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

Let Γ be a finitely generated discrete group and let M˜ be a Galois Γ-covering of a smooth compact manifold M. Let u:XBΓ be the associated classifying map. Finally, let SΓ∗(M˜) be the analytic structure group, a K-theory group appearing in the Higson-Roe exact sequence ⋯→SΓ∗(M˜)→K∗(M)→K∗(C∗Γ)→⋯. Under suitable assumptions on Γ we construct two pairings, first between SΓ∗(M˜) and the delocalized part of the cyclic cohomology of CΓ, and secondly between SΓ∗(M˜) and the relative cohomology H∗(MBΓ). Both are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class ρ(D˜)∈SΓ∗(M˜) of an invertible Γ-equivariant Dirac type operator on M˜. Regarding the first pairing we establish in fact a more general result, valid without additional assumptions on Γ: indeed, we prove that it is possible to map the Higson-Roe analytic surgery sequence to the long exact sequence in noncommutative de Rham homology ⋯−→jH∗−1(AΓ)→ιHdel∗−1(AΓ)→δHe∗(AΓ)−→j∗⋯ with AΓ a dense homomorphically closed subalgebra of CrΓ and Hdel∗(AΓ) and He∗(AΓ) denoting versions of the delocalized homology and the homology localized at the identity element, respectively.

 

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick, Xie.

 

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

Using standard results from higher (secondary) index theory, we prove that the positive scalar curvature bordism groups of a cartesian product GxZ are infinite in dimension 4n if n>0 G a group with non-trivial torsion. We construct representatives of each of these classes which are connected and with fundamental group GxZ. We get the same result in dimension 4n+2 (n>0) if G is finite and contains an element which is not conjugate to its inverse. This generalizes the main result of Kazaras, Ruberman, Saveliev, "On positive scalar curvature cobordism and the conformal Laplacian on end-periodic manifolds" to arbitrary even dimensions and arbitrary groups with torsion.

 

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy RP^5 has infinitely many path components. We also show that in each dimension 4k+1 there are at least 2^{2k} homotopy RP^{4k+1}s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4k+3≥7.

 

Journalto appear in Transactions of the AMS
Link to preprint version

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index, and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractiblegeometric realizations.

 

Related project(s):
8Parabolics and invariants

In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.

 

Related project(s):
1Hitchin components for orbifolds

For d = 4, 5, 6, 7, 8, we exhibit examples of \(\mathrm{AdS}^{d,1}\) strictly GHC-regular groups which are not quasi-isometric to the hyperbolic space \(\mathbb{H}^d\), nor to any symmetric space. This provides a negative answer to Question 5.2 in a work of Barbot et al. and disproves Conjecture 8.11 of Barbot-Mérigot [Groups Geom. Dyn. 6 (2012), pp. 441-483].   We construct those examples using the Tits representation of well-chosen Coxeter groups. On the way, we give an alternative proof of Moussong's hyperbolicity criterion (Ph.D. Thesis) for Coxeter groups built on Danciger-Guéritaud-Kassel's 2017 work and find examples of Coxeter groups W such that the space of strictly GHC-regular representations of W into \(\mathrm{PO}_{d,2}(\mathbb{R})\) up to conjugation is disconnected.

 

JournalTransactions of the American Mathematical Society
Volume372
Pages153-186
Link to preprint version
Link to published version

Related project(s):
1Hitchin components for orbifolds

We study the Thurston–Parreau boundary both of the Hitchinand of the maximal character varieties and determine therein an open setof discontinuity for the action of the mapping class group. This result isobtained as consequence of a canonical decomposition of a geodesic cur-rent on a surface of finite type arising from a topological decompositionof the surface along special geodesics. We show that each componenteither is associated to a measured lamination or has positive systole. Fora current with positive systole, we show that the intersection function onthe set of closed curves is bi-Lipschitz equivalent to the length functionwith respect to a hyperbolic metric.

 

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

In this paper we investigate the Hausdorff dimension of limitsetsof Anosov representations. In this context we revisit and extend the frameworkof hyperconvex representations and establish a convergence property for them,analogue to a differentiability property. As an applicationof this convergence,we prove that the Hausdorff dimension of the limit set of a hyperconvex rep-resentation is equal to a suitably chosen critical exponent. In the appendix, incollaboration with M. Bridgeman, we extend a classical result on the Hessianof the Hausdorff dimension on purely imaginary directions.

 

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

We study Anosov representation for which the image of the bound-ary map is the graph of a Lipschitz function, and show that theorbit growthrate with respect to an explicit linear function, the unstable Jacobian, is inte-gral. Several applications to the orbit growth rate in the symmetric space areprovided.

 

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

In this article, we are interested in the question whether any complete contractible 3-manifold of positive scalar curvature is homeomorphic to \(\mathbb{R}^3\). We study the fundamental group at infinity, \(\pi^\infty_1\), and its relationship with the existence of complete metrics of positive scalar curvature. We prove that a complete contractible 3-manifold with positive scalar curvature and trivial \(\pi_1^\infty\) is homeomorphic to \(\mathbb{R}^3\).

 

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

In this work we prove that the Whitehead manifold has no complete metric of positive scalar curvature. This result can be generalized to the genus one case. Precisely, we show that no contractible genus one 3-manifold admits a complete metric of positive scalar curvature.

 

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods.

 

JournalJournal de Mathématiques Pures et Appliquées. (9)
Volume126
Pages109--143
Link to preprint version
Link to published version

Related project(s):
19Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive
scalar curvature metrics on closed simply connected manifolds in dimensions at least 
five appears on spin manifolds, and is given by the non-vanishing of the \(\alpha\)-genus 
of Hitchin. 
When unobstructed we will in this paper realise  a positive scalar curvature metric by an 
immersion into Euclidean space whose dimension is uniformly close to the classical Whitney 
upper-bound for smooth immersions. Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure 
for constructing positive scalar curvature metrics.

 

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

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.

 

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

We consider solutions to Ricci flow defined on manifolds M for a time interval (0,T) whose Ricci curvature is bounded uniformly in time

from below, and for which the norm of the full curvature tensor at time t  is bounded by c/t for some fixed constant c>1 for all t in (0,T).

From previous works, it is known that if the solution is complete for all times t>0, then there is a limit metric space (M,d_0), as time t approaches zero. We show : if there is a region V on which (V,d_0) is smooth, then the solution can be extended smoothly to time zero on V.

 

Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.

 

JournalAmer. J. Math.
PublisherJohn Hopkins Univ. Press
Volume141 (5)
Pages1421-1455
Link to preprint version
Link to published version

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

We establish a one-to-one correspondence between Finsler structures on the \(2\)-sphere with constant curvature \(1\) and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $$\mathbb{CP}(a_1,a_2)\to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)$$ of weighted projective spaces provide examples of Finsler \(2\)-spheres of constant curvature and all geodesics closed.

 

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

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