## Prof. Dr. Bernd Ammann

### Project leader

Universität Regensburg

E-mail: ammann(at)berndammann.de

Telephone: +49 941 943-2769

Homepage: http://www.mathematik.uni-regensburg.de/…

## Project

**3**Geometric operators on a class of manifolds with bounded geometry
**35**Geometric operators on singular domains

## Publications within SPP2026

We show that the space introduced by Vasy in order to construct a pseudodifferential calculus adapted to the N-body problem can be obtained as the primitive ideal spectrum of one of the N-body algebras considered by Georgescu. In the process, we provide an alternative description of the iterated blow-up space of a manifold with corners with respect to a clean semilattice of adapted submanifolds (i.e. p-submanifolds). Since our constructions and proofs rely heavily on manifolds with corners and their submanifolds, we found it necessary to clarify the various notions of submanifolds of a manifold with corners.

**Related project(s):****3**Geometric operators on a class of manifolds with bounded geometry

We prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains. Our class of domains contains the class of domains with isolated oscillating conical singularities. Our results thus generalize the classical results of Kondratiev on domains with conical singularities. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier results on manifolds with boundary and bounded geometry.

Journal | Comptes Rendus Mathématique Sér. I 357 487-493 (2019) |

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Link to published version |

**Related project(s):****3**Geometric operators on a class of manifolds with bounded geometry

Let M be a smooth manifold with (smooth) boundary ∂M and bounded geometry and ∂<sub>D</sub>M ⊂ ∂M be an open and closed subset. We prove the well-posedness of the mixed Robin boundary value problem Pu = f in M, u = 0 on ∂<sub>D</sub>M, ∂<sup>P</sup><sub>ν</sub> u + bu = 0 on ∂M \ ∂<sub>D</sub>M under the following assumptions. First, we assume that P satisfies the strong Legendre condition (which reduces to the uniformly strong ellipticity condition in the scalar case) and that it has totally bounded coefficients (that is, that the coefficients of P and all their derivatives are bounded). Let ∂<sub>R</sub>M ⊂ ∂M \ ∂<sub>D</sub>M be the set where b≠ 0.

Journal | Rev. Roumaine Math. Pures Appl. 64 85-111 (2019) |

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**Related project(s):****3**Geometric operators on a class of manifolds with bounded geometry

Let M be a manifold with boundary and bounded geometry. We assume that M has "finite width", that is, that the distance from any point to the boundary is bounded uniformly. Under this assumption, we prove that the Poincaré inequality for vector valued functions holds on M. We also prove a general regularity result for uniformly strongly elliptic equations and systems on general manifolds with boundary and bounded geometry. By combining the Poincaré inequality with the regularity result, we obtain-as in the classical case-that uniformly strongly elliptic equations and systems are well-posed on M in Hadamard's sense between the usual Sobolev spaces associated to the metric. We also provide variants of these results that apply to suitable mixed Dirichlet-Neumann boundary conditions. We also indicate applications to boundary value problems on singular domains.

Journal | Mathematische Nachrichten |

Publisher | Wiley |

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**Related project(s):****3**Geometric operators on a class of manifolds with bounded geometry

We discuss a method to construct Dirac-harmonic maps developed by J. Jost, X. Mo and M. Zhu. The method uses harmonic spinors and twistor spinors, and mainly applies to Dirac-harmonic maps of codimension 1 with target spaces of constant sectional curvature. Before the present article, it remained unclear when the conditions of the theorems in the publication by Jost, Mo and Zhu were fulfilled. We show that for isometric immersions into spaceforms, these conditions are fulfilled only under special assumptions. In several cases we show the existence of solutions.

Journal | Lett. Math. Phys. |

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Link to published version |

**Related project(s):****3**Geometric operators on a class of manifolds with bounded geometry