## Prof. Dr. Boris Vertman

### Project leader

Carl-von-Ossietzky-Universität Oldenburg

E-mail: boris.vertman(at)uni-oldenburg.de

Telephone: +49 441 798-3234

Homepage: https://www.uni-oldenburg.de/boris-vertm…

## Project

**23**Spectral geometry, index theory and geometric flows on singular spaces
**64**Spectral geometry, index theory and geometric flows on singular spaces II

## Publications within SPP2026

We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibred boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibred boundary (aka $\phi$-) pseudodifferential operator when the resolvent parameter tends to zero.

This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibres are points. The new feature in the case of non-trivial fibres is that the resolvent has different asymptotic behavior on the subspace of forms that are fibrewise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate 'split' pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on $\phi$-manifolds.

**Related project(s):****13**Analysis on spaces with fibred cusps**23**Spectral geometry, index theory and geometric flows on singular spaces**49**Analysis on spaces with fibred cusps II

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 consider a generalized Dirac operator on a compact stratified space with an iterated cone-edge metric. Assuming a spectral Witt condition, we prove its essential self-adjointness and identify its domain and the domain of its square with weighted edge Sobolev spaces. This sharpens previous results where the minimal domain is shown only to be a subset of an intersection of weighted edge Sobolev spaces. Our argument does not rely on microlocal techniques and is very explicit. The novelty of our approach is the use of an abstract functional analytic notion of interpolation scales. Our results hold for the Gauss-Bonnet and spin Dirac operators satisfying a spectral Witt condition.

Journal | JOURNAL OF SPECTRAL THEORY |

Volume | Volume 8, Issue 4, 2018, pp. 1295–1348 |

Link to preprint version | |

Link to published version |

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

Let (M,g) be a compact smoothly stratified pseudomanifold with an iterated cone-edge metric satisfying a spectral Witt condition. Under these assumptions the Hodge-Laplacian Δ is essentially self-adjoint. We establish the asymptotic expansion for the resolvent trace of Δ. Our method proceeds by induction on the depth and applies in principle to a larger class of second-order differential operators of regular-singular type, e.g., Dirac Laplacians. Our arguments are functional analytic, do not rely on microlocal techniques and are very explicit. The results of this paper provide a basis for studying index theory and spectral invariants in the setting of smoothly stratified spaces and in particular allow for the definition of zeta-determinants and analytic torsion in this general setup.

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