hence the Gabber--Galil expander, is coarsely distinct from the Selberg expander. For the proof, we use the concept of piecewise asymptotic dimension and show a coarse rigidity result: A coarse embedding of either R2 or H2 into either R2 or H2 is a quasi-isometry.

Related project(s):
60Property (T)

The present book deals with the spectral geometry of infinite graphs. This topic involves the interplay of three different subjects: geometry, the spectral theory of Laplacians and the heat flow of the underlying graph. These three subjects are brought together under the unifying perspective of Dirichlet forms. The spectral geometry of manifolds is a well-established field of mathematics. On manifolds, the focus is on how Riemannian geometry, the spectral theory of the Laplace–Beltrami operator, Brownian motion and heat evolution interact. In the last twenty years large parts of this theory have been subsumed within the framework of strongly local Dirichlet forms. Indeed, this point of view has proven extremely fruitful.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

We study planar graphs with large negative curvature outside of a finite set and the spectral theory of Schrödinger operators on these graphs. We obtain estimates on the first and second order term of the eigenvalue asymptotics. Moreover, we prove a unique continuation result for eigenfunctions and decay properties of general eigenfunctions. The proofs rely on a detailed analysis of the geometry which employs a Copy-and-Paste procedure based on the Gauß-Bonnet theorem.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

In this article we prove upper bounds for the k-th Laplace eigenvalues below the essential spectrum for strictly negatively curved Cartan–Hadamard manifolds. Our bound is given in terms of k^2 and specific geometric data of the manifold. This applies also to the particular case of non‐compact manifolds whose sectional curvature tends to minus infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

We construct secondary cup and cap products on coarse (co-)homology theories from given cross and slant products. They are defined for coarse spaces relative to weak generalized controlled deformation retracts.

On ordinary coarse cohomology, our secondary cup product agrees with a secondary product defined by Roe. For coarsifications of topological coarse (co-)homology theories, our secondary cup and cap products correspond to the primary cup and cap products on Higson dominated coronas via transgression maps. And in the case of coarse $\mathrm{K}$-theory and -homology, the secondary products correspond to canonical primary products between the $\mathrm{K}$-theories of the stable Higson corona and the Roe algebra under assembly and co-assembly.

Related project(s):
10Duality and the coarse assembly map78Duality and the coarse assembly map II

We construct a slant product on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer, which is the domain of the co-assembly map. In fact, we obtain such products simultaneously on the entire Higson-Roe sequence. The existence of these products implies injectivity results for external product maps on each term of the Higson-Roe sequence. Our results apply in particular to taking products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the general class of manifolds where this method applies, we introduce the notion of Higson-essentialness. A complete spin-c manifold is Higson-essential if its fundamental class is detected by the stable Higson corona via the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. Finally, we draw conclusions for positive scalar curvature metrics on product spaces, in particular on non-compact manifolds. We also construct suitable equivariant versions of these slant products and discuss related problems of exactness and amenability of the stable Higson corona.

Related project(s):
10Duality and the coarse assembly map

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible \(\sigma\)-compact space in which the corona sits as a \(\mathbb{Z}\)-set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space \(BG\), then our constructions yield a \(\mathbb{Z}\)-structure for the group.

Related project(s):
10Duality and the coarse assembly map

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry

In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prove that the entropies are monotone along the singular Ricci de Turck flow. We employ these entropies to show that in the singular setting, Ricci solitons are gradient and that steady or expanding Ricci solitons are Einstein.

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry23Spectral geometry, index theory and geometric flows on singular spaces

 We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold M×R, where M is asymptotically flat. If the initial hypersurface F⊂M×R is uniformly spacelike and asymptotic to M×{s} for some s∈R at infinity, we show that the mean curvature flow starting at F0 exists for all times and converges uniformly to M×{s} as t→∞.

Related project(s):
23Spectral geometry, index theory and geometric flows on singular spaces

We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel Lp−Lq decay estimates are established, allowing the Schrödinger operator to have a non-trivial L2-kernel. We also prove new decay estimates for spatial derivatives of arbitrary order, in a general geometric setting. Our main motivation is the application to stability of non-linear geometric equations, primarily Ricci flow, which will be presented in a companion paper. The arguments in this paper use that many geometric Schrödinger operators can be written as the square of Dirac type operators. By a remarkable result of Wang, this is even true for the Lichnerowicz Laplacian, under the assumption of a parallel spinor. Our analysis is based on a novel combination of the Fredholm theory for Dirac type operators on ALE manifolds and recent advances in the study of the heat kernel on non-compact manifolds.

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry

We compute the spectra of the Laplace-Beltrami operator, the connection Laplacian on 1-forms and the Einstein operator on symmetric 2-tensors on the sine-cone over a positive Einstein manifold $(M,g)$. We conclude under which conditions on $(M,g)$, the sine-cone is dynamically stable under the singular Ricci-de Turck flow and rigid as a singular Einstein manifold

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry64Spectral geometry, index theory and geometric flows on singular spaces II

We obtain new lower bounds for the first non-zero eigenvalue of the scalar

sub-Laplacian for 3-Sasaki metrics, improving Lichnerowicz-Obata type

estimates by Ivanov et al. The limiting eigenspace is fully decribed in

terms of the automorphism algebra. Our results can be thought of as an

analogue of the Lichnerowicz-Matsushima estimate for Kähler-Einstein

metrics. In dimension 7, if the automorphism algebra is non-vanishing,

we also compute the second eigenvalue for the sub-Laplacian and construct

explicit eigenfunctions. In addition, for all metrics in the canonical

variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension.

We also strengthen a result pertaining to the growth rate of harmonic

functions, due to Conlon, Hein and Sun, in the case of hyperkähler

cones. In this setup we also describe the space of holomorphic functions.

Related project(s):
74Rigidity, stability and deformations in nearly parallel G2-geometry

We initiate a systematic study of the deformation theory of the second Einstein

metric \(g_{1/\sqrt{5}}\)  respectively the proper nearly G2 structure  \(\phi_{1/\sqrt{5}}\) of a 3-Sasaki manifold \((M^7,g)\). We show that infinitesimal Einstein deformations for  \(g_{1/\sqrt{5}}\)   coincide with infinitesimal  \(G_2\) deformations for \(\phi_{1/\sqrt{5}}\). The latter are showed to be parametrised by eigenfunctions of the basic Laplacian of g, with eigenvalue twice the Einstein constant of the base 4-dimensional orbifold, via an explicit differential operator. In terms of this parametrisation we determine those infinitesimal \(G_2\) deformations which are unobstructed to second order.

Related project(s):
74Rigidity, stability and deformations in nearly parallel G2-geometry

We describe the second order obstruction to deformation for nearly G_2 structures on compact manifolds. Building on work of B. Alexandrov and U. Semmelmann this allows proving rigidity under deformation for the proper nearly G_2 structure on the Aloff-Wallach space N(1,1).

Related project(s):
74Rigidity, stability and deformations in nearly parallel G2-geometry