## JProf. Dr. Klaus Kröncke

### Project leader

Universität Hamburg

E-mail: klaus.kroencke(at)uni-hamburg.de

Telephone: +49 40 42838-5183

Homepage: https://www.math.uni-hamburg.de/home/kro…

## Project

**21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry
**63**Uniqueness in mean curvature flow
**64**Spectral geometry, index theory and geometric flows on singular spaces II

## Publications within SPP2026

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 Einstein manifold

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry**64**Spectral geometry, index theory and geometric flows on singular spaces II

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 $p\in [1,\frac{n}{n-2})$, generalizing a result by Appleton.

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry

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.

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry

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, mirroring the standard property of Ricci flow on compact manifolds. The analytic difficulty is the a priori low regularity of scalar curvature at the conical tip along the flow, so that the maximum principle does not apply. We view this work as a first step toward studying positivity of the curvature operator along the singular Ricci flow.

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry**23**Spectral geometry, index theory and geometric flows on singular spaces

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold M×R, where M is asymptotically flat. If the initial hypersurface F⊂M×R is uniformly spacelike and asymptotic to M×{s} for some s∈R at infinity, we show that the mean curvature flow starting at F0 exists for all times and converges uniformly to M×{s} as t→∞.

Journal | recently accepted for publication at Journal of Geometric Analysis |

Link to preprint version |

**Related project(s):****23**Spectral geometry, index theory and geometric flows on singular spaces

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 monotone along the singular Ricci de Turck flow. We employ these entropies to show that in the singular setting, Ricci solitons are gradient and that steady or expanding Ricci solitons are Einstein.

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry**23**Spectral geometry, index theory and geometric flows on singular spaces

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, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied.

Journal | Calc. Var. Part. Differ. Eq. |

Publisher | Springer |

Volume | 58 |

Pages | 75 |

Link to preprint version | |

Link to published version |

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry**23**Spectral geometry, index theory and geometric flows on singular spaces

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 for ALE Calabi-Yau manifolds which implies that they are dynamically stable.

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry