Publications

Let K be a number field with ring of integers D and let G be a Chevalley group scheme not of type E8, F4 or G2. We use the theory of Tits buildings and a result of Tóth on Steinberg modules to prove that H^vcd(G(D);Q)=0 if D is Euclidean.

Related project(s):
62A unified approach to Euclidean buildings and symmetric spaces of noncompact type

Non-local boundary conditions, such as the Atiyah-Patodi-Singer (APS) conditions, for Dirac operators on Riemannian manifolds are well under\-stood while not much is known for such operators on spacetimes with timelike boundary. We define a class of Lorentzian boundary conditions that are local in time and non-local in the spatial directions and show that they lead to a well-posed Cauchy problem for the Dirac operator. This applies in particular to the APS conditions imposed on each level set of a given Cauchy temporal function.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds

We establish a one-to-one correspondence between Finsler structures on the \(2\)-sphere with constant curvature \(1\) and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $$\mathbb{CP}(a_1,a_2)\to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)$$ of weighted projective spaces provide examples of Finsler \(2\)-spheres of constant curvature and all geodesics closed.

Related project(s):
68Minimal Lagrangian connections and related structures

We associate a flow \(\phi\) to a solution of the vortex equations on a closed oriented Riemannian 2-manifold \((M,g)\) of negative Euler characteristic and investigate its properties. We show that \(\phi\) always admits a dominated splitting and identify special cases in which \(\phi\) is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of \((M,g)\).

Related project(s):
68Minimal Lagrangian connections and related structures

Given a parabolic geometry on a smooth manifold \(M\), we study a natural affine bundle \(A \to M\), whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on \(A\), which induces an almost bi-Lagrangian structure on \(A\) and a compatible linear connection on \(TA\). We prove that the split-signature metric given by the almost bi-Lagrangian structure is Einstein with non-zero scalar curvature, provided the parabolic geometry is torsion-free and \(|1|\)-graded. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in \(A\). For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. We also show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge-Ampere equation and thus to properly convex projective structures.

Related project(s):
68Minimal Lagrangian connections and related structures

We present an Eilenberg--Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories with already existing or newly introduced equivariance fit into this setup.

Furthermore, a new and more flexible notion of coarse homotopy is given which is more in the spirit of topological homotopies. Some, but not all, coarse (co-)homology theories are even invariant under these new homotopies. They also led us to a meaningful concept of topological actions of locally compact groups on coarse spaces.

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

We prove the following Lipschitz rigidity result in scalar curvature geometry. Let $M$ be a closed smooth connected spin manifold of even dimension $n$, let $g$ be a Riemannian metric of regularity $W^{1,p}$, $p > n$, on $M$ whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by $n(n-1)$, and let  $f \colon (M,g) \to \mathbb{S}^n$ be a $1$-Lipschitz continuous (not necessarily smooth) map of non-zero degree to the unit  $n$-sphere. Then $f$ is a metric isometry. This generalizes a result of Llarull (1998) and answers in the affirmative a question of Gromov (2019) in his "four lectures". Our proof is based on spectral properties of Dirac operators for  low regularity Riemannian metrics and twisted with Lipschitz bundles, and on the theory of quasiregular maps due to Reshetnyak.

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II58Profinite perspectives on l2-cohomology73Geometric Chern characters in p-adic equivariant K-theory

The relative index theorem is proved for general first-order elliptic operators that are complete and coercive at infinity over measured manifolds. This extends the original result by Gromov-Lawson for generalised Dirac operators as well as the result of Bär-Ballmann for Dirac-type operators. The theorem is seen through the point of view of boundary value problems, using the graphical decomposition of elliptically regular boundary conditions for general first-order elliptic operators due to Bär-Bandara. Splitting, decomposition and the Phi-relative index theorem are proved on route to the relative index theorem.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds

We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of N-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity.

Related project(s):
61At infinity of symmetric spaces

Related project(s):
43Singular Riemannian foliations and collapse

The non-abelian Hodge correspondence  is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions of the self-duality equation. In this paper we construct such solutions for strongly parabolic $\mathfrak{sl}(2,\mathbb C)$ Higgs fields on a $4$-punctured sphere with parabolic weights $t \sim 0$ using loop groups methods through an implicit function theorem argument. We identify the rescaled limit hyper-K\"ahler moduli space at $t=0$ to be (the completion of) the nilpotent orbit in $\mathfrak{sl}(2, \mathbb C)$ equipped the Eguchi-Hanson metric. Our methods and computations are based on the twistor approach to the self-duality equations using Deligne and Simpson's  $\lambda$-connections interpretation. Due to the implicit function theorem, Taylor expansions of these quantities can be computed at $t=0$. By construction they have closed form expressions in terms of Multiple-Polylogarithms and their geometric properties lead to some identities of $\Omega$-values which we believe deserve further investigations.

Related project(s):
55New hyperkähler spaces from the the self-duality equations56Large genus limit of energy minimizing compact minimal surfaces in the 3-sphere

We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago--Ivanov--Polterovich. As a key tool we construct a hyperbolic graph on which these groups act, which is the analog of the curve graph for the mapping class group.

Related project(s):
38Geometry of surface homeomorphism groups

We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.

The main application is a general approximation result by sections which have very restrictive local properties an open dense subsets. This shows, for instance, that given any K∈R every manifold of dimension at least two carries a complete C^1,1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course this is impossible for C^2-metrics in general.

Related project(s):
5Index theory on Lorentzian manifolds15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

We prove that the normal metric on the homogeneous space E7/PSO(8) is stable

with respect to the Einstein-Hilbert action, thereby exhibiting the first

known example of a non-symmetric metric of positive scalar curvature with this property.

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

In this article we study the stability problem for the Einstein metrics on

Sasaki Einstein and on complete nearly parallel G2 manifolds. In the Sasaki

case we show linear instability if the second Betti number is positive.

Similarly we prove that nearly parallel G2 manifolds with positive third

Betti number are linearly unstable. Moreover, we prove linear instability

for the Berger space SO(5)/SO(3)_irr which is a 7-dimensional homology

sphere with a proper nearly parallel G2 structure.

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

In this article we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation, and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.

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

We study the spectral properties of the Laplace operator associated to a hyperbolic surface in the presence of a unitary representation of the fundamental group. Following the approach by Guillopé and Zworski, we establish a factorization formula for the twisted scattering determinant and describe the behavior of the scattering matrix in a neighborhood of \(1/2\).

Related project(s):
70Spectral theory with non-unitary twists

Using non-Abelian Hodge theory for parabolic Higgs bundles,

we construct infinitely many non-congruent  hyperbolic affine spheres modeled on a thrice-punctured sphere with monodromy in ${\bf SL}_3(\Z)$. These give rise to non-isometric semi-flat Calabi--Yau metrics on special Lagrangian torus bundles over an open ball in $\R^{3}$  with a Y-vertex deleted, thereby answering a question raised by Loftin, Yau, and Zaslow.

Related project(s):
55New hyperkähler spaces from the the self-duality equations

The Lawson surfaces $\xi_{1,g}$ of genus $g$ are constructed by rotating and reflecting the Plateau solution $f_t$ with respect to a particular geodesic $4$-gon $\Gamma_t$ along its boundary, where $t= \tfrac{1}{2g+2}$ is an angle of  $\Gamma_t$.  In this paper we combine the existence and regularity of the Plateau solution $f_t$ in $t \in (0, \tfrac{1}{4})$ with topological information about the moduli space of Fuchsian systems on the 4-puncture sphere to obtain existence of a Fuchsian DPW potential $\eta_t$ for every $f_t$ with $t\in(0, \tfrac{1}{4}]$. Moreover, the coefficients of $\eta_t$ are shown to depend real analytically on $t$. This implies that the Taylor approximation of the DPW potential $\eta_t$ and of the area obtained at $t=0$ found in \cite{HHT2} determines these quantities for  all $\xi_{1,g}$. In particular, this leads to an algorithm to conformally parametrize all Lawson surfaces $\xi_{1,g}$.

Related project(s):
55New hyperkähler spaces from the the self-duality equations56Large genus limit of energy minimizing compact minimal surfaces in the 3-sphere

We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in Sp(n,R), G2(2), E8(C), F4(C), and G2(C) are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.

Related project(s):
58Profinite perspectives on l2-cohomology