Publications

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

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

In this paper we develop the theory of homological geometric invariants (following Bieri-Neumann-Strebel-Renz) for locally compact Hausdorff groups. The homotopical version is treated elsewhere. Both versions are connected by a Hurewicz-like theorem.

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

We show that the covolume of an irreducible lattice in a higher rank semisimple Lie group with the congruence subgroup property is determined by the profinite completion. Without relying on CSP, we additionally show that volume is a profinite invariant of octonionic hyperbolic congruence manifolds.

Related project(s):
58Profinite perspectives on l2-cohomology61At infinity of symmetric spaces

In this paper we study n-dimensional Ricci flows  (M,g(t)), t in [0,T),   where  T is finite,  and   potentially a singular time, and for which the spatial L^p norm,  p>n/2,  of the scalar curvature is uniformly  bounded on [0,T). 

 In the case that M is closed, we show that non-collapsing  and non-inflating estimates hold. If we further assume   that  n=4 or that  M^n is Kähler, we explain how  these non-inflating/non-collapsing estimates can be combined  with    integral bounds on the Ricci and full curvature tensor of  the  prequel paper   to   obtain  an improved space time integral bound of the Ricci curvature.  

  As an application of these estimates,  we show  that if we further restrict to n=4, then  the solution convergences to an orbifold as t approaches T and  that the flow can be extended    using the Orbifold Ricci flow to the time interval    [0,T+a)$ for some a>0.

  We also prove local versions of many of the  results mentioned above. 

Related project(s):
75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

 In this   paper we  prove   localised weighted curvature   integral estimates for solutions to the Ricci flow 
in the setting of a  smooth four dimensional Ricci flow or a closed n-dimensional Kähler Ricci flow. 
These integral   estimates improve and extend  the integral curvature estimates shown by the second author  in an earlier paper. If  
the scalar curvature is uniformly bounded in the spatial L^p sense for some p>2, then the estimates imply a uniform bound on the spatial L^2 norm of the Riemannian curvature  tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition.    
In a sequel paper, we show that in many natural settings,  a   non-inflating condition holds.

Related project(s):
75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

We extend the K-cowaist inequality to generalized Dirac operators in the sense of Gromov and Lawson and study applications to manifolds with boundary.

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

In this paper, we combine and generalize to higher dimensions the approaches to proving the uniqueness of connected (3+1)-dimensional static vacuum asymptotically flat black hole spacetimes by Müller zum Hagen--Robinson--Seifert and by Robinson. Applying these techniques, we prove and/or reprove geometric inequalities for connected (n+1)-dimensional static vacuum asymptotically flat spacetimes with either black hole or equipotential photon surface or specifically photon sphere inner boundary. In particular, assuming a natural upper bound on the total scalar curvature of the boundary, we recover and extend the well-known uniqueness results for such black hole and equipotential photon surface spacetimes. We also relate our results and proofs to existing results, in particular to those by Agostiniani--Mazzieri and by Nozawa--Shiromizu--Izumi--Yamada.

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

It is a well-known fact that the Schwarzschild spacetime admits a maximal spacetime extension in null coordinates which extends the exterior Schwarzschild region past the Killing horizon, called the Kruskal-Szekeres extension. This method of extending the Schwarzschild spacetime was later generalized by Brill-Hayward to a class of spacetimes of "profile h" across non-degenerate Killing horizons. Circumventing analytical subtleties in their approach, we reconfirm this fact by reformulating the problem as an ODE, and showing that the ODE admits a solution if and only if the naturally arising Killing horizon is non-degenerate. Notably, this approach lends itself to discussing regularity across the horizon for non-smooth metrics.

We will discuss applications to the study of photon surfaces, extending results by Cederbaum-Galloway and Cederbaum-Jahns-Vičánek-Martínez beyond the Killing horizon. In particular, our analysis asserts that photon surfaces approaching the Killing horizon must necessarily cross it.

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

We present a new proof of the Willmore inequality for an arbitrary bounded domain Ω⊂ℝ^n with smooth boundary. Our proof is based on a parametric geometric inequality involving the electrostatic potential for the domain Ω; this geometric inequality is derived from a geometric differential inequality in divergence form. Our parametric geometric inequality also allows us to give new proofs of the quantitative Willmore-type and the weighted Minkowski inequalities by Agostiniani and Mazzieri.

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