Publications

We provide a coarse version of the relative index of Gromov and Lawson and thoroughly establish all of its basic properties. As an application, we discuss a general procedure to construct wrong way maps on the \(K\)-theory of the Roe algebra mapping the coarse index class of the Dirac operator of a manifold to the one of a suitably embedded submanifold of arbitrary codimension, thereby establishing an abstract machinery to find obstructions to uniform positive scalar curvature coming from these submanifolds.

Related project(s):
45Macroscopic invariants of manifolds78Duality and the coarse assembly map II

We construct a slant product \(\mathrm{S}^{G\times H}_p(X\times Y)\otimes \mathrm{K}_{-q}(\bar{\mathfrak{c}}^{\mathrm{red}} Y\rtimes H)\to \mathrm{K}_{p-q}(\mathrm{C}^\ast_G X)\) on the analytic structure group of Higson and Roe and the K-theory of the stable Higson compactification taking values in the (equivariant) Roe algebra. This complements the slant products constructed in earlier work of Engel and the authors ( arXiv:1909.03777 [math.KT] ). The distinguishing feature of our new slant product is that it specializes to a duality pairing \(\mathrm{S}^H_p(Y) \otimes \mathrm{K}_{-p}(\bar{\mathfrak{c}}^{\mathrm{red}} (Y)\rtimes H)\to \mathbb{Z}\) which can be used to extract numerical invariants out of elements in the analytic structure group such as rho-invariants associated to positive scalar curvature metrics.

Related project(s):
78Duality and the coarse assembly map II

We establish a duality between harmonic maps from Riemann surfaces to hyperbolic 3-space $\mathbb{H}^3$ and harmonic maps from Riemann surfaces to de Sitter three-space $\mathrm{dS}_3$, best viewed as a generalized Gauß map. On the gauge theoretic side, it matches $\mathrm{SU}(2)$ and $\mathrm{SU}(1,1)$ solutions of Hitchin's self-duality equations via a signature flip along an eigenline of the Higgs field. Reversing this operation typically produces singular solutions, occurring where the eigenline becomes lightlike. Motivated by explicit model examples and this singular behavior, we extend this duality to a class of \emph{transgressive} harmonic maps $f:M\to \mathbb S^3$: these are harmonic on the hemispheres equipped with the hyperbolic metric, intersect the equator orthogonally, and have vanishing Hopf differential along the crossing set. We construct large families by gluing and analyze their regularity, and as an application obtain $\tau$-real negative sections of the Deligne--Hitchin moduli space of arbitrarily large energy that are not twistor lines.

Related project(s):
55New hyperkähler spaces from the the self-duality equations77Asymptotic geometry of the Higgs bundle moduli space II

Given a non-compact semisimple real Lie group G and an Anosov subgroup Γ, we utilize the correspondence between ℝ-valued additive characters on Levi subgroups L of G and ℝ-affine homogeneous line bundles over G/L to systematically construct families of non-empty domains of proper discontinuity for the Γ-action. If Γ is torsion-free, the analytic dynamical systems on the quotients are Axiom A, and assemble into a single partially hyperbolic multiflow. Each Axiom A system admits global analytic stable/unstable foliations with non-wandering set a single basic set on which the flow is conjugate to Sambarino's refraction flow, establishing that all refraction flows arise in this fashion. Furthermore, the ℝ-valued additive character is regular if and only if the associated Axiom A system admits a compatible pseudo-Riemannian metric and contact structure, which we relate to the Poisson structure on the dual of the Lie algebra of G.

Related project(s):
65Resonances for non-compact locally symmetric spaces

A three-dimensional quasi-Fuchsian Lorentzian manifold M is a globally hyperbolic spacetime diffeomorphic to Σ×(−1,1) for a closed orientable surface Σ of genus ≥2. It is the quotient M=Γ∖ΩΓ of an open set ΩΓ⊂AdS3 by a discrete group Γ of isometries of AdS3 which is a particular example of an Anosov representation of π1(Σ). We first show that the spacelike geodesic flow of M is Axiom A, has a discrete Ruelle resonance spectrum with associated (co-)resonant states, and that the Poincaré series for Γ extend meromorphically to ℂ. This is then used to prove that there is a natural notion of resolvent of the pseudo-Riemannian Laplacian ◻ of M, which is meromorphic on ℂ with poles of finite rank, defining a notion of quantum resonances and quantum resonant states related to the Ruelle resonances and (co-)resonant states by a quantum-classical correspondence. This initiates the spectral study of convex co-compact pseudo-Riemannian locally symmetric spaces.

Related project(s):
65Resonances for non-compact locally symmetric spaces

The fermionic relative entropy in two-dimensional Rindler spacetime is studied using both modular theory and the reduced one-particle density operators. The methods and results are compared. A formula for the relative entropy for general Gaussian states is derived. As an application, the relative entropy is computed for a class of non-unitary excitations.

Related project(s):
47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces

The fermionic von Neumann entropy, the fermionic entanglement entropy and the fermionic relative entropy are defined for causal fermion systems. Our definition makes use of entropy formulas for quasi-free fermionic states in terms of the reduced one-particle density operator. Our definitions are illustrated in various examples for Dirac spinors in two- and four-dimensional Minkowski space, in the Schwarzschild black hole geometry and for fermionic lattices. We review area laws for the two-dimensional diamond and a three-dimensional spatial region in Minkowski space. The connection is made to the computation of the relative entropy using modular theory.

Related project(s):
47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces

We show the existence of a solution to the Ricci flow with a compact length space of bounded curvature, i.e., a space that has curvature bounded above and below in the sense of Alexandrov, as its initial condition. We show that this flow converges in the \(C^{1,\alpha}\)-sense to a \(C^{1,\alpha}\)-continuous Riemannian manifold which is isometric to the original metric space. Moreover, we prove that the flow is uniquely determined by the initial condition, up to isometry.

Related project(s):
43Singular Riemannian foliations and collapse79Alexandrov geometry in the light of symmetry and topology

We give an extension of Cheeger's deformation techniques for smooth Lie group actions on manifolds to the setting of singular Riemannian foliations induced by Lie groupoids actions. We give an explicit description of the sectional curvature of our generalized Cheeger deformation.

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

We prove that the group of isometries preserving a metric foliation on a closed Alexandrov space \(X\), or a singular Riemannian foliation on a manifold \(M\) is a closed subgroup of the isometry group of \(X\) in the case of a metric foliation, or of the isometry group of \(M\) for the case of a singular Riemannian foliation. We obtain a sharp upper bound for the dimension of these subgroups and show that, when equality holds, the foliations that realize this upper bound are induced by fiber bundles whose fibers are round spheres or projective spaces. Moreover, singular Riemannian foliations that realize the upper bound are induced by smooth fiber bundles whose fibers are round spheres or projective spaces.

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

We prove that the Patterson-Sullivan and Wigner distributions on the unit sphere bundle of a convex-cocompact hyperbolic surface are asymptotically identical. This generalizes results in the compact case by Anantharaman-Zelditch and Hansen-Hilgert-Schröder.

Related project(s):
65Resonances for non-compact locally symmetric spaces

By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Guéritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.

Related project(s):
65Resonances for non-compact locally symmetric spaces

The fermionic Rényi entanglement entropy is studied for causal diamonds in two-dimensional Minkowski spacetime. Choosing the quasi-free state describing the Minkowski vacuum with an ultraviolet regularization, a logarithmically enhanced area law is derived.

Related project(s):
47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces

In earlier work we have shown that for certain geometric structures on a smooth manifold \(M\) of dimension \(n\), one obtains an almost para-Kähler--Einstein metric on a manifold \(A\) of dimension \(2n\) associated to the structure on \(M\). The geometry also associates a diffeomorphism between \(A\) and \(T^*M\) to any torsion-free connection compatible with the geometric structure. Hence we can use this construction to associate to each compatible connection an almost para-Kähler--Einstein metric on \(T^*M\). In this short article, we discuss the relation of these metrics to Patterson--Walker metrics and derive explicit formulae for them in the cases of projective, conformal and Grassmannian structures.

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

We introduce the notion of a flat extension of a connection \(\theta\) on a principal bundle. Roughly speaking, \(\theta\) admits a flat extension if it arises as the pull-back of a component of a Maurer--Cartan form. For trivial bundles over closed oriented \(3\)-manifolds, we relate the existence of certain flat extensions to the vanishing of the Chern–Simons invariant associated with \(\theta\). As an application, we recover the obstruction of Chern--Simons for the existence of a conformal immersion of a Riemannian \(3\)-manifold into Euclidean \(4\)-space. In addition, we obtain corresponding statements for a Lorentzian \(3\)-manifold, as well as a global obstruction for the existence of an equiaffine immersion into \(\mathbb{R}^4\) of a \(3\)-manifold that is equipped with a torsion-free connection preserving a volume form.

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

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

We present the Laplace operator associated to a hyperbolic surface \(\Gamma\backslash\mathbb{H}\) and a unitary representation of the fundamental group \(\Gamma\), extending the previous definition for hyperbolic surfaces of finite area to those of infinite area. We show that the resolvent of this operator admits a meromorphic continuation to all of \(\mathbb{C}\) by constructing a parametrix for the Laplacian, following the approach by Guillopé and Zworski. We use the construction to provide an optimal upper bound for the counting function of the poles of the continued resolvent.

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

The goal of this note is to demonstrate how existing results can be adapted

to establish the following result: A locally metric measure homogeneous RCD(????, ????)

space is isometric to, after multiplying a positive constant to the reference measure,

a smooth Riemannian manifold with the Riemannian volume measure.

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We prove a local version of the index theorem for Lorentzian Dirac-type operators on globally hyperbolic Lorentzian manifolds with Cauchy boundary. In case the Cauchy hypersurface is compact we do not assume self-adjointness of the Dirac operator on the spacetime or the associated elliptic Dirac operator on the boundary. In this case integration of our local index theorem results in a generalization of previously known index theorems for globally hyperbolic spacetimes that allows for twisting bundles associated with non-compact gauge groups.

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

We develop an algorithm for recognizing whether a character belongs to \(\Sigma^m\). In order to apply it we just need to know that the ambient group is of type \(\mathrm{FP}_m\) or of type \( \mathrm{F}_2\) and that the word problem is solvable for this group. Then finite data is sufficient proof of membership in \(\Sigma^m\), not just for the given character but also for a neighborhood of it.

Related project(s):
39Geometric invariants of discrete and locally compact groups