Publications

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 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 study four-dimensional asymptotically flat electrostatic electro-vacuum spacetimes with a connected black hole, photon sphere, or equipotential photon surface inner boundary. Our analysis, inspired by the potential theory approach by Agostiniani–Mazzieri, allows to give self-contained proofs of known uniqueness theorems of the sub-extremal, extremal, and super-extremal Reissner–Nordström spacetimes. We also obtain new results for connected photon spheres and equipotential photon surfaces in the extremal case. Finally, we provide, up to a restrictionon the range of their radii, the uniqueness result for connected (both non-degenerate and degenerate) equipotential photon surfaces in the super-extremal case, not yet treated in the literature.

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

We extend the classical theory of homotopical Σ-sets, defined by Bieri, Neumann, Renz and Strebel for abstract groups, to  locally compact Hausdorff groups. Given such a group G, our geometric invariants are sets of continuous homomorphisms G→R ("characters"). They match the classical Σ-sets if G is discrete, and refine the homotopical compactness properties of Abels and Tiemeyer. Moreover, our theory recovers the definition of low-dimensional geometric invariants for topological gropus proposed by Kochloukova.

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

In this paper, we consider the stability of the generalized Lagrangian mean curvature flow of graph case in the cotangent bundle, which is first defined by Smoczyk-Tsui-Wang [14]. By new estimates of derivatives along the flow, we weaken the initial condition and remove the positive curvature condition in [14]. More precisely, we prove that if the graph induced by a closed $1$-form is a special Lagrangian submanifold in the cotangent bundle of a Riemannian manifold, then the generalized Lagrangian mean curvature flow is stable near it.

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in \(\mathbb R^{n+1}\), and prove the existence and regularity of the flow before extincting to a point in finite time.

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

In this paper, we show the relation between the existence of twisted conical Kähler-Ricci solitons and the greatest log Bakry-Emery-Ricci lower bound on Fano manifolds. This is based on our proofs of some openness theorems on the existence of twisted conical Kähler-Ricci solitons, which generalize Donaldson's existence conjecture and openness theorem of the conical Kähler-Einstein metrics to the conical soliton case.

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II