Publications

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

In this paper we make a detailed analysis of conservation principles in the context of a family of fourth-order gravitational theories generated via a quadratic Lagrangian. In particular, we focus on the associated notion of energy and start a program related to its study. We also exhibit examples of solutions which provide intuitions about this notion of energy which allows us to interpret it, and introduce several study cases where its analysis seems tractable. Finally, positive energy theorems are presented in restricted situations.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

In this paper we address the existence of preferred asymptotic coordinates on asymptotically Euclidean (AE) manifolds \((M^3,g)\) such that \(g\) admits an asymptotically Schwarzschildian first order expansion, based purely on a priori geometric conditions, which will then be used to establish geometric criteria guaranteeing the convergence of the ADM center of mass (COM). This problem is analysed relating it to the study of the regularity of conformal compactifications of such manifolds, which is itself explored via elliptic theory for operators with coefficients of very limited regularity. With these related problems in mind, we first establish Sobolev regularity properties of conformally compactified AE 3-manifolds via the decay of the Cotton tensor, improving on previous results. This allows us to construct preferred asymptotic coordinates on such AE manifolds where the metric has a first order Schwarzschildian expansion, which in turn allows us to address a version of a conjecture posed by C. Cederbaum and A. Sakovich concerning the convergence of the COM of such manifolds.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

The classical singularity theorems of R. Penrose and S. Hawking from the 1960s show that, given a pointwise energy condition (and some causality as well as initial assumptions), spacetimes cannot be geodesically complete. Despite their great success, the theorems leave room for physically relevant improvements, especially regarding the classical energy conditions as essentially any quantum field theory necessarily violates them. While singularity theorems with weakened energy conditions exist for worldline integral bounds, so-called worldvolume bounds are in some cases more applicable than the worldline ones, such as the case of some massive free fields. In this paper, we study integral Ricci curvature bounds based on worldvolume quantum strong energy inequalities. Under the additional assumption of a—potentially very negative—global timelike Ricci curvature bound, a Hawking-type singularity theorem is proved. Finally, we apply the theorem to a cosmological scenario proving past geodesic incompleteness in cases where the worldline theorem was inconclusive.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

Given an extendible spacetime one may ask how much, if any, uniqueness can in general be expected of the extension. Locally, this question was considered and comprehensively answered in a recent paper of Sbierski, where he obtains local uniqueness results for anchored spacetime extensions of similar character to earlier work for conformal boundaries by Chruściel. Globally, it is known that non-uniqueness can arise from timelike geodesics behaving pathologically in the sense that there exist points along two distinct timelike geodesics which become arbitrarily close to each other interspersed with points which do not approach each other. We show that this is in some sense the only obstruction to uniqueness of maximal future boundaries: Working with extensions that are manifolds with boundary we prove that, under suitable assumptions on the regularity of the considered extensions and excluding the existence of such “intertwined timelike geodesics”, extendible spacetimes admit a unique maximal future boundary extension. This is analogous to results of Chruściel for the conformal boundary.

Related project(s):
41Geometrically defined asymptotic coordinates in general relativity

We utilize a condition for algebraic curvature operators called surgery stability as suggested by the work of Hoelzel to investigate the space of riemannian metrics over closed manifolds satisfying these conditions. Our main result is a parametrized Gromov–Lawson construction with not necessarily trivial normal bundles and shows that the homotopy type of this space of metrics is invariant under surgeries of a suitable codimension. This is a generalization of a well-known theorem by Chernysh and Walsh for metrics of positive scalar curvature. As an application of our method, we show that the space of metrics of positive scalar curvature on quaternionic projective spaces are homotopy equivalent to that on spheres.

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

The twisted suspension of a manifold is obtained by surgery along the fibre of a principal circle bundle over the manifold. It generalizes the spinning operation for knots and preserves various topological properties. In this article, we show that Riemannian metrics of positive Ricci curvature can be lifted along twisted suspensions. As an application we show that the maximal symmetry rank of a closed, simply connected Riemannian manifold of positive Ricci curvature is $(n-2)$ in all dimensions $n\geq 4$. Further applications include simply connected 6-manifolds whose homology has torsion, (rational) homology spheres in all dimensions at least 4, and manifolds with prescribed third homology.

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

The twisted suspension of a manifold is obtained by surgery along the fibre of a principal circle bundle over the manifold. It generalizes the spinning operation for knots and preserves various topological properties. In this article, we show that Riemannian metrics of positive Ricci curvature can be lifted along twisted suspensions. As an application we show that the maximal symmetry rank of a closed, simply connected Riemannian manifold of positive Ricci curvature is $(n-2)$ in all dimensions $n\geq 4$. Further applications include simply connected 6-manifolds whose homology has torsion, (rational) homology spheres in all dimensions at least 4, and manifolds with prescribed third homology.

We consider smooth vector bundles over smooth manifolds equipped with non-smooth geometric data. For nilpotent differential operators acting on these bundles, we show that the kernels of induced Hodge-Dirac-type operators remain isomorphic under uniform perturbations of the geometric data. We consider applications of this to the Hodge-Dirac operator on differential forms induced by so-called rough Riemannian metrics, which can be of only measurable coefficient in regularity, on both compact and non-compact settings. As a consequence, we show that the kernel of the associated non-smooth Hodge-Dirac operator with respect to a rough Riemannian metric remains isomorphic to smooth and singular cohomology when the underlying manifold is compact.

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

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov. This includes optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of non-vanishing \(\widehat{\mathrm{A}}\)-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor. Furthermore, we establish scalar- and mean curvature extremality results for certain log-concave warped products. The latter includes annuli in all simply-connected space forms. On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions.

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

Let \(\mathcal{E}\) be an asymptotically Euclidean end in an otherwise arbitrary complete and connected Riemannian spin manifold \((M,g)\). We show that if \(\mathcal{E}\) has negative ADM-mass, then there exists a constant \(R > 0\), depending only on \(\mathcal{E}\), such that \(M\) must become incomplete or have a point of negative scalar curvature in the \(R\)-neighborhood around \(\mathcal{E}\) in \(M\). This gives a quantitative answer to Schoen and Yau's question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions \(\leq 7\) assuming Schwarzschild asymptotics on the end \(\mathcal{E}\). We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end \(\mathcal{E}\). Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature.

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

We prove a positive mass theorem for spin initial data sets \((M,g,k)\) that contain an asymptotically flat end and a shield of dominant energy (a subset of M on which the dominant energy scalar \(μ−|J|\) has a positive lower bound). In a similar vein, we show that for an asymptotically flat end \(\mathcal{E}\) that violates the positive mass theorem (i.e. \(\mathrm{E}<|\mathrm{P}|\)), there exists a constant \(R > 0\), depending only on \(\mathcal{E}\), such that any initial data set containing \(\mathcal{E}\) must violate the hypotheses of Witten's proof of the positive mass theorem in an \(R\)-neighborhood of \(\mathcal{E}\). This implies the positive mass theorem for spin initial data sets with arbitrary ends, and we also prove a rigidity statement. Our proofs are based on a modification of Witten's approach to the positive mass theorem involving an additional independent timelike direction in the spinor bundle.

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

For every \(n \geq 4\) we construct infinitely many mutually not homotopic closed manifolds of dimension \(n\) which admit a negatively curved Einstein metric but no locally symmetric metric.

Related project(s):
51The geometry of locally symmetric manifolds via natural maps

We prove a new criterion for the existence and completeness of the wave operators corresponding to the Laplace-Beltrami operators corresponding to two Riemannian metrics on a fixed noncompact manifold. Our result relies on recent estimates on the heat semigroup and its derivative, that are valid if the negative part of the Ricci curvature is in the Kato class - so called Kato-Ricci manifolds.

Related project(s):
50Probabilistic and spectral properties of weighted Riemannian manifolds with Kato bounded Bakry-Emery-Ricci curvature

The main result of this paper is that when M_0, M_1 are two simply connected spin manifolds of the same dimension d≥5 which both admit a metric of positive scalar curvature, the spaces \mathcal{M}^+(M0) and \mathcal{M}^+(M_1) of such metrics are homotopy equivalent. This supersedes a previous result of Chernysh and Walsh which gives the same conclusion when M_0 and M_1 are also spin cobordant.

We also prove an analogous result for simply connected manifolds which do not admit a spin structure; we need to assume that d≠8 in that case.

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

We derive Gaussian heat kernel bounds on graphs with respect to a fixed origin for large times under the assumption of a Sobolev inequality and volume doubling on large balls. The upper bound from our previous work [KR22] is affected by a new correction term measuring the distance to the origin. The main result is then applied to anti-trees with unbounded vertex degree, yielding Gaussian upper bounds for this class of graphs for the first time. In order to prove this, we show that isoperimetric estimates with respect to intrinsic metrics yield Sobolev inequalities. Finally, we prove that anti-trees are Ahlfors regular and that they satisfy an isoperimetric inequality of a larger dimension.

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

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