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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  • 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

We prove an analog of Schoen-Yau univalentness theorem for saddle maps between discs.

 

Related project(s):
24Minimal surfaces in metric spaces

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the (n+1)-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for n=2 we obtain a Minkowski-type inequality and for n=3 we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

 

Related project(s):
22Willmore functional and Lagrangian surfaces

In this paper, by providing the uniform gradient estimates for approximating equations, we prove the existence, uniqueness and regularity of conical parabolic complex Monge-Ampère equation with weak initial data. As an application, we obtain a regularity estimate, that is, any $L^{\infty}$-solution of the conical complex Monge-Ampère equation admits the $C^{2,\alpha,\beta}$-regularity.

 

JournalCalculus of Variations and Partial Differential Equations
Link to preprint version

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

Given two weighted graphs $(X,b_k,m_k)$, $k=1,2$ with $b_1\sim b_2$ and $m_1\sim m_2$, we prove a weighted $L^1$-criterion for the existence and completeness of the wave operators $W_{\pm}(H_{2},H_1, I_{1,2})$, where $H_k$ denotes the natural Laplacian in $\ell^2(X,m_k)$ w.r.t. $(X,b_k,m_k)$ and $I_{1,2}$ the trivial identification of $\ell^2(X,m_1)$ with $\ell^2(X,m_2)$. In particular, this entails a general criterion for the absolutely continuous spectra of $H_1$ and $H_2$ to be equal.

 

JournalMath. Phys. Anal. Geom.
Pages21-28
Link to preprint version

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

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite dimensional totally geodesic subspace on which the action is maximal. In the opposite direction we construct examples of geometrically dense maximal representation in the infinite dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, that we are able to construct in low ranks or under some suitable Zariski-density assumption, circumventing the lack of local compactness in the infinite dimensional setting.

 

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

Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. We show, nevertheless, that the abundance of new examples is restricted to even dimensions. As one key ingredient we provide a characterization of orientable manifolds among orientable orbifolds in terms of characteristic classes.

 

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

We prove that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n \geqslant 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n\geqslant 5$.

 

We also provide explicit lower bounds for the Kazhdan constants of $\mathrm{SAut}(F_n)$ (with $n \geqslant 6$) and of $\mathrm{SL}_n(\mathbb Z)$ (with $n \geqslant 3$) with respect to natural generating sets.

In the latter case, these bounds improve upon previously known lower bounds whenever $n> 6$.

 

Related project(s):
8Parabolics and invariants

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an application, simply-connected, rationally-elliptic manifolds admitting slice-maximal torus actions are classified up to equivariant rational homotopy. The case where the rational-ellipticity hypothesis is replaced by non-negative curvature is also discussed, and the Bott Conjecture in the presence of a slice-maximal torus action is proved.

 

JournalInt. Math. Res. Not. IMRN
Volume18
Pages5786--5822
Link to preprint version
Link to published version

Related project(s):
11Topological and equivariant rigidity in the presence of lower curvature bounds15Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds

In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected 5-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension 4 are connected sums of \(S^3\)-bundles over \(S^2\). Furthermore, the Betti numbers of the 5-manifolds and of the quotient 4-manifolds are related by a simple formula involving the number of fixed-point components. We also investigate semi-free \(S^3\) actions on simply connected 8-manifolds with quotient a 5-manifold and show, in particular, that the Pontrjagin classes, the \(\hat A\) -genus and the signature of the 8-manifold must all necessarily vanish.

 

Related project(s):
11Topological and equivariant rigidity in the presence of lower curvature bounds

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces

 

JournalGeom. Topol.
Volume22
Pages591-644
Link to preprint version

Related project(s):
24Minimal surfaces in metric spaces

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.

 

JournalActa Math.
Volume221
Pages159-202
Link to preprint version
Link to published version

Related project(s):
24Minimal surfaces in metric spaces

We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.

 

JournalMath. Ann.
VolumeOnline First
Link to preprint version

Related project(s):
24Minimal surfaces in metric spaces

We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity of the metric structure on a large part.

 

JournalGeom. Funct. Anal.
VolumeTo appear
Link to preprint version

Related project(s):
24Minimal surfaces in metric spaces

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems.

 

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

Geometric structures on manifolds became popular when Thurston used them in his work on the Geometrization Conjecture. They were studied by many people and they play an important role in Higher Teichmüller Theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin's equations. Baraglia has shown in his Thesis that this technique can be used to construct geometric structures in interesting cases. Here we will survey some recent results in this direction, which are joint work with Qiongling Li.

 

Related project(s):
1Hitchin components for orbifolds

 Let X be a compact manifold, D a real elliptic operator on X, G a Lie group, P a principal G-bundle on X, and B_P the infinite-dimensional moduli space of all connections on P modulo gauge, as a topological stack. For each connection \nabla_P, we can consider the twisted elliptic operator on X. This is a continuous family of elliptic operators over the base B_P, and so has an orientation bundle O^D_P over B_P, a principal Z_2-bundle parametrizing orientations of KerD^\nabla_Ad(P) + CokerD^\nabla_Ad(P) at each \nabla_P. An orientation on (B_P,D) is a trivialization of O^D_P.

 

In gauge theory one studies moduli spaces M of connections \nabla_P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X, g). Under good conditions M is a smooth manifold, and orientations on (B_P,D) pull back to

orientations on M in the usual sense of differential geometry.

 

This is important in areas such as Donaldson theory, where one needs an orientation on M

to define enumerative invariants.

 

We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (B_P,D), after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds,

instantons, the Kapustin-Witten equations, and the Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds.

 

Related project(s):
33Gerbes in renormalization and quantization of infinite-dimensional moduli spaces

 Suppose (X, g) is a compact, spin Riemannian 7-manifold, with Dirac operator D. Let G be SU(m) or U(m), and E be a rank m complex bundle with G-structure on X. Write  B_E for the infinite-dimensional moduli space of connections on E, modulo gauge. There is a natural principal Z_2-bundle O^D_E on B_E parametrizing orientations of det D_Ad A for twisted elliptic operators D_Ad A at each [A] in  B_E. A theorem of Walpuski shows O^D_E is trivializable.

 

We prove that if we choose an orientation for det D, and a flag structure on X in the sense of Joyce arXiv:1610.09836, then we can define canonical trivializations of O^D_E for all such bundles E on X, satisfying natural compatibilities.

 

Now let (X,\varphi,g) be a compact G_2-manifold, with d(*\varphi)=0. Then we can consider moduli spaces  M_E^G_2 of G_2-instantons on E over X, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general. The restriction of O^D_E to M_E^G_2 is the Z_2-bundle of orientations on M_E^G_2. Thus, our theorem induces canonical orientations on all such G_2-instanton moduli spaces  M_E^G_2.

 

This contributes to the Donaldson-Segal programme arXiv:0902.3239, which proposes defining enumerative invariants of G_2-manifolds (X,\varphi,g) by counting moduli spaces  M_E^G_2, with signs depending on a choice of orientation. This paper is a sequel to Joyce-Tanaka-Upmeier arXiv:1811.01096, which develops the general theory of orientations on gauge-theoretic moduli spaces, and gives applications in dimensions 3,4,5 and 6.

 

Related project(s):
33Gerbes in renormalization and quantization of infinite-dimensional moduli spaces

We show that the complex of free factors of a free group of rank n > 1 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n > 1, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent to the complex of free factor systems and moreover is (n-2)-connected. In addition, we show that for every non-trivial free factor system of a free group, the corresponding relative free splitting complex is contractible.

 

Related project(s):
8Parabolics and invariants

We develop the theory of agrarian invariants, which are algebraic counterparts to $L^2$-invariants. Specifically, we introduce the notions of agrarian Betti numbers, agrarian acyclicity, agrarian torsion and agrarian polytope.

 

We use the agrarian invariants to solve the torsion-free case of 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, for such groups, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl--Tillmann and Friedl--Lück--Tillmann. Finally, we prove that for agrarian groups of deficiency $1$, the agrarian polytope admits a marking of its vertices which controls the Bieri--Neumann--Strebel invariant of the group, improving a result of the second author and partially answering a question of Friedl--Tillmann.

 

Related project(s):
8Parabolics and invariants

We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers.

Secondly, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an application, we calculate the asymptotic dimension of a warped cone and improve bounds by Szabo, Wu, and Zacharias and by Bartels on the amenability dimension of actions of virtually nilpotent groups.

In the appendix, we justify optimality of our result on general quasi-isometries by showing that quasi-isometric warped cones need not come from quasi-isometric groups, contrary to the case of box spaces.

 

JournalProc. Lond. Math. Soc.
PublisherWiley
Link to preprint version
Link to published version

Related project(s):
8Parabolics and invariants

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