## 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“.

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

We define a functional \({\cal J}(h)\) for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that \({\cal J}(h)\) is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We derive a formula relating \({\cal J}(h)\) and another functional \({\cal I}(h)\), closely related to the Yang-Mills-Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of \({\cal J}(h)\), which is expressed as a certain \(L^{2}\)-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of \({\cal J}(h)\) if and only if the corresponding Hitchin-Simpson mean curvature is parallel with respect to the Hitchin-Simpson connection.

Journal | International Journal of Geometric Methods in Modern Physics |

Publisher | World Scientific |

Volume | 17(13) |

Line | art. no. 2050200 |

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**Related project(s):****32**Asymptotic geometry of the Higgs bundle moduli space**69**Wall-crossing and hyperkähler geometry of moduli spaces

We construct smooth bundles with base and fiber products of two spheres whose total spaces have nonvanishing A-hat-genus. We then use these bundles to locate nontrivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all Spin manifolds of dimension 6 or at least 10, which admit such a metric and are a connected sum of some manifold and \(S^n\times S^n\) or \(S^n\times S^{n+1}\), respectively. We also construct manifolds M whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order that lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of M.

Journal | International Mathematics Research Notices |

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**Related project(s):****9**Diffeomorphisms and the topology of positive scalar curvature**15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.

**Related project(s):****36**Cohomogeneity, curvature, cohomology

In each dimension $4k+1\geq 9$, we exhibit infinite families of closed manifolds with fundamental group $\mathbb Z_2$ for which the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with this property were known before only in dimension $5$ and dimensions $4k+3\geq 7$.

**Related project(s):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds**52**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We show that a properly convex projective structure \(\mathfrak{p}\) on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if \(\mathfrak{p}\) is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that \(\mathfrak{p}\) admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this non-linear PDE into a transport equation, we obtain our result by applying methods from geometric inverse problems. In particular, we use an extension of a remarkable \(L^2\)-energy identity known as Pestov's identity to prove a vanishing theorem for the relevant transport equation.

Journal | Analysis & PDE |

Publisher | Mathematical Sciences Publishers |

Volume | 13 |

Pages | 1073--1097 |

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**Related project(s):****26**Projective surfaces, Segre structures and the Hitchin component for PSL(n,R)

Let *M* be a simply connected spin manifold of dimension at least six which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on *M* has non-trivial higher homotopy groups.

Moreover, denote by *\mathcal{M}^*+_0(*M*) the moduli space of positive scalar cuvature metrics on *M* associated to the group of orientation-preserving diffeomorphisms of *M*. We show that if *M* belongs to a certain class of manifolds which includes (2*n*−2)-connected (4*n*−2)-dimensional manifolds, then the fundamental group of *\mathcal{M}^*+_0(*M*) is non-trivial.

Journal | Int. Math. Res. Not. IMRN |

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**Related project(s):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

The main result of this paper is that when *M*_0, *M_*1 are two simply connected spin manifolds of the same dimension *d*≥5 which both admit a metric of positive scalar curvature, the spaces *\mathcal{M}^*+(*M*0) and *\mathcal{M}^*+(*M_*1) of such metrics are homotopy equivalent. This supersedes a previous result of Chernysh and Walsh which gives the same conclusion when *M_*0 and *M*_1 are also spin cobordant.

We also prove an analogous result for simply connected manifolds which do not admit a spin structure; we need to assume that *d*≠8 in that case.

**Related project(s):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of \(\mathbb{R}\) or \(\mathbb{C}\). We prove an adelic version of superrigidity which implies that two such commensurability classes define the same profinite commensurability class if and only if the algebraic groups are adelically isomorphic. We discuss noteworthy consequences on profinite rigidity questions.

**Related project(s):****58**Profinite perspectives on l2-cohomology

We investigate which higher rank simple Lie groups admit profinitely but not abstractly commensurable lattices. We show that no such examples exist for the complex forms of type \(E_8\), \(F_4\), and \(G_2\). In contrast, there are arbitrarily many such examples in all other higher rank Lie groups, except possibly \(\mathrm{SL}_{2n+1}(\mathbb{R})\), \(\mathrm{SL}_{2n+1}(\mathbb{C})\), \(\mathrm{SL}_n(\mathbb{H})\), or groups of type~\(E_6\).

**Related project(s):****58**Profinite perspectives on l2-cohomology

We define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on p-adic simplicial volumes. We investigate the dependence on the prime and establish homology bounds in terms of p-adic simplicial volumes. As the main examples, we compute the weightless and p-adic simplicial volumes of surfaces. This is based on an alternative way to calculate classical simplicial volume of surfaces without hyperbolic straightening and shows that surfaces satisfy mod p and p-adic approximation of simplicial volume.

Journal | Glasgow Math. Journal |

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**Related project(s):****58**Profinite perspectives on l2-cohomology

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

A geodesic *γ* in an abstract reflection space *X* (in the sense of Loos, without any assumption of differential structure) is known to canonically admits an action of a 1-parameter subgroup of the group of transvections of *X*. In this article, we prove an analog of this result stating that, if *X* contains an embedded hyperbolic plane, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL(2,R). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact type embedded in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac-Moody symmetric space G/K for an algebraically simply connected two-spherical Kac-Moody group G satisfies a universal property similar to the universal property that the group G satisfies itself.

Journal | Adv. Geometry |

Volume | 20 |

Pages | 499-506 |

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**Related project(s):****61**At infinity of symmetric spaces

In the present article we introduce and study a class of topological reflection spaces that we call Kac-Moody symmetric spaces. These generalize Riemannian symmetric spaces of non-compact type. We observe that in a non-spherical Kac-Moody symmetric space there exist pairs of points that do not lie on a common geodesic; however, any two points can be connected by a chain of geodesic segments. We moreover classify maximal flats in Kac-Moody symmetric spaces and study their intersection patterns, leading to a classification of global and local automorphisms. Unlike Riemannian symmetric spaces, non-spherical non-affine irreducible Kac-Moody symmetric spaces also admit an invariant causal structure. For causal and anti-causal geodesic rays with respect to this structure we find a notion of asymptoticity, which allows us to define a future and past boundary of such Kac-Moody symmetric space. We show that these boundaries carry a natural polyhedral structure and are cellularly isomorphic to the halves of the geometric realization of the twin buildings of the underlying split real Kac-Moody group. We also show that every automorphism of the symmetric space is uniquely determined by the induced cellular automorphism of the future and past boundary. The invariant causal structure on a non-spherical non-affine irreducible Kac-Moody symmetric space gives rise to an invariant pre-order on the underlying space, and thus to a subsemigroup of the Kac-Moody group. We conclude that while in some aspects Kac-Moody symmetric spaces closely resemble Riemannian symmetric spaces, in other aspects they behave similarly to ordered affine hovels, their non-Archimedean cousins.

Journal | Münster J. Math. |

Volume | 13 |

Pages | 1-114 |

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**Related project(s):****61**At infinity of symmetric spaces

We construct explicit geometric models for moduli spaces of stable parabolic Higgs bundles on the Riemann sphere, in the case of rank two, four marked points, any degree, and arbitrary weights. The construction mechanism relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on carefully crafted spaces. These techniques are not exclusive to the case we examine. Therefore, this work elucidates a general approach to construct arbitrary moduli spaces of stable parabolic Higgs bundles in genus 0, which is encoded into the combinatorics of weight polytopes. Moreover, we present a comprehensive analysis of the geometric models' behavior under variation of weights and wall-crossing. This analysis is concentrated on their nilpotent cones, and is applicable to the study of the hyperkähler geometry of Hitchin metrics as gravitational instantons of ALG type.

**Related project(s):****69**Wall-crossing and hyperkähler geometry of moduli spaces

We prove a local version of the index theorem for Lorentzian Dirac-type operators on globally hyperbolic Lorentzian manifolds with Cauchy boundary. In case the Cauchy hypersurface is compact we do not assume self-adjointness of the Dirac operator on the spacetime or the associated elliptic Dirac operator on the boundary. In this case integration of our local index theorem results in a generalization of previously known index theorems for globally hyperbolic spacetimes that allows for twisting bundles associated with non-compact gauge groups.

**Related project(s):****37**Boundary value problems and index theory on Riemannian and Lorentzian manifolds

Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.

**Related project(s):****37**Boundary value problems and index theory on Riemannian and Lorentzian manifolds**52**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

These are the refereed proceedings of the 2019 'Australian-German Workshop on Differential Geometry in the Large' which represented a cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks treating geometric evolution equations, structures on manifolds, non-negative curvature and topics in Kähler, Alexandrov and Sasaki geometry as well as differential topology.

Journal | London Mathematical Society Lecture Notes Series |

Publisher | Cambridge University Press |

Book | Differential Geometry in the Large |

Volume | 463 |

Pages | 398 |

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**Related project(s):****52**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak--Wenger, which satisfies a related maximality condition.

Journal | Calc. Var. Partial Differential Equations |

Volume | 59 |

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**Related project(s):****24**Minimal surfaces in metric spaces

Assume you are given a finite configuration $\Gamma$ of disjoint rectifiable Jordan curves in $\mathbb{R}^n$. The Plateau-Douglas problem asks whether there exists a minimizer of area among all compact surfaces of genus at most $p$ which span $\Gamma$. While the solution to this problem is well-known, the classical approaches break down if one allows for singular configurations $\Gamma$ where the curves are potentially non-disjoint or self-intersecting. Our main result solves the Plateau-Douglas problem for such potentially singular configurations. Moreover, our proof works not only in $\mathbb{R}^n$ but in general proper metric spaces. Thus we are also able to extend previously known existence results of Jürgen Jost as well as of the second author together with Stefan Wenger for regular configurations. In particular, existence is new for disjoint configurations of Jordan curves in general complete Riemannian manifolds. A minimal surface of fixed genus $p$ bounding a given configuration $\Gamma$ need not always exist, even in the most regular settings. Concerning this problem, we also generalize the approach for singular configurations via minimal sequences satisfying conditions of cohesion and adhesion to the setting of metric spaces.

**Related project(s):****24**Minimal surfaces in metric spaces