## 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 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

A Calder\'on projector for an elliptic operator $P$ on a manifold with boundary $X$ is a projection from general boundary data to the set of boundary data of solutions $u$ of $Pu=0$. Seeley proved in 1966 that for compact $X$ and for $P$ uniformly elliptic up to the boundary there is a Calder\'on projector which is a pseudodifferential operator on $\partial X$. We generalize this result to the setting of fibred cusp operators, a class of elliptic operators on certain non-compact manifolds having a special fibred structure at infinity.

This applies, for example, to the Laplacian on certain locally symmetric spaces

or on particular singular spaces, such as a domain with cusp singularity or the complement of two touching smooth strictly convex domains in Euclidean space. Our main technical tool is the $\phi$-pseudodifferential calculus introduced by Mazzeo and Melrose.

In our presentation we provide a setting that may be useful for doing analogous constructions for other types of singularities.

**Related project(s):****13**Analysis on spaces with fibred cusps**49**Analysis on spaces with fibred cusps II

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 construct secondary cup and cap products on coarse (co-)homology theories from given cross and slant products. They are defined for coarse spaces relative to weak generalized controlled deformation retracts.

On ordinary coarse cohomology, our secondary cup product agrees with a secondary product defined by Roe. For coarsifications of topological coarse (co-)homology theories, our secondary cup and cap products correspond to the primary cup and cap products on Higson dominated coronas via transgression maps. And in the case of coarse $\mathrm{K}$-theory and -homology, the secondary products correspond to canonical primary products between the $\mathrm{K}$-theories of the stable Higson corona and the Roe algebra under assembly and co-assembly.

**Related project(s):****10**Duality and the coarse assembly map**78**Duality and the coarse assembly map 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

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

We study the structure of the branch set of solutions to Plateau's problem in metric spaces satisfying a quadratic isoperimetric inequality. In our first result, we give examples of spaces with isoperimetric constant arbitrarily close to the Euclidean isoperimetric constant $(4\pi)^{−1}$ for which solutions have large branch set. This complements recent results of Lytchak--Wenger and Stadler stating, respectively, that any space with Euclidean isoperimetric constant is a CAT(0) space and solutions to Plateau's problem in a CAT(0) space have only isolated branch points. We also show that any planar cell-like set can appear as the branch set of a solution to Plateau's problem. These results answer two questions posed by Lytchak and Wenger. Moreover, we investigate several related questions about energy-minimizing parametrizations of metric disks: when such a map is quasisymmetric, when its branch set is empty, and when it is unique up to a conformal diffeomorphism.

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

In this paper, we study curve shortening flow on Riemann surfaces with singular metrics. It turns out that this flow is governed by a degenerate quasilinear parabolic equation. Under natural geometric assumptions, we prove short-time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and prove some collapsing and convergence results.

**Related project(s):****23**Spectral geometry, index theory and geometric flows on singular spaces**30**Nonlinear evolution equations on singular manifolds

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

We present an Eilenberg--Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories with already existing or newly introduced equivariance fit into this setup.

Furthermore, a new and more flexible notion of coarse homotopy is given which is more in the spirit of topological homotopies. Some, but not all, coarse (co-)homology theories are even invariant under these new homotopies. They also led us to a meaningful concept of topological actions of locally compact groups on coarse spaces.

**Related project(s):****10**Duality and the coarse assembly map

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

We study $\ell^2$ Betti numbers, coherence, and virtual fibring of random groups in the few-relator model. In particular, random groups with negative Euler characteristic are coherent, have $\ell^2$ homology concentrated in dimension 1, and embed in a virtually free-by-cyclic group with high probability. Similar results are shown with positive probability in the zero Euler characteristic case.

**Related project(s):****8**Parabolics and invariants

We show that a finitely generated residually finite rationally solvable (or RFRS) group *G* is virtually fibred, in the sense that it admits a virtual surjection to $\mathbb Z$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of *G* vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of 3-manifolds.

Journal | J. Amer. Math. Soc |

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**Related project(s):****8**Parabolics and invariants

We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of Sikorav.

We offer several applications: we reprove Thurston's theorem on the existence of a polytope controlling the BNS invariants of a 3-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincare duality groups of type F in dimension 3 and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl.

We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the *L*2-torsion polytope of Friedl-Lueck is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl-Lueck-Tillmann.

Journal | Invent. Math. |

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**Related project(s):****8**Parabolics and invariants

For every Lie group *G*, we compute the maximal *n* such that an *n*-fold product of nonabelian free groups embeds into *G*.

**Related project(s):****18**Analytic L2-invariants of non-positively curved spaces