## Publications of SPP2026

On this site you find preprints and publications produced within the projects and with the support of the DFG priority programme „Geometry at Infinity“.

In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces \( \Sigma_R\) converging for \(R\searrow 0\) to a surface \( \Sigma_0\) with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration \(\mathcal M_d \to \Sigma_R\) gives rise to a graph-continuous Fredholm family, the index of it being stable when passing to the limit. We also report on similarities and differences between properties of the Hitchin fibration in this degeneration and in the limit of large Higgs fields as studied in Mazzeo et al. (Duke Math. J. 165(12):2227–2271, 2016).

Journal | Abh. Math. Semin. Univ. Hambg. |

Publisher | Springer Berlin Heidelberg |

Volume | 86 |

Pages | 189--201 |

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**Related project(s):****32**Asymptotic geometry of the Higgs bundle moduli space

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the *η*-invariant of the Cauchy hypersurfaces.

Journal | Commun. Math. Phys. |

Publisher | Springer |

Volume | 347 |

Pages | 703-721 |

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**Related project(s):****5**Index theory on Lorentzian manifolds

We associate to each stable Higgs pair \((A_0,\Phi_0)\) on a compact Riemann surface *X* a singular limiting configuration \((A_\infty,\Phi_\infty)\), assuming that \(\det\Phi\) has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions \((A_t,\Phi_t) \) to Hitchin's equations which converge to this limiting configuration as \(t\to\infty\). This provides a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.

Journal | Duke Math. J. |

Publisher | Duke University Press |

Volume | 165 |

Pages | 2227-2271 |

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**Related project(s):****32**Asymptotic geometry of the Higgs bundle moduli space

An upper bound is obtained on the rank of a torus which can act smoothly and effectively on a smooth, closed, simply connected, rationally elliptic manifold. In the maximal-rank case, the manifolds admitting such actions are classified up to equivariant rational homotopy type.

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds

We present a new technique that employs partial differential equations in order to explicitly construct primitives in the continuous bounded cohomology of Lie groups. As an application, we prove a vanishing theorem for the continuous bounded cohomology of SL(2,R) in degree 4, establishing a special case of a conjecture of Monod.

Journal | Geometry & Topology |

Volume | 19 |

Pages | 3603–3643 |

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**Related project(s):****27**Invariants and boundaries of spaces

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.

Journal | Amer. J. Math (to appear) |

Publisher | John Hopkins Univ. Press |

Link to preprint version |

**Related project(s):****5**Index theory on Lorentzian manifolds

Let \(M^n, n \in \{4,5,6\}\), be a compact, simply connected *n*-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on \(M^n\) by a torus \(T^{n-2}\) is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.

Journal | Math. Z. |

Volume | 276 |

Pages | 133--152 |

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Link to published version |

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds