## 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 associate a flow $\phi$ to a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi$ always admits a dominated splitting and identify special cases in which $\phi$ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$.

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

Given a generic stable strongly parabolic $SL(2,\mathbb{C})$-Higgs bundle

$(\mathcal{E}, \varphi)$, we describe the family of harmonic metrics $h_t$ for

the ray of Higgs bundles $(\mathcal{E}, t \varphi)$ for $t\gg0$ by perturbing

from an explicitly constructed family of approximate solutions

$h_t^{\mathrm{app}}$. We then describe the natural hyperK\"ahler metric on

$\mathcal{M}$ by comparing it to a simpler "semi-flat" hyperK\"ahler metric. We

prove that $g_{L^2} - g_{\mathrm{sf}} = O(\mathrm{e}^{-\gamma t})$ along a

generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture.

Our results extend to weakly parabolic $SL(2,\mathbb{C})$-Higgs bundles as

well.

In the case of the four-puncture sphere, we describe the moduli space and

metric more explicitly. In this case, we prove that the hyperk\"ahler metric is

ALG and show that the rate of exponential decay is the conjectured optimal one,

$\gamma=4L$, where $L$ is the length of the shortest geodesic on the base curve

measured in the singular flat metric $|\mathrm{det}\, \varphi|$.

Pages | 73 pages |

Link to preprint version |

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

Let X be a compact Calabi-Yau 3-fold, and write \(\mathcal{M}, \overline{\mathcal{M}}\) for the moduli stacks of objects in coh(X) and the derived category D^b coh(X). There are natural line bundles \(K_{\mathcal{M}} \to \mathcal{M}, K_{\overline{\mathcal{M}}} \to \overline{\mathcal{M}}\) analogues of canonical bundles. Orientation data is an isomorphism class of square root line bundles \(K_{\mathcal{M}}^{1/2}, K_{\overline{\mathcal{M}}}^{1/2}\), satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman in their theory of motivic Donaldson-Thomas invariants, and is also important in categorifying Donaldson-Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi-Yau 3-folds X, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds X that admit a spin smooth projective compactification.

**Related project(s):****33**Gerbes in renormalization and quantization of infinite-dimensional moduli spaces

The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [2016] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, à la Mantoulidis and Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of an ad-hoc notion of charged Barnik mass for suitable charged minimal Bartnik data.

Journal | ADV. THEOR. MATH. PHYS |

Volume | 23 (0) |

Pages | 1951--1980 |

Link to preprint version |

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

Relying on the theory of agrarian invariants introduced in previous work, we solve a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, when the groups are additionally torsion-free, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Lück-Tillmann.

Journal | To appear in J. Lond. Math. Soc. |

Link to preprint version |

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

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.

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

We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of such approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy equivalence relations --- such as the so-called thin homotopy --- and the resulting interpretation of gauge fields as group homomorphisms to a Lie group *G* satisfying a suitable smoothness condition, encoding the holonomy of a gauge orbit of smooth connections on a principal *G*-bundle. We also prove several structural results on thin homotopy, and in particular we clarify the difference between thin equivalence and retrace equivalence for piecewise-smooth based loops on a smooth manifold, which are often used interchangeably in the physics literature. We conclude by listing a set of questions on topological and functional analytic aspects of groups of based loops, which we consider to be fundamental to establish a rigorous differential geometric foundation of the holonomy formulation of gauge theory.

Pages | 1-20 |

Link to preprint version |

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

Parallel transport as dictated by a gauge field determines a collection of local reference systems. Comparing local reference systems in overlapping regions leads to an ensemble of algebras of relational kinematical observables for gauge theories including general relativity. Using an auxiliary cellular decomposition, we propose a discretization of the gauge field based on a decimation of the mentioned ensemble of kinematical observables. The outcome is a discrete ensemble of local subalgebras of 'macroscopic observables' characterizing a measuring scale. A set of evaluations of those macroscopic observables is called an extended lattice gauge field because it determines a *G*-bundle over *M* (and over submanifolds of *M* that inherit a cellular decomposition) together with a lattice gauge field over an embedded lattice. A physical observable in our algebra of macroscopic observables is constructed. An initial study of aspects of regularization and coarse graining, which are special to this description of gauge fields over a combinatorial base, is presented. The physical relevance of this extension of ordinary lattice gauge fields is discussed in the context of quantum gravity.

Journal | Classical and Quantum Gravity |

Publisher | Inst. Phys. |

Volume | 36, no. 23 |

Link to preprint version | |

Link to published version |

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

We provide a unified treatment of several results concerning full groups of ample groupoids and paradoxical decompositions attached to them. This includes a criterion for the full group of an ample groupoid being amenable as well as comparison of its orbit, Koopman and groupoid-left-regular representations. Besides that, we unify several recent results about paradoxicality in semigroups and groupoids, relating embeddings of Thompson's group V into full groups of ample étale groupoids.

**Related project(s):****2**Asymptotic geometry of sofic groups and manifolds

We show a rigidity result for subfactors that are normalized by a representation of a lattice Γ in a higher rank simple Lie group with trivial center into a finite factor. This implies that every subfactor of *L*Γ which is normalized by the natural copy of Γ is trivial or of finite index.

**Related project(s):****2**Asymptotic geometry of sofic groups and manifolds

We give a characterization of those Alexandrov spaces admitting a cohomogeneity one action of a compact connected Lie group G for which the action is Cohen--Macaulay. This generalizes a similar result for manifolds to the singular setting of Alexandrov spaces where, in contrast to the manifold case, we find several actions which are not Cohen--Macaulay. In fact, we present results in a slightly more general context. We extend the methods in this field by a conceptual approach on equivariant cohomology via rational homotopy theory using an explicit rational model for a double mapping cylinder.

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

Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces ξm,k of high genus by deforming the corresponding DPW potential. As a byproduct, we obtain for fixed m estimates on the area of ξm,k in terms of their genus g=mk≫1.

Pages | 31 |

Link to preprint version |

**Related project(s):****16**Minimizer of the Willmore energy with prescribed rectangular conformal class

We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below 8π. In particular, every constrained Willmore torus with Willmore energy below 8π and non-rectangular conformal class is non-degenerated.

Pages | 19 |

Link to preprint version |

**Related project(s):****16**Minimizer of the Willmore energy with prescribed rectangular conformal class

We show that the of 2-lobed Delaunay tori are stable as constrained Willmore surfaces in the 3-sphere.

Pages | 14 |

Link to preprint version |

**Related project(s):****16**Minimizer of the Willmore energy with prescribed rectangular conformal class

In a previous work, a six-parameter family of highly connected 7-manifolds which admit an $\mathrm{SO}(3)$-invariant metric of non-negative sectional curvature was constructed. Each member of this family is the total space of a Seifert fibration with generic fibre $\mathbb S^3$ and, in particular, has the cohomology of an $\mathbb S^3$-bundle over $\mathbb S^4$. In the present article, the linking form of these manifolds is computed and used to demonstrate that the family contains infinitely many manifolds which are not even homotopy equivalent to an $\mathbb S^3$-bundle over $\mathbb S^4$, the first time that any such spaces have been shown to admit non-negative sectional curvature.

**Related project(s):****4**Secondary invariants for foliations**11**Topological and equivariant rigidity in the presence of lower curvature bounds

In this short note we observe the existence of free, isometric actions of finite cyclic groups on a family of 2-connected 7-manifolds with non-negative sectional curvature. This yields many new examples including fake, and possible exotic, lens spaces.

**Related project(s):****4**Secondary invariants for foliations**11**Topological and equivariant rigidity in the presence of lower curvature bounds

We explain how the construction of the real numbers using quasimorphisms can be transformed into a general method to construct the completion of a field with respect to an absolute value.

Journal | P-Adic Numbers Ultrametric Anal. Appl. |

Volume | 11 |

Pages | 335 - 337 |

Link to preprint version | |

Link to published version |

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

We define a variant of Benjamini-Schramm convergence for finite simplicial complexes with the action of a fixed finite group G which leads to the notion of random rooted simplicial G-complexes. For every random rooted simplicial G-complex we define a corresponding *ℓ*2-homology and the *ℓ*2-multiplicity of an irreducible representation of G in the homology. The *ℓ*2-multiplicities generalize the *ℓ*2-Betti numbers and we show that they are continuous on the space of sofic random rooted simplicial G-complexes. In addition, we study induction of random rooted complexes and discuss the effect on *ℓ*2-multiplicities.

Pages | 20 |

Link to preprint version |

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

Several formulas for computing coarse indices of twisted Dirac type operators are introduced. One type of such formulas is by composition product in \(E\)-theory. The other type is by module multiplications in *\(K\)*-theory, which also yields an index theoretic interpretation of the duality between Roe algebra and stable Higson corona.

Journal | Journal of Topology and Analysis |

Publisher | World Scientific Publishing |

Volume | 11(4) |

Pages | 823-873 |

Link to preprint version | |

Link to published version |

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