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Klaus Kroencke

Publications and source records attributed to Klaus Kroencke.

At least 19 recordsLinked to original sources
arXiv2026

$L^p$ cohomology and Hodge decomposition for ALE manifolds

We relate the dimensions of $L^p$ reduced cohomology spaces in degree k of an ALE manifold to the dimension of some spaces of decaying harmonic forms, depending both on p and on k. In this class of manifolds, this provides an extension to $p\neq 2$ of the well-known result of Hodge. In particular, we prove that for fixed $k\notin\left\{1,n-1\right\}$, the dimension of the $L^p$ reduced cohomology spaces in degree k is independent of $p\in (1,\infty)$, while for $k\in\{1,n-1\}$, the dimension jumps exactly once by a factor N-1 (N being the number of ends) when $p$ varies in $(1,\infty)$. We also prove $L^p$ Hodge decompositions for k-forms on such manifolds, for the optimal values of k and p. When these are not available, we provide a substitute (a modified Hodge decomposition).

math.DG↗
arXiv2026

Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity

We prove stability of integrable ALE manifolds with a parallel spinor under Ricci flow, given an initial metric which is close in $L^p \cap L^\infty$, for any $p \in (1, n)$, where $n$ is the dimension of the manifold. In particular, our result applies to all known examples of $4$-dimensional gravitational instantons. Our decay rates are strong enough to prove positive scalar curvature rigidity in $L^p$, for each $p \in \left[1, \frac{n}{n-2}\right)$, generalizing a result by Appleton.

math.DG↗
arXiv2026

On the uniqueness of surfaces of constant spacetime mean curvature in asymptotically Schwarzschildean lightcones

In this paper, we address the uniqueness of surfaces of constant spacetime mean curvature in an asymptotically Schwarzschildean lightcone of mass $m>0$. We prove that there exists a unique asymptotically flat foliation by surfaces of constant spacetime mean curvature for a fairly generic notion of asymptotic flatness. This foliation has Bondi energy $m$ and vanishing Bondi linear momentum. The authors have already established the existence of such a foliation in previous work, but proven uniqueness only in a very restrictive class of surfaces. Although this restrictive class of surfaces was necessary for the construction, here we show that the foliation is a posteriori unique under significantly weaker assumptions.

math.DG↗
arXiv2026

Some isolation and stability results for Einstein manifolds

We prove new isolation and stability results for Einstein manifolds in a variety of settings. Imposing conditions on the Weyl tensor, we establish new stability criteria for compact, asymptotically hyperbolic (AH) and asymptotically locally Euclidean (ALE) manifolds and an isolation result in the latter setting. For compact Kähler and Sasaki $η$-Einstein manifolds, we provide similar results which involve the Bochner tensor and the contact Bochner tensor, respectively.

math.DG↗
arXiv2026

Static Vacuum Spacetimes with $Λ<0$ as Attractors of the Ricci-Harmonic Flow

We prove dynamical stability and instability theorems for asymptotically hyperbolic static solutions of Einstein's equation with $Λ<0$, viewed as self-similar solutions of the Ricci-harmonic flow. More precisely, we show that static metrics are dynamically stable if and only if a positive mass type theorem holds for nearby metrics. Our key tool is a new variant of the expander entropy for the Ricci-harmonic flow.

math.DG↗
arXiv2026

On pp-waves with lightlike parallel spinors

We parametrize pp-wave spacetimes with compact codimension 2 hypersurfaces. In the vacuum case, we show that these spacetimes are locally in one-to-one correspondence with smooth curves of Riemannian Ricci-flat metrics modulo smooth curves of diffeomorphisms. We also prove that this one-to-one correspondence extends to pp-waves with prescribed null Ricci curvature. Moreover, the pp-wave spacetime carries a lightlike parallel spinor if and only if one (and hence all) of the Ricci-flat metrics carries a parallel spinor.

math.DG↗
arXiv2025

Ricci flow and the scalar curvature rigidity of Einstein manifolds

We review recent results relating linear stability to dynamical stability and the scalar curvature rigidity of Einstein manifolds. We discuss closed and open Einstein manifolds as well as complete noncompact Einstein manifolds which are asymptotically locally Euclidean and asymptotically hyperbolic. For these classes, the relation to the positive mass theorem will also be explained.

math.DG↗
arXiv2025

The Volume-Renormalized Mass from a Hamiltonian Perspective

We demonstrate that the volume-renormalized mass for asymptotically hyperbolic manifolds recently introduced by the authors can be deduced from a reduced Hamiltonian perspective. In order to do this, we first use Michel's formalism of mass invariants to extend the definition of the volume-renormalized mass to initial data sets. We consider spacetimes that are foliated by asymptotically Poincaré--Einstein Riemannian manifolds in the spirit of the Milne model of cosmology and reduce the ADM Hamiltonian to an unconstrained Hamiltonian system, analogous to the work of Fischer and Moncrief for spatially compact spacetimes. We find that the reduced Hamiltonian in this case recovers the volume-renormalized mass. We then analyze the first and second variation of the reduced Hamiltonian and demonstrate that it is non-increasing over the evolution and constant only for self-similar spacetimes.

math.DG↗
arXiv2025

Green functions and a positive mass theorem for asymptotically hyperbolic $3$-manifolds

We prove a new positive mass theorem for three-dimensional manifolds which are asymptotically hyperboloidal of order greater than $1$. The mass quantity under consideration is the volume-renormalized mass recently introduced in a paper by Dahl, McCormick and the first author. The proof is based on a monotonicity formula holding along the level sets of the Green function for the Laplace operator centered at an arbitrary point. In order for this argument to work out, we require that the second homology of the manifold does not contain any spherical classes.

math.DG↗
arXiv2024

Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature

We construct asymptotic foliations of asymtotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (STCMC). Our construction is motivated by the approach of Huisken-Yau for the Riemannian setting in employing a geometric flow. We prove that initial data within a sufficient a-priori class converges exponentially to an STCMC surface under area preserving null mean curvature flow. Further, we show that the resulting STCMC surfaces form an asymptotic foliation that is unique within the a-priori class.

math.DG↗
arXiv2024

On stability and scalar curvature rigidity of quaternion-Kähler manifolds

We show that every quaternion-Kähler manifold of negative scalar curvature is stable as an Einstein manifold and therefore scalar curvature rigid. In particular, this implies that every irreducible nonpositive Einstein manifold of special holonomy is stable. In contrast, we demonstrate that there exist quaternion-Kähler manifolds of positive scalar curvature which are not scalar curvature rigid even though they are semi-stable.

math.DG↗
arXiv2024

The asymptotic expansion of the spacetime metric at the event horizon

Hawking's local rigidity theorem, proven in the smooth setting by Alexakis-Ionescu-Klainerman, says that the event horizon of any stationary non-extremal black hole is a non-degenerate Killing horizon. In this paper, we prove that the full asymptotic expansion of any smooth vacuum metric at a non-degenerate Killing horizon is determined by the geometry of the horizon. This gives a new perspective on the black hole uniqueness conjecture. In spacetime dimension $4$, we also prove an existence theorem: Given any non-degenerate horizon geometry, Einstein's vacuum equations can be solved to infinite order at the horizon in a unique way (up to isometry). The latter is a gauge invariant version of Moncrief's classical existence result, without any restriction on the topology of the horizon. In the real analytic setting, the asymptotic expansion is shown to converge and we get well-posedness of this characteristic Cauchy problem.

math.DG↗
arXiv2024

A volume-renormalized mass for asymptotically hyperbolic manifolds

We define a geometric quantity for asymptotically hyperbolic manifolds, which we call the volume-renormalized mass. It is essentially a linear combination of the ADM mass surface integral and a renormalization of the volume. We show that the volume-renormalized mass is well-defined and diffeomorphism invariant under weaker fall-off conditions than required to ensure that the renormalized volume and the ADM mass surface integral are well-defined separately. We prove several positivity results for the volume-renormalized mass. We also use it to define a renormalized Einstein--Hilbert action and a renormalized expander entropy which is nondecreasing under the Ricci flow. Further, we show that local maximizers of the entropy are local minimizers of the volume-renormalized mass.

math.DG↗
arXiv2023

Dynamical Stability and Instability of Poincaré--Einstein Manifolds

We prove dynamical stability and instability theorems for Poincaré-Einstein metrics under the Ricci flow. Our key tool is a variant of the expander entropy for asymptotically hyperbolic manifolds, which Dahl, McCormick and the first author established in a recent article. It allows us to characterize stability and instability in terms of a local positive mass theorem and in terms of volume comparison for nearby metrics.

math.DG↗
arXiv2022

Local and global scalar curvature rigidity of Einstein manifolds

An Einstein manifold is called scalar curvature rigid if there are no compactly supported volume-preserving deformation of the metric which increase the scalar curvature. We give various characterizations of scalar curvature rigidity for open Einstein manifolds as well as for closed Einstein manifolds. As an application, we construct mass-decreasing perturbations of the Riemannian Schwarzschild metric and the Taub-Bolt metric.

math.DG↗
arXiv2022

Optimal coordinates for Ricci-flat conifolds

We compute the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and give a detailed description of the corresponding radially homogeneous tensor fields in its kernel. For a Ricci-flat conifold $(M,g)$ which may have asymptotically conical as well as conically singular ends, we compute at each end a lower bound for the order with which the metric converges to the tangent cone. As a special subcase of our result, we show that any Ricci-flat ALE manifold $(M^n,g)$ is of order $n$ and thereby close a small gap in a paper by Cheeger and Tian.

math.DG↗
arXiv2021

Long-time estimates for heat flows on ALE manifolds

We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel $L^p - L^q$ decay estimates are established, allowing the Schrödinger operator to have a non-trivial $L^2$-kernel. We also prove new decay estimates for spatial derivatives of arbitrary order, in a general geometric setting. Our main motivation is the application to stability of non-linear geometric equations, primarily Ricci flow, which will be presented in a companion paper. The arguments in this paper use that many geometric Schrödinger operators can be written as the square of Dirac type operators. By a remarkable result of Wang, this is even true for the Lichnerowicz Laplacian, under the assumption of a parallel spinor. Our analysis is based on a novel combination of the Fredholm theory for Dirac type operators on ALE manifolds and recent advances in the study of the heat kernel on non-compact manifolds.

math.AP↗