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
  • 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

We prove that the group of isometries preserving a metric foliation on a closed Alexandrov space \(X\), or a singular Riemannian foliation on a manifold \(M\) is a closed subgroup of the isometry group of \(X\) in the case of a metric foliation, or of the isometry group of \(M\) for the case of a singular Riemannian foliation. We obtain a sharp upper bound for the dimension of these subgroups and show that, when equality holds, the foliations that realize this upper bound are induced by fiber bundles whose fibers are round spheres or projective spaces. Moreover, singular Riemannian foliations that realize the upper bound are induced by smooth fiber bundles whose fibers are round spheres or projective spaces.

 

Related project(s):
43Singular Riemannian foliations and collapse

In this paper, we consider the stability of the generalized Lagrangian mean curvature flow of graph case in the cotangent bundle, which is first defined by Smoczyk-Tsui-Wang [14]. By new estimates of derivatives along the flow, we weaken the initial condition and remove the positive curvature condition in [14]. More precisely, we prove that if the graph induced by a closed $1$-form is a special Lagrangian submanifold in the cotangent bundle of a Riemannian manifold, then the generalized Lagrangian mean curvature flow is stable near it.

 

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in \(\mathbb R^{n+1}\), and prove the existence and regularity of the flow before extincting to a point in finite time.

 

JournalAdvanced Nonlinear Studies
Link to preprint version
Link to published version

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

In this paper, we show the relation between the existence of twisted conical Kähler-Ricci solitons and the greatest log Bakry-Emery-Ricci lower bound on Fano manifolds. This is based on our proofs of some openness theorems on the existence of twisted conical Kähler-Ricci solitons, which generalize Donaldson's existence conjecture and openness theorem of the conical Kähler-Einstein metrics to the conical soliton case.

 

JournalScience China Mathematics
Link to preprint version
Link to published version

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II

For vector-valued Maass cusp forms for SL(2,Z) with real weight k∈R and spectral parameter s∈C, Res∈(0,1), s≢±k/2 mod 1, we propose a notion of vector-valued period functions, and we establish a linear isomorphism between the spaces of Maass cusp forms and period functions by means of a cohomological approach. The period functions are a generalization of those for the classical Maass cusp forms, being solutions of a finite-term functional equation or, equivalently, eigenfunctions with eigenvalue 1 of a transfer operator deduced from the geodesic flow on the modular surface. We apply this result to deduce a notion of period functions and related linear isomorphism for Jacobi Maass forms of weight k+1/2 for the semi-direct product of SL2(Z) with the integer points Hei(Z) of the Heisenberg group.

 

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

For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences. Some are known and include Llarull's scalar curvature rigidity of the standard metric on the sphere, Geroch's conjecture on the impossibility of positive scalar curvature on tori and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler. New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for Kähler manifolds, Kähler-Einstein manifolds, quaternionic Kähler manifolds and manifolds with a harmonic 1-form of constant length.

 

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds

We investigate groups that act amenably on their Higson corona (also known as bi-exact groups) and we provide reformulations of this in relation to the stable Higson corona, nuclearity of crossed products and to positive type kernels. We further investigate implications of this in relation to the Baum-Connes conjecture, and prove that Gromov hyperbolic groups have isomorphic equivariant K-theories of their Gromov boundary and their stable Higson corona.

 

Related project(s):
45Macroscopic invariants of manifolds

We present a new method to construct finitely generated, residually finite, infinite torsion groups. In contrast to known constructions, a profinite perspective enables us to control finite quotients and normal subgroups of these torsion groups. As an application, we describe the first examples of residually finite, hereditarily just-infinite groups with positive first 2-Betti-number. In addition, we show that these groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.

 

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

We prove that every finitely generated, residually finite group G embeds into a finitely generated perfect branch group such that many properties of G are preserved under this embedding. Among those are the properties of being torsion, being amenable and not containing a non-abelian free group. As an application, we construct a finitely generated, non-amenable torsion branch group.

 

JournalBull. Lond. Math. Soc.
Volume56
Link to preprint version
Link to published version

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

We study \(\mathsf{RCD}\)-spaces \((X,d,\mathfrak{m})\) with group actions by isometries preserving the reference measure \(\mathfrak{m}\) and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that each slice at a point is homeomorphic to a non-negatively curved \(\mathsf{RCD}\)-space. Under the assumption that \(X\) is non-collapsed we further show that the slices are homeomorphic to metric cones over homogeneous spaces with \(\mathrm{Ric}\geq 0\). As a consequence we obtain complete topological structural results and a principal orbit representation theorem. Conversely, we show how to construct new \(\mathsf{RCD}\)-spaces from a cohomogeneity one group diagram, giving a complete description of \(\mathsf{RCD}\)-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed \(\mathsf{RCD}\)-spaces of essential dimension at most 4.

 

Related project(s):
43Singular Riemannian foliations and collapse

We show that the sign of the Euler characteristic of an S-arithmetic subgroup of a simple k-group over a number field k depends on the S-congruence completion only. Consequently, the sign is a profinite invariant for such S-arithmetic groups with the congruence subgroup property. This generalizes previous work of the first author with Kionke-Raimbault-Sauer.

 

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

We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped  with   strictly log-concave warping functions. This generalizes earlier results of  Cecchini-Zeidler to all dimensions. Moreover, we show scalar curvature rigidity of round spheres of dimension at least $3$ minus two antipodal points, thus resolving a problem in Gromov's  ``Four Lectures'' in all dimensions. Our arguments are based on spin geometry.

 

Journal Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume20
Pagesarticle 035, 26 pages
Link to preprint version
Link to published version

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We prove that there exist ????????(3)-invariant metrics on Aloff-Wallach spaces W^7_{k1,k2}, as well as ????????(5)-invariant metrics on the Berger space B^{13}, which have positive sectional curvature and evolve under the Ricci flow to metrics with non-positively curved planes.

 

Related project(s):
79Alexandrov geometry in the light of symmetry and topology

By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Guéritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.

 

Related project(s):
65Resonances for non-compact locally symmetric spaces

For negatively curved symmetric spaces it is known that the poles of the scattering matrices defined via the standard intertwining operators for the spherical principal representations of the isometry group are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. We extend this result to classical locally symmetric spaces of negative curvature with convex-cocompact fundamental group using results of Bunke and Olbrich. The method of proof forces us to exclude the spectral parameters corresponding to singular Poisson transforms.

 

Related project(s):
65Resonances for non-compact locally symmetric spaces

We present how to collapse a manifold equipped with a closed flat regular Riemannian foliation with leaves of positive dimension on a compact manifold, while keeping the sectional curvature uniformly bounded from above and below. From this deformation, we show that a closed flat regular Riemannian foliation with leaves of positive dimension on a compact simply-connected manifold is given by torus actions. This gives a geometric characterization of aspherical regular Riemannian foliations given by torus actions.

 

Related project(s):
43Singular Riemannian foliations and collapse

The fine curve graph was introduced as a geometric tool to study the homeomorphisms of surfaces. In this paper we study the Gromov boundary of this space and the local topology near points associated with minimal measurable foliations. We then give several applications including finding explicit elements with positive stable commutator length, and proving a Tits alternative for subgroups of the homemorphism group of a closed surface containing a pseudo-Anosov map, generalizing a result of Hurtado-Xue.

 

Related project(s):
38Geometry of surface homeomorphism groups

In the framework of infinite ergodic theory, we derive equidistribution results for suitable weighted sequences of cusp points of Hecke triangle groups encoded by group elements of constant word length with respect to a set of natural generators. This is a generalization of the corresponding results for the modular group, for which we rely on advanced results from infinite ergodic theory and transfer operator techniques developed for AFN-maps.

 

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

We consider first-order elliptic differential operators acting on vector bundles over smooth manifolds with smooth boundary, which is permitted to be noncompact. Under very mild assumptions, we obtain a regularity theory for sections in the maximal domain. Under additional geometric assumptions, and assumptions on an adapted boundary operator, we obtain a trace theorem on the maximal domain. This allows us to systematically study both local and nonlocal boundary conditions. In particular, the Atiyah-Patodi-Singer boundary condition occurs as a special case. Furthermore, we study contexts which induce semi-Fredholm and Fredholm extensions.

 

Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds

Every finite collection of oriented closed geodesics in the modular surface has a canonically associated link in its unit tangent bundle coming from the periodic orbits of the geodesic flow. We study the volume of the associated link complement with respect to its unique complete hyperbolic metric. We provide the first lower volume bound that is linear in terms of the number of distinct exponents in the code words corresponding to the collection of closed geodesics.

 

Related project(s):
38Geometry of surface homeomorphism groups

This website uses cookies

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