Publications

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“.

all projects
  • all projects
  • 01Hitchin components for orbifolds
  • 02Asymptotic geometry of sofic groups and manifolds
  • 03Geometric operators on a class of manifolds with bounded geometry
  • 04Secondary invariants for foliations
  • 05Index theory on Lorentzian manifolds
  • 06Spectral Analysis of Sub-Riemannian Structures
  • 07Asymptotic geometry of moduli spaces of curves
  • 08Parabolics and invariants
  • 09Diffeomorphisms and the topology of positive scalar curvature
  • 10Duality and the coarse assembly map
  • 11Topological and equivariant rigidity in the presence of lower curvature bounds
  • 12Anosov representations and Margulis spacetimes
  • 13Analysis on spaces with fibred cusps
  • 14Boundaries of acylindrically hyperbolic groups and applications
  • 15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds
  • 16Minimizer of the Willmore energy with prescribed rectangular conformal class
  • 17Existence, regularity and uniqueness results of geometric variational problems
  • 18Analytic L2-invariants of non-positively curved spaces
  • 19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces
  • 20Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings
  • 21Stability and instability of Einstein manifolds with prescribed asymptotic geometry
  • 22Willmore functional and Lagrangian surfaces
  • 23Spectral geometry, index theory and geometric flows on singular spaces
  • 24Minimal surfaces in metric spaces
  • 25The Willmore energy of degenerating surfaces and singularities of geometric flows
  • 26Projective surfaces, Segre structures and the Hitchin component for PSL(n,R)
  • 27Invariants and boundaries of spaces
  • 28Rigidity, deformations and limits of maximal representations
  • 29Curvature flows without singularities
  • 30Nonlinear evolution equations on singular manifolds
  • 31Solutions to Ricci flow whose scalar curvature is bounded in Lp.
  • 32Asymptotic geometry of the Higgs bundle moduli space
  • 33Gerbes in renormalization and quantization of infinite-dimensional moduli spaces
  • 34Asymptotic geometry of sofic groups and manifolds II
  • 35Geometric operators on singular domains
  • 36Cohomogeneity, curvature, cohomology
  • 37Boundary value problems and index theory on Riemannian and Lorentzian manifolds
  • 38Geometry of surface homeomorphism groups
  • 39Geometric invariants of discrete and locally compact groups
  • 40Construction of Riemannian manifolds with scalar curvature constraints and applications to general relativity
  • 41Geometrically defined asymptotic coordinates in general relativity
  • 42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds
  • 43Singular Riemannian foliations and collapse
  • 44Actions of mapping class groups and their subgroups
  • 45Macroscopic invariants of manifolds
  • 46Ricci flows for non-smooth spaces, monotonic quantities, and rigidity
  • 47Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces
  • 48Profinite and RFRS groups
  • 49Analysis on spaces with fibred cusps II
  • 50Probabilistic and spectral properties of weighted Riemannian manifolds with Kato bounded Bakry-Emery-Ricci curvature
  • 51The geometry of locally symmetric manifolds via natural maps
  • 52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II
  • 53Gauge-theoretic methods in the geometry of G2-manifolds
  • 54Cohomology of symmetric spaces as seen from infinity
  • 55New hyperkähler spaces from the the self-duality equations
  • 56Large genus limit of energy minimizing compact minimal surfaces in the 3-sphere
  • 57Existence, regularity and uniqueness results of geometric variational problems II
  • 58Profinite perspectives on l2-cohomology
  • 59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces
  • 60Property (T)
  • 61At infinity of symmetric spaces
  • 62A unified approach to Euclidean buildings and symmetric spaces of noncompact type
  • 63Uniqueness in mean curvature flow
  • 64Spectral geometry, index theory and geometric flows on singular spaces II
  • 65Resonances for non-compact locally symmetric spaces
  • 66Minimal surfaces in metric spaces II
  • 67Asymptotics of singularities and deformations
  • 68Minimal Lagrangian connections and related structures
  • 69Wall-crossing and hyperkähler geometry of moduli spaces
  • 70Spectral theory with non-unitary twists
  • 71Rigidity, deformations and limits of maximal representations II
  • 72Limits of invariants of translation surfaces
  • 74Rigidity, stability and deformations in nearly parallel G2-geometry
  • 75Solutions to Ricci flow whose scalar curvature is bounded in L^p II
  • 76Singularities of the Lagrangian mean curvature flow
  • 77Asymptotic geometry of the Higgs bundle moduli space II
  • 78Duality and the coarse assembly map II
  • 79Alexandrov geometry in the light of symmetry and topology

For a smooth manifold \(M\), possibly with boundary and corners, and a Lie group \(G\), we consider a suitable description of gauge fields in terms of parallel transport, as groupoid homomorphisms from a certain path groupoid in \(M\) to \(G\).  Using a cotriangulation \(\mathscr{C}\) of \(M\), and collections of finite-dimensional families of paths relative to \(\mathscr{C}\), we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains further local topological information sufficient to reconstruct a principal \(G\)-bundle on \(M\) up to equivalence. The space of ELG fields of a given pair \((M,\mathscr{C})\) is a covering for the space of fields in Lattice Gauge Theory, whose connected components parametrize equivalence classes of principal \(G\)-bundles on \(M\). We give a criterion to determine when ELG fields over different cotriangulations define equivalent bundles.

 

JournalAdvances in Theoretical and Mathematical Physics
PublisherInternational Press
Volume23(8)
Pages2207 – 2254
Link to preprint version
Link to published version

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

Photon surfaces are timelike, totally umbilic hypersurfaces of Lorentzian spacetimes. In the first part of this paper, we locally characterize all possible photon surfaces in a class of static, spherically symmetric spacetimes that includes Schwarzschild, Reissner--Nordström, Schwarzschild-anti de Sitter, etc., in n+1dimensions. In the second part, we prove that any static, vacuum, "asymptotically isotropic" n+1-dimensional spacetime that possesses what we call an "equipotential" and "outward directed" photon surface is isometric to the Schwarzschild spacetime of the same (necessarily positive) mass, using a uniqueness result by the first named author.

 

Journalaccepted in JMP
Link to preprint version

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

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):
30Nonlinear 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.

 

JournalADV. THEOR. MATH. PHYS
Volume23 (0)
Pages1951--1980
Link to preprint version

Related project(s):
5Index 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.

 

JournalTo appear in J. Lond. Math. Soc.
Link to preprint version

Related project(s):
8Parabolics 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→∞.

 

Journalrecently accepted for publication at Journal of Geometric Analysis
Link to preprint version

Related project(s):
23Spectral 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.

 

Related project(s):
32Asymptotic 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.

 

JournalClassical and Quantum Gravity
PublisherInst. Phys.
Volume36, no. 23
Link to preprint version
Link to published version

Related project(s):
32Asymptotic 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):
2Asymptotic 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 LΓ which is normalized by the natural copy of Γ is trivial or of finite index.

 

Related project(s):
2Asymptotic 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):
15Spaces 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.

 

Related project(s):
16Minimizer 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.

 

Related project(s):
16Minimizer 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):
4Secondary invariants for foliations11Topological 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):
4Secondary invariants for foliations11Topological 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.

 

JournalP-Adic Numbers Ultrametric Anal. Appl.
Volume11
Pages335 - 337
Link to preprint version
Link to published version

Related project(s):
18Analytic 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 2-homology and the 2-multiplicity of an irreducible representation of G in the homology. The 2-multiplicities generalize the 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 2-multiplicities.

 

Related project(s):
18Analytic 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.

 

JournalJournal of Topology and Analysis
PublisherWorld Scientific Publishing
Volume11(4)
Pages823-873
Link to preprint version
Link to published version

Related project(s):
10Duality 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):
28Rigidity, deformations and limits of maximal representations

This website uses cookies

By using this page, browser cookies are set. Read more ›