## 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“.

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

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

When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules.

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

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

We introduce coordinates on the space of Lagrangian decorated and framed representations of the fundamental group of a surface with punctures into the symplectic group Sp(2n,R). These coordinates provide a non-commutative generalization of the parametrizations of the spaces of representations into SL(2,R) given by Thurston, Penner, and Fock-Goncharov. With these coordinates, the space of framed symplectic representations provides a geometric realization of the non-commutative cluster algebras introduced by Berenstein-Retakh. The locus of positive coordinates maps to the space of decorated maximal representations. We use this to determine the homotopy type of the space of decorated maximal representations, and its homeomorphism type when n=2

**Related project(s):****28**Rigidity, deformations and limits of maximal representations

We show that, for each $n\geqslant 1$, there exist infinitely many spin and non-spin diffeomorphism types of closed, smooth, simply-connected $(n+4)$-manifolds with a smooth, effective action of a torus $T^{n+2}$ and a metric of positive Ricci curvature invariant under a $T^{n}$-subgroup of $T^{n+2}$. As an application, we show that every closed, smooth, simply-connected $5$- and $6$-manifold admitting a smooth, effective torus action of cohomogeneity two supports metrics with positive Ricci curvature invariant under a circle or $T^2$-action, respectively.

Journal | Proc. Amer. Math. Soc. |

Volume | In press. |

Link to preprint version |

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

We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus Central Limit Theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case.

**Related project(s):****11**Topological and equivariant rigidity in the presence of lower curvature bounds**15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

Abstract: For an immersed Lagrangian submanifold, let \check{A} be the Lagrangian trace-free second fundamental form. In this note we consider the equation \nabla^*T=0 on Lagrangian surfaces immersed in \mathbb{C}^2, where T=-2\nabla^*(\check{A}\lrcornerω), and we prove a gap theorem for the Whitney sphere as a solution to this equation.

**Related project(s):****22**Willmore functional and Lagrangian surfaces

In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.

**Related project(s):****22**Willmore functional and Lagrangian surfaces

In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. This solves completely a long-standing open problem. In the proof one of crucial ingredients is a new Minkowski type formula. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.

Journal | Math. Ann. |

Volume | 374 |

Pages | 1845--1882 |

Link to preprint version | |

Link to published version |

**Related project(s):****22**Willmore functional and Lagrangian surfaces

In this note, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the free boundary hypersurfaces and capillary hypersurfaces. Then we present some interesting minimal Lagrangian submanifolds examples satisfying this boundary condition and we prove a Lagrangian version of Nitsche (or Hopf) type theorem. Some problems are proposed at the end of this note.

**Related project(s):****22**Willmore functional and Lagrangian surfaces

We study immersed surfaces in R3 which are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane orthogonally along the boundary, and when the boundary is contained in a line. In both cases we derive weak forms of the resulting free boundary conditions and prove regularity by reflection.

**Related project(s):****22**Willmore functional and Lagrangian surfaces