## 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“.

Let (Mi,gi)i∈N be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower dimensional Riemannian manifold (B,h) in the Gromov-Hausdorff topology. Lott showed that the spectrum converges to the spectrum of a certain first order elliptic differential operator D on B. In this article we give an explicit description of D. We conclude that D is self-adjoint and characterize the special case where D is the Dirac operator on B.

Journal | Annals of Global Analysis and Geometry |

Publisher | Springer |

Volume | 57 |

Pages | 121-151 |

Link to preprint version | |

Link to published version |

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

In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds in R*n*+*m*. We first provide, following Cabré \cite{Cabre2008}, an ABP proof of the relative isoperimetric inequality proved in Choe-Ghomi-Ritoré \cite{CGR07}, by generalizing ideas of restricted normal cones given in \cite{CGR06}. Then we prove a relative isoperimetric inequalities for minimal submanifolds in R*n*+*m*, which is optimal when the codimension *m*≤2. In other words we obtain a relative version of isoperimetric inequalities for minimal submanifolds proved recently by Brendle \cite{Brendle2019}. When the codimension *m*≤2, our result gives an affirmative answer to an open problem proposed by Choe in \cite{Choe2005}, Open Problem 12.6. As another application we prove an optimal logarithmic Sobolev inequality for free boundary submanifolds in the Euclidean space following a trick of Brendle in \cite{Brendle2019b}.

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

In this work we describe horofunction compactifications of metric spaces and finite dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, ultrapowers of the spaces at hand. The compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

**Related project(s):****20**Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings

In this paper we consider a Ricci de Turck flow of spaces with isolated conical singularities, which preserves the conical structure along the flow. We establish that a given initial regularity of Ricci curvature is preserved along the flow. Moreover under additional assumptions, positivity of scalar curvature is preserved under such a flow, mirroring the standard property of Ricci flow on compact manifolds. The analytic difficulty is the a priori low regularity of scalar curvature at the conical tip along the flow, so that the maximum principle does not apply. We view this work as a first step toward studying positivity of the curvature operator along the singular Ricci flow.

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry**23**Spectral geometry, index theory and geometric flows on singular spaces

We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $\mathfrak{p}$ and a volume form $\sigma$ on an oriented surface $M$ equip the total space of a certain disk bundle $Z \to M$ with a pair $(J_{\mathfrak{p}},\mathfrak{J}_{\mathfrak{p},\sigma})$ of almost complex structures. A conformal structure on $M$ corresponds to a section of $Z\to M$ and $\mathfrak{p}$ is metrisable by the metric $g$ if and only if $[g] : M \to Z$ is a pseudo-holomorphic curve with respect to $J_{\mathfrak{p}}$ and $\mathfrak{J}_{\mathfrak{p},dA_g}$.

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

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

We show \(R\)-sectoriality for the fractional powers of possibly non-invertible \(R\)-sectorial operators. Applications concern existence, uniqueness and maximal \(L^{q}\)-regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided and its relation to the local geometry is established. Our method extends the freezing-of-coefficients method to the case of non-local operators that are expressed as linear combinations of terms in the form of a product of a function and a fractional power of a local operator.

**Related project(s):****30**Nonlinear evolution equations on singular manifolds

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 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