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“.

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  • 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
  • 73Geometric Chern characters in p-adic equivariant K-theory
  • 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
  • 80Nonlocal boundary problems: Index theory and semiclassical asymptotics

In this article we study the stability problem for the Einstein metrics on

Sasaki Einstein and on complete nearly parallel G2 manifolds. In the Sasaki

case we show linear instability if the second Betti number is positive.

Similarly we prove that nearly parallel G2 manifolds with positive third

Betti number are linearly unstable. Moreover, we prove linear instability

for the Berger space SO(5)/SO(3)_irr which is a 7-dimensional homology

sphere with a proper nearly parallel G2 structure.

 

 

Journalto appear in Int. J. Math.
Link to preprint version

Related project(s):
74Rigidity, stability and deformations in nearly parallel G2-geometry

In this article we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation, and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.

 

Journal J. Geom. Anal. (2022) 32:137
Link to preprint version

Related project(s):
74Rigidity, stability and deformations in nearly parallel G2-geometry

We study the spectral properties of the Laplace operator associated to a hyperbolic surface in the presence of a unitary representation of the fundamental group. Following the approach by Guillopé and Zworski, we establish a factorization formula for the twisted scattering determinant and describe the behavior of the scattering matrix in a neighborhood of \(1/2\).

 

Related project(s):
70Spectral theory with non-unitary twists

We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.

 

Related project(s):
70Spectral theory with non-unitary twists

Using non-Abelian Hodge theory for parabolic Higgs bundles,

 

we construct infinitely many non-congruent  hyperbolic affine spheres modeled on a thrice-punctured sphere with monodromy in ${\bf SL}_3(\Z)$. These give rise to non-isometric semi-flat Calabi--Yau metrics on special Lagrangian torus bundles over an open ball in $\R^{3}$  with a Y-vertex deleted, thereby answering a question raised by Loftin, Yau, and Zaslow.

 

Related project(s):
55New hyperkähler spaces from the the self-duality equations

The Lawson surfaces $\xi_{1,g}$ of genus $g$ are constructed by rotating and reflecting the Plateau solution $f_t$ with respect to a particular geodesic $4$-gon $\Gamma_t$ along its boundary, where $t= \tfrac{1}{2g+2}$ is an angle of  $\Gamma_t$.  In this paper we combine the existence and regularity of the Plateau solution $f_t$ in $t \in (0, \tfrac{1}{4})$ with topological information about the moduli space of Fuchsian systems on the 4-puncture sphere to obtain existence of a Fuchsian DPW potential $\eta_t$ for every $f_t$ with $t\in(0, \tfrac{1}{4}]$. Moreover, the coefficients of $\eta_t$ are shown to depend real analytically on $t$. This implies that the Taylor approximation of the DPW potential $\eta_t$ and of the area obtained at $t=0$ found in \cite{HHT2} determines these quantities for  all $\xi_{1,g}$. In particular, this leads to an algorithm to conformally parametrize all Lawson surfaces $\xi_{1,g}$.

 

Related project(s):
55New hyperkähler spaces from the the self-duality equations56Large genus limit of energy minimizing compact minimal surfaces in the 3-sphere

We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in Sp(n,R), G2(2), E8(C), F4(C), and G2(C) are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.

 

Related project(s):
58Profinite perspectives on l2-cohomology

This article presents a method for proving upper bounds for the first \(\ell^2\)-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first \(\ell^2\)-Betti number.

Our approach extends to generalizations of \(\ell^2\)-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.

 

Related project(s):
58Profinite perspectives on l2-cohomology

The present book deals with the spectral geometry of infinite graphs. This topic involves the interplay of three different subjects: geometry, the spectral theory of Laplacians and the heat flow of the underlying graph. These three subjects are brought together under the unifying perspective of Dirichlet forms. The spectral geometry of manifolds is a well-established field of mathematics. On manifolds, the focus is on how Riemannian geometry, the spectral theory of the Laplace–Beltrami operator, Brownian motion and heat evolution interact. In the last twenty years large parts of this theory have been subsumed within the framework of strongly local Dirichlet forms. Indeed, this point of view has proven extremely fruitful.

 

PublisherSpringer
BookGrundlehren der mathematischen Wissenschaften
Volume358
Pages668
Link to preprint version
Link to published version

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

We study planar graphs with large negative curvature outside of a finite set and the spectral theory of Schrödinger operators on these graphs. We obtain estimates on the first and second order term of the eigenvalue asymptotics. Moreover, we prove a unique continuation result for eigenfunctions and decay properties of general eigenfunctions. The proofs rely on a detailed analysis of the geometry which employs a Copy-and-Paste procedure based on the Gauß-Bonnet theorem.

 

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

In this article we prove upper bounds for the k-th Laplace eigenvalues below the essential spectrum for strictly negatively curved Cartan–Hadamard manifolds. Our bound is given in terms of k^2 and specific geometric data of the manifold. This applies also to the particular case of non‐compact manifolds whose sectional curvature tends to minus infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.

 

JournalMathematische Nachrichten
PublisherWiley
Volume294
Pages1134-1139
Link to preprint version
Link to published version

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

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.

 

JournalResearch in the Mathematical Sciences
Volume8, Article number: 36
Pages64p
Link to preprint version
Link to published version

Related project(s):
10Duality and the coarse assembly map78Duality and the coarse assembly map II

We construct a slant product on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer, which is the domain of the co-assembly map. In fact, we obtain such products simultaneously on the entire Higson-Roe sequence. The existence of these products implies injectivity results for external product maps on each term of the Higson-Roe sequence. Our results apply in particular to taking products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the general class of manifolds where this method applies, we introduce the notion of Higson-essentialness. A complete spin-c manifold is Higson-essential if its fundamental class is detected by the stable Higson corona via the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. Finally, we draw conclusions for positive scalar curvature metrics on product spaces, in particular on non-compact manifolds. We also construct suitable equivariant versions of these slant products and discuss related problems of exactness and amenability of the stable Higson corona.

 

JournalAnnales de l'Institut Fourier
Volume71 (2021) no. 3
Pages913-1021
Link to preprint version
Link to published version

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

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible \(\sigma\)-compact space in which the corona sits as a \(\mathbb{Z}\)-set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space \(BG\), then our constructions yield a \(\mathbb{Z}\)-structure for the group.

 

JournalJournal of Topology and Analysis
Volumeonline ready
Pages83p
Link to preprint version
Link to published version

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

In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prove that the entropies are monotone along the singular Ricci de Turck flow. We employ these entropies to show that in the singular setting, Ricci solitons are gradient and that steady or expanding Ricci solitons are Einstein.

 

JournalTrans. Amer. Math. Soc.
Volume374
Pages2873-2908
Link to preprint version
Link to published version

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry23Spectral geometry, index theory and geometric flows on singular 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→∞.

 

JournalJ. Geom. Anal.
Volume31
Pages5451–5479
Link to preprint version
Link to published version

Related project(s):
23Spectral geometry, index theory and geometric flows on singular spaces

We consider the heat equation associated to Schrödinger operators acting on vector bundles on asymptotically locally Euclidean (ALE) manifolds. Novel LpLq decay estimates are established, allowing the Schrödinger operator to have a non-trivial L2-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):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry

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