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

Given a manifold $M$, we completely determine which rational $\kappa$-classes are non-trivial for (fiber homotopy trivial) $M$-bundles over the $k$-sphere, provided that the dimension of $M$ is large compared to $k$. We furthermore study the vector space of these spherical $\kappa$-classes and give an upper and a lower bound on its dimension. The proof is based on the classical approach to studying diffeomorphism groups via block bundles and surgery theory and we make use of ideas developed by Krannich--Kupers--Randal-Williams.

 

As an application, we show the existence of elements of infinite order in the homotopy groups of the spaces $\mathcal R_{\mathrm{Ric}>0}(M)$ and $\mathcal R_{\mathrm{sec}>0}(M)$ of positive Ricci and positive sectional curvature, provided that $M$ is $\mathrm{Spin}$, has a non-trivial rational Pontryagin class and admits such a metric. This is done by showing that the $\kappa$-class associated to the $\hat{\mathcal A}$-class is spherical for such a manifold.

 

In the appendix co-authored by Jens Reinhold it is (partially) determined which classes of the rational oriented cobordism ring contain an element that fibers over a sphere of a given dimension.

 

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

In this paper we study spaces of Riemannian metrics with lower bounds on intermedi- ate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional Spin-manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.

 

Related project(s):
15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We construct and study an H-space multiplication on \(\mathcal{R}^+(M)\) for manifolds M which are nullcobordant in their own tangential 2-type. This is applied to give a rigidity criterion for the action of the diffeomorphism group on \(\mathcal{R}^+(M)\) via pullback. We also compare this to other known multiplicative structures on \(\mathcal{R}^+(M)\).

 

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

We present a rigidity theorem for the action of the mapping class group \(\pi_0(\mathrm{Diff}(M))\)on the space \(\mathcal{R}^+(M)\) of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional Spin-manifolds.

 

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

After R. Schoen completed the solution of the Yamabe problem, compact manifolds could be allocated in three classes depending on whether they admit a metric with positive, non-negative or only negative scalar curvature. Here we follow Yamabe's first attempt to solve his problem through variational methods and provide an analogous equivalent classification for manifolds equipped with actions by non-discrete compact Lie groups. Moreover, we apply the method, and the results to classify total spaces of fibre bundles with compact structure groups (concerning scalar curvature), to conclude density results, and compare realizable scalar curvature functions between some exotic manifolds their standard counterpart. We also provide an extended range of prescribed scalar curvature functions of warped products, especially with Calabi--Yau manifolds, providing an upper bound for the first positive eigenvalue of the Laplacian under relatively mild conditions.

 

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II

 In this paper  we construct solutions to Ricci DeTurck flow in four dimensions on closed manifolds which are instantaneously smooth but whose initial  values \(g\) are (possibly) non-smooth Riemannian metrics whose components  in smooth coordinates   belong to \(W^{2,2}\)(M) and satisfy  \(\frac{1}{a}h \leq g \leq ah\)  for some \(1<a<\infty\) and some smooth Riemannian metric  \(h\) on M. A Ricci flow related solution is constructed whose initial value is isometric in a weak sense to the initial value of the Ricci DeTurck solution. Results  for a related non-compact setting are also  presented. Various \(L^p\) estimates for Ricci flow, which we require for some of the main results, are also derived. As an application we present a possible definition of scalar curvature \(\geq k\) for  \(W^{2,2}\)(M)  metrics  \(g\)  on closed four manifolds which are bounded in the \(L^{\infty}\) sense by \(\frac{1}{a}h \leq g \leq ah\)  for some \(1<a<\infty\)  and some smooth Riemannian metric \(h\) on  M.

 

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

Let \(\mathcal{E}\) be an asymptotically Euclidean end in an otherwise arbitrary complete and connected Riemannian spin manifold \((M,g)\). We show that if \(\mathcal{E}\) has negative ADM-mass, then there exists a constant \(R > 0\), depending only on \(\mathcal{E}\), such that \(M\) must become incomplete or have a point of negative scalar curvature in the \(R\)-neighborhood around \(\mathcal{E}\) in \(M\). This gives a quantitative answer to Schoen and Yau's question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions \(\leq 7\) assuming Schwarzschild asymptotics on the end \(\mathcal{E}\). We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end \(\mathcal{E}\). Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature.

 

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

For every $g \gg 1$, we show the existence of a complete and smooth family  of closed constant mean curvature surfaces $f_\varphi^g,$ $ \varphi \in [0, \tfrac{\pi}{2}],$ in the round $3$-sphere deforming the Lawson surface $\xi_{1, g}$ to a doubly covered geodesic 2-sphere with monotonically increasing Willmore energy. To do so we use an implicit function theorem argument in the parameter $s= \tfrac{1}{2(g+1)}$. This allows us to give an iterative algorithm to compute the power series expansion of the DPW potential and area of $f_\varphi^g$  at $s= 0$ explicitly.  In particular,  we obtain for large genus Lawson surfaces $\xi_{1,g}$, due to the real analytic dependence of its area  and DPW potential on $s,$  a scheme to explicitly compute the coefficients of the power series in $s$ in terms of multilogarithms. Remarkably, the third order coefficient of the area expansion coincides numerically with $\tfrac{9}{4}\zeta(3),$ where $\zeta$ is the Riemann $\zeta$ function (while the first and second order term were shown to be $\log(2)$  and $0$ respectively in \cite{HHT}).

 

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

For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $\Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_{\Sigma}$ admits holomorphic connections with $\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. This also answers a question of \cite{CDHL}. The construction carries over to all very stable and compatible real holomorphic structures over the topologically trivial rank two bundle on $\Sigma$, and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.

 

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 study the holomorphic symplectic geometry of (the smooth locus of) the space of holomorphic sections of a twistor space with rotating circle action. The twistor space has a meromorphic connection constructed by Hitchin. We give an interpretation of Hitchin's meromorphic connection in the context of the Atiyah--Ward transform of the corresponding hyperholomorphic line bundle. It is shown that the residue of the meromorphic connection serves as a moment map for the induced circle action, and its critical points are studied. Particular emphasis is given to the example of Deligne--Hitchin moduli spaces.

 

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

We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n^{β+1}) where n is the size of the matrix and O(n^β) is the cost of multiplying n×n-matrices, β∈[2,2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.

 

JournalLinear Algebra and its Applications
Volume630
Pages39-55
Link to preprint version
Link to published version

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

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 $(M,g)$.

 

JournalErgodic Theory and Dynamical Systems
PublisherCambridge University Press
Link to preprint version
Link to published version

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

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 $\mathfrak{J}_{\mathfrak{p},dA_g}$.

 

JournalMathematische Zeitschrift
PublisherSpringer
Link to preprint version
Link to published version

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

We construct a natural transformation between two versions of G-equivariant K-homology with coefficients in a G-C*-category for a countable discrete group G. Its domain is a coarse geometric K-homology and its target is the usual analytic K-homology. Following classical terminology, we call this transformation the Paschke transformation. We show that under certain finiteness assumptions on a G-space X, the Paschke transformation is an equivalence on X. As an application, we provide a direct comparison of the homotopy theoretic Davis–Lück assembly map with Kasparov’s analytic assembly map appearing in the Baum–Connes conjecture.

 

Related project(s):
45Macroscopic invariants of manifolds

In this paper, we establish the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in $\mathbb{R}^3$ as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex surfaces whose metrics converge uniformly to the metric of the initial surface in intrinsic sense.

 

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

This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability.

We construct a finitely generated, residually finite, amenable group A and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to A. All of these groups are branch groups.

Moreover, picking up  Grothendieck's problem, the group A embeds in these groups such that the inclusion induces an isomorphism of profinite completions.  

 

In addition, we review the concept of uniform amenability, a strengthening of amenability introduced in the 70's, and we prove that uniform amenability indeed is detectable from the profinite completion.

 

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

Building on work of Bowden-Hensel-Webb, we study the action of the homeomorphism group of a surface $S$ on the fine curve graph $\mathcal{C}^\dagger(S)$.  While the definition of $\mathcal{C}^\dagger(S)$ parallels the classical curve graph for mapping class groups, we show that the dynamics of the action of $\mathrm{Homeo}(S)$ on $\mathcal{C}^\dagger(S)$ is much richer:  homeomorphisms induce parabolic isometries in addition to elliptics and hyperbolics, and all positive reals are realized as asymptotic translation lengths. 

When the surface $S$ is a torus, we relate the dynamics of the action of a homeomorphism on $\mathcal{C}^\dagger(S)$ to the dynamics of its action on the torus via the classical theory of {\em rotation sets}.  We characterize homeomorphisms acting hyperbolically, show asymptotic translation length provides a lower bound for the area of the rotation set, and, while no characterisation purely in terms of rotation sets is possible, we give sufficient conditions for elements to be elliptic or parabolic.  

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

This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of Bär-Ballmann to first order elliptic operators. The space of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calderón projectors which, in the first order case, is equivalent to results of Bär-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven.

 

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

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we thereby give refined answers to questions on metric inequalities recently proposed by Gromov. This includes optimal estimates for Riemannian bands and for the long neck problem. In the case of bands over manifolds of non-vanishing \(\widehat{\mathrm{A}}\)-genus, we establish a rigidity result stating that any band attaining the predicted upper bound is isometric to a particular warped product over some spin manifold admitting a parallel spinor. Furthermore, we establish scalar- and mean curvature extremality results for certain log-concave warped products. The latter includes annuli in all simply-connected space forms. On a technical level, our proofs are based on new spectral estimates for the Dirac operator augmented by a Lipschitz potential together with local boundary conditions.

 

Related project(s):
42Spin obstructions to metrics of positive scalar curvature on nonspin manifolds78Duality and the coarse assembly map II

For a mean curvature flow of complete graphical hypersurfaces M_t=graph u(⋅,t) defined over domains Ω_t, the enveloping cylinder is ∂Ω_t×R. We prove the smooth convergence of M_th e_{n+1} to the enveloping cylinder under certain circumstances. Moreover, we give examples demonstrating that there is no uniform curvature bound in terms of the inital curvature and the geometry of Ω_t. Furthermore, we provide an example where the hypersurface increasingly oscillates towards infinity in both space and time. It has unbounded curvature at all times and is not smoothly asymptotic to the enveloping cylinder. We also prove a relation between the initial spatial asymptotics at the boundary and the temporal asymptotics of how the surface vanishes to infinity for certain rates in the case Ω_t are balls.

 

Related project(s):
29Curvature flows without singularities

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