
Klaus KrönckeKTH Royal Institute of Technology | KTH · Department of Mathematics (SCI-MAT)
Klaus Kröncke
Dr. rer. nat.
About
50
Publications
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337
Citations
Introduction
Additional affiliations
October 2015 - December 2021
October 2014 - September 2015
January 2014 - September 2014
Education
October 2010 - December 2013
October 2005 - September 2010
Publications
Publications (50)
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 S...
We show that every quaternion-K\"ahler 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-K\"ahler manifolds of positive sc...
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 po...
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 geom...
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 whic...
We define a geometric quantity for asymptotically hyperbolic manifolds, which we call the volume-renormalized mass. It is essentially a linear combination of a renormalization of the volume and the standard ADM mass integral. We show that the volume-renormalized mass is well-defined and diffeomorphism invariant under weaker fall-off conditions than...
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\...
An Einstein manifold is called scalar curvature rigid if there are no compactly supported volume-preserving deformations 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-decreasi...
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 whi...
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 ge...
We prove a geometric characterization of all possible 4-dimensional real analytic vacuum spacetimes near non-degenerate Killing horizons. It is known that any such horizon admits a canonically induced real analytic Riemannian metric with a Killing vector field of constant length. In this paper we prove the converse statement: Every real analytic Ri...
Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a spa...
We compute the spectra of the Laplace-Beltrami operator, the connection Laplacian on 1-forms and the Einstein operator on symmetric 2-tensors on the sine-cone over a positive Einstein manifold (M,g). We conclude under which conditions on (M,g), the sine-cone is dynamically stable under the singular Ricci-de Turck flow and rigid as a singular Einste...
In this survey we provide an overview of our recent results concerning Ricci de Turck flow on spaces with isolated conical singularities. The crucial characteristic of the flow is that it preserves the conical singularity. Under certain conditions, Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is pr...
We compute the spectra of the Laplace-Beltrami operator, the connection Laplacian on 1-forms and the Einstein operator on symmetric 2-tensors on the sine-cone over a positive Einstein manifold $(M, g)$. We conclude under which conditions on $(M,g)$, the sine-cone is dynamically stable under the singular Ricci-de Turck flow and rigid as a singular E...
The 2019 'Australian-German Workshop on Differential Geometry in the Large' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures o...
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 eno...
We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold \(M\times \mathbb {R}\), where M is asymptotically flat. If the initial hypersurface \(F_0\subset M\times \mathbb {R}\) is uniformly spacelike and asymptotic to \(M\times \left\{ s\right\} \) for some \(s\in \mathbb {R}\) at...
We consider the heat equation associated to Schr\"{o}dinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel $L^p - L^q$ decay estimates are established, allowing the Schr\"{o}dinger operator to have a non-trivial $L^2$-kernel. We also prove new decay estimates for spatial derivatives of arbitrary order,...
In this paper we consider a Ricci de Turck flow of spaces with isolated conical singularities, which preserves the conical structure along the flow. We establish that a given initial regularity of Ricci curvature is preserved along the flow. Moreover under additional assumptions, positivity of scalar curvature is preserved under such a flow, mirror...
We consider the Einstein-flow on a product manifold with one factor being a compact quotient of 3-dimensional hyperbolic space without boundary and the other factor being a flat torus of fixed arbitrary dimension. We consider initial data symmetric with respect to the toroidal directions. We obtain effective Einsteinian field equations coupled to a...
In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability,...
We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold $M\times\mathbb{R}$, where $M$ is asymptotically flat. If the initial hypersurface $F_0\subset M\times\mathbb{R}$ is uniformly spacelike and asymptotic to $M\times\left\{s\right\}$ for some $s\in\mathbb{R}$ at infinity, we sh...
Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a spa...
In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prove that the entropies are monot...
On a closed connected oriented manifold M we study the space \(\mathcal {M}_\Vert (M)\) of all Riemannian metrics which admit a non-zero parallel spinor on the universal covering. Such metrics are Ricci-flat, and all known Ricci-flat metrics are of this form. We show the following: The space \(\mathcal {M}_\Vert (M)\) is a smooth submanifold of the...
We complement a recent work on the stability of fixed points of the CMC-Einstein-Λ flow. In particular, we modify the utilized gauge for the Einstein equations and remove a restriction on the fixed points whose stability we are able to prove by this method, and thereby generalize the stability result. In addition, we consider the notion of the redu...
We prove the global existence of Dirac-wave maps with curvature term with small initial data on globally hyperbolic manifolds of arbitrary dimension which satisfy a suitable growth condition. In addition, we also prove a global existence result for wave maps under similar assumptions.
We complement a recent work on the stability of fixed points of the CMC-Einstein-$\Lambda$ flow. In particular, we modify the utilized gauge for the Einstein equations and remove a restriction on the fixed points whose stability we are able to prove by this method, and thereby generalize the stability result. In addition, we consider the notion of...
We consider the Einstein flow on a product manifold with one factor being a compact quotient of 3-dimensional hyperbolic space without boundary and the other factor being a flat torus of fixed arbitrary dimension. We consider initial data symmetric with respect to the toroidal directions. We obtain effective Einsteinian field equations coupled to a...
In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability,...
We prove the global existence of wave maps with small initial data on globally hyperbolic manifolds of arbitrary dimension which satisfy a suitable growth condition. In addition, we also prove a global existence result for Dirac-wave maps with curvature term under similar assumptions.
We prove that if an ALE Ricci-flat manifold $(M,g)$ is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to $g$. By adapting Tian's approach in the closed case, we show that integrability holds f...
We prove that if an ALE Ricci-flat manifold $(M,g)$ is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to $g$. By adapting Tian's approach in the closed case, we show that integrability holds f...
We consider the vacuum Einstein flow with a positive cosmological constant λ on spatial manifolds of product form M = M1 × M2. In dimensions n = dim M ≥ 4 we show the existence of continuous families of recollapsing models whenever at least one of the factors M1 or M2 admits a Riemannian Einstein metric with positive Einstein constant. We moreover...
This work concerns stability and instability of Einstein warped products with an Einsteinian fiber of codimension 1. We study the cases where the scalar curvature of the warped product and of the base are either both positive or both negative to complement the results in [Kr\"o16]. Up to a small gap in the case of sin-cones, the stability propertie...
We consider the vacuum Einstein flow with a positive cosmological constant on spatial manifolds of product form. In spatial dimension at least four we show the existence of continuous families of recollapsing models whenever at least one of the factors or admits a Riemannian Einstein metric with positive Einstein constant. We moreover show that the...
The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also holds for connections with metric torsion in the case of non-positive Euler characteristic. Our main tool is an ad...
The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also holds for connections with metric torsion in the case of non-positive Euler characteristic. Our main tool is an ad...
We give a survey on the stability problem of compact Einstein manifolds and on infinitesimal Einstein deformations. We review some important results from this topic, including recent work of the author. Moreover, we discuss applications in mathematical physics.
In this article, we systematically investigate the stability properties of
certain warped product Einstein manifolds. We characterize stability of these
metrics in terms of an eigenvalue condition of the Einstein operator on the
base manifold. In particular, we prove that all complete manifolds carrying
imaginary Killing spinors are strictly stable...
In this paper, an obstruction against the integrability of certain
infinitesimal solitonic deformations is given. Using this obstruction, we show
that the complex projective spaces of even complex dimension are rigid as Ricci
solitons although they have infinitesimal solitonic deformations.
We give a concise proof of nonlinear stability for a large class of solutions
to the Einstein equations with a positive cosmological constant and compact
spatial topology, where the spatial metric is Einstein with either positive or
negative Einstein constant. The proof uses the CMC Einstein flow and stability
follows by an energy argument. We prov...
We study infinitesimal Einstein deformations on compact flat manifolds and on
product manifolds. Moreover, we prove refinements of results by Koiso and
Bourguignon which yield obstructions on the existence of infinitesimal Einstein
deformations under certain curvature conditions.
This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second...
We consider the volume-normalized Ricci flow close to compact shrinking Ricci
solitons. We show that if a compact Ricci soliton $(M,g)$ is a local maximum of
Perelman's shrinker entropy, any normalized Ricci flow starting close to it
exists for all time and converges towards a Ricci soliton. If $g$ is not a
local maximum of the shrinker entropy, we...
We prove dynamical stability and instability theorems for compact Einstein metrics under the Ricci flow. We give a nearly complete charactarization of dynamical stability and instability in terms of the conformal Yamabe invariant and the Laplace spectrum. In particular, we prove dynamical stability of some classes of Einstein manifolds for which it...