Publications

The positive energy theorems are a fundamental pillar in mathematical general relativity. Originally proved by Schoen-Yau and later Witten, these theorems were established for asymptotically flat manifolds where the metric tends to the standard Euclidean metric and whose second fundamental form decays to zero at infinity. This ansatz on the metric and second fundamental form is motivated by the desire to model an isolated gravitational system with a Minkowski space background for the spacetime. However, actual astrophysical massive objects are not truly isolated but rather exist within an expanding cosmological universe, where the second fundamental form is umbilic. With this in mind, we seek a notion of energy for initial data sets with an umbilic second fundamental form. In this work, we present a definition of energy in such an expanding cosmological setting. Instead of Minkowski space, we take de Sitter space as the background spacetime, which, when written in flat-expanding coordinates, is foliated by umbilic hypersurfaces each isometric to Euclidean 3-space. This cosmological setting necessitates a quasi-local energy definition, as the presence of a cosmological horizon in de Sitter space obstructs a global one. We define energy in this quasi-local setting by adapting the Liu-Yau energy to our framework and establish positivity of this energy for certain bounded values of the cosmological constant.

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

In this paper we prove some rigidity theorems associated to \(Q\)-curvature analysis on asymptotically Euclidean (AE) manifolds, which are inspired by the analysis of conservation principles within fourth order gravitational theories. A central object in this analysis is a notion of fourth order energy, previously analysed by the authors, which is subject to a positive energy theorem. We show that this energy can be more geometrically rewritten in terms of a fourth order analogue to the Ricci tensor, which we denote by \(J_g\). This allows us to prove that Yamabe positive \(J\)-flat AE manifolds must be isometric to Euclidean space. As a by product, we prove that this \(J\)-tensor provides a geometric control for the optimal decay rates at infinity. This last result reinforces the analogy of \(J\) as a fourth order analogue to the Ricci tensor.

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

We analyse the geometry and topology of the trapped photon region in the domain of outer communication of sub-extremal Kerr-Newman and Kerr-Sen spacetimes. Specifically, we show that its projection to the (co-)tangent bundle forms a five-dimensional submanifold with topology $SO(3)\times\mathbb{R}^2$ in each setup. The proof adapts the method of Cederbaum and Jahns for sub-extremal Kerr spacetime.

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

We review and announce recent results on the asymptotic behavior of asymptotically Euclidean relativistic initial data sets and asymptotic foliations thereof. In particular, we discuss the geometrization of asymptotic flatness and of asymptotic geometric (in-)\-variants such as mass, energy, linear momentum, angular momentum and center of mass as well as their relations to certain geometric asymptotic foliations such as the CMC- and STCMC-foliations.

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

We consider a smooth compact manifold with boundary, M,  embedded in a smooth manifold of the same dimension on which an amenable group \(\Gamma\) acts by isometries. We do not assume M to be invariant under \(\Gamma\). This results in a partial action of \(\Gamma\) on: For $g\in \Gamma$ we let \(M^\circ_g = g(M^\circ)\cap M^\circ\) and obtain diffeomorphisms \(g:M^\circ_{g^{-1}} \to

M^\circ_g

\).

We assume that any two images of \(\partial M\) under \(\Gamma\) either coincide or are disjoint and that only finitely many lie in M. The spherical blow-up of these images of \(\partial M\) in M yields a

manifold Y with  boundary consisting of finitely many components. Moreover, Y inherits a partial action of \(\Gamma\).

We can then define the C*-algebra \(\mathcal

A=\overline{\Psi_\Gamma(Y,\partial Y)}\) of operators on \(L^2(Y)\oplus

L^2(\partial Y)\), generated by the algebra \(\Psi(Y,\partial Y)\) of operators of order and type zero in Boutet de Monvel's calculus on Y and partial isometries associated with the partial action. Denote by

\(\Sigma=\overline{\Psi(Y,\partial Y)}/\mathcal K

\) the symbol space. If the partial action of \(\Gamma\) on Prim(\(\Sigma\)) is topologically

free, we find a criterion for the Fredholm property of the operators in \(\overline{\Psi_\Gamma(Y,\partial Y)}\).

Moreover, we obtain the classification of the elliptic elements in

\(\overline{\Psi_\Gamma(Y,\partial Y)}\) modulo stable homotopies: For \(\mathcal A_0= C(Y\sqcup \partial Y)\rtimes\Gamma\)

\(Ell(\mathcal A_0,\mathcal A)\cong K_0(C_0(T^*Y^\circ)\rtimes\Gamma

)\oplus K_0(C(\partial Y)\rtimes \Gamma).\)

If \(\Gamma\) is finitely generated and of polynomial growth, then the elements associated with the second summand do not contribute to the index.

Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics

Let G be a compact Lie group that acts smoothly on a closed manifold M. Using a general Simonenko principle, we derive a novel criterion for the Fredholm property of G-pseudodifferential operators acting on Sobolev spaces of sections of vector bundles over M. In case the group is finite, we obtain a further characterization of the Fredholm property of G-pseudodifferential operators in terms of the invertibility of suitable symbols.

Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics

We describe various ways of obtaining the Hadamard coefficients associated to a normally hyperbolic operator from the corresponding Green's operators. As the Hadamard expansion on its own is not enough for this, we include additional information either by considering something like a resolvent or powers of Green's operators or by looking at a product of the original manifold with the real line.

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

We prove a Riemannian positive mass theorem for asymptotically flat spin manifolds with hypersurface singularities.

    Unlike earlier results, some components of the singular set may be mean-concave, provided that other components of the singular set are sufficiently mean-convex. 

    Our proof uses initial data sets where a suitably chosen second fundamental form transfers convexity defects between different singularity components. 

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 Gromov's conjecture on the total mean curvature of fill-ins in various cases.

Our methods are based on surgery to reduce the statement to fill-ins of spheres, which can be treated by instances of the positive mass theorem. 

For spin fill-ins, where we permit the mean curvature to take negative values, we build on a recent positive mass theorem with creases by Kazaras--Khuri--Lin.

For non-spin fill-ins of spin manifolds, where we assume the mean curvature to be non-negative, we develop a novel quantitative surgery process to reduce the general situation to a result of Shi--Wang--Wei.

We also treat the case of fill-ins of non-spin manifolds, provided there is a fixed positive lower bound on the mean curvature.

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 provide a complete classification of groups that can be realized as isometry groups of a translation surface M with non-finitely generated fundamental group and no planar ends. Furthermore, we demonstrate that if S has no non-displaceable subsurfaces and its space of ends is self-similar, then every countable subgroup of GL+(2,R) can be realized as the Veech group of a translation surface X homeomorphic to M. The latter result generalizes and improves upon the previous findings of Przytycki-Valdez-Weitze-Schmithüsen and Maluendas-Valdez. To prove these results, we adapt ideas from the work of Aougab-Patel-Vlamis, which focused on hyperbolic surfaces, to translation surfaces.

Related project(s):
72Limits of invariants of translation surfaces

We determine the distribution of the number of saddle connections on a random translation surface of large genus. More specifically, for genus g tending to infinity, the number of saddle connections with lengths in a given interval [a/g, b/g] converges in distribution to a Poisson distributed random variable. Furthermore, the numbers of saddle connections associated to disjoint intervals of lengths are independent.

Related project(s):
72Limits of invariants of translation surfaces

We study the Cauchy problem for symmetric hyperbolic systems on Lorentzian manifolds with timelike boundary, where initial values are prescribed on a spacelike slice and nonlocal boundary conditions are imposed on the timelike boundary. This extends the classical theory of local boundary conditions to include matching conditions and slicewise Atiyah--Patodi--Singer type boundary conditions.We prove existence of weak solutions and, under suitable regularity assumptions on the Cauchy data, establish existence and uniqueness of strong solutions.

An interesting phenomenon occurs for finite propagation speed: for nonlocal boundary conditions, when a wave first encounters the boundary, the entire boundary instantaneously radiates back everywhere. This allows signals to propagate faster than with any prescribed velocity.

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

Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes we use approximations by manifolds with smooth boundary.

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

We develop a formula for the equivariant index of a twisted Dirac operator on a compact globally hyperbolic spacetime with timelike boundary on which a group acts isometrically, subject to APS boundary conditions. The formula is the same as in the Riemannian case: the equivariant index for a group element is an integral over the fixed point set of that element plus some boundary terms. The proof uses a surprisingly simple technique for reducing from the equivariant to the non-equivariant regime in order to show an equivariant version of the Lorentzian "index = spectral flow" formula.

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

For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences. Some are known and include Llarull's scalar curvature rigidity of the standard metric on the sphere, Geroch's conjecture on the impossibility of positive scalar curvature on tori and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler. New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for Kähler manifolds, Kähler-Einstein manifolds, quaternionic Kähler manifolds and manifolds with a harmonic 1-form of constant length.

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

We study the heat equation associated to the Hodge Laplacian on simplicial complexes. Using recently developed techniques for magnetic Schrödinger operators, we prove Davies-Gaffney-Grigoryan type estimates for the kernel of the heat semigroup on  $\ell^2$   which we then use to extend the semigroup to  $\ell^p$   for $p\in[1,\infty] $  under suitable curvature and volume growth conditions. Furthermore, we establish  -independence of the Hodge Laplacian spectrum under the assumption of form bounded curvature and uniform subexponential volume growth. While the main focus of the paper is the Hodge Laplacian on simplicial complexes, the results are indeed proven for general positive magnetic Schrödinger operators on graphs.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

We characterize all semigroups sandwiched between the semigroup of a Dirichlet form and the semigroup of its active main part. In case the Dirichlet form is regular, we give a more explicit description of the quadratic forms of the sandwiched semigroups in terms of pairs consisting of an open set and a measure on an abstract boundary.

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

We survey recent results on graphs and their Laplacians related to the behavior of the graph at large. In particular, we focus on Liouville theorems, recurrence and characterizations of Dirichlet forms via boundary terms.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

We study the complex property $\partial\partial=0$ of the boundary operator $\partial$ on a weighted, infinite, and possibly non-locally finite simplicial complex. We give a characterization of this property in $\ell^2$   in terms of the recurrence of the links of simplices. The complex property is essential to ensure that Hodge Laplacians $\Delta^H$  indeed act as $\delta\partial+\partial\delta$  and to decompose $\Delta^H$    into a direct sum of operators acting on  $k$-forms. Furthermore, it allows us to define relative cohomology classes, show a respective weak Hodge decomposition, and prove the existence of harmonic Dirichlet eigenforms. We also discuss a transience property for simplicial complexes, that was introduced by Parzanchevski and Rosenthal.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

We investigate the equivalence of Sobolev inequalities and the conjunction of Gaussian upper heat kernel bounds and volume doubling on large scales on graphs. For the normalizing measure, we obtain the equivalence up to constants. If arbitrary measures are considered, we incorporate a new local regularity condition. Furthermore, new correction functions for the Gaussian, doubling, and Sobolev dimension are introduced. For the Gaussian and doubling, the variable correction functions always tend to one at infinity. Moreover, the variable Sobolev dimension can be related to the doubling dimension and the vertex degree growth.

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