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

We consider the evolution of hypersurfaces in R^{n+1} with normal velocity given by a positive power of the mean curvature. The hypersurfaces under consideration are assumed to be strictly mean convex (positive mean curvature), complete, and given as the graph of a function. Long-time existence of the H-flow is established by means of approximation by bounded problems.

 

Related project(s):
29Curvature flows without singularities

By a symmetric double graph we mean a hypersurface which is mirror-symmetric and the two symmetric parts are graphs over the hyperplane of symmetry. We prove that there is a weak solution of mean curvature flow that preserves these properties and singularities only occur on the hyperplane of symmetry. The result can be used to construct smooth solutions to the free Neumann boundary problem on a supporting hyperplane with singular boundary. For the construction we introduce and investigate a notion named "vanity" and which is similar to convexity. Moreover, we rely on Sáez' and Schnürer's "mean curvature flow without singularities" to approximate weak solutions with smooth graphical solutions in one dimension higher.

 

Related project(s):
29Curvature flows without singularities

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products.

 

Using this theory we construct positive scalar curvature metrics on closed smooth manifolds of dimensions at least five which have odd order abelian fundamental groups, are non-spin and  atoral. This solves the Gromov-Lawson-Rosenberg conjecture for a new class of manifolds with finite fundamental groups.

 

 

 

JournalGeometry & Topology
PublisherMSP
Volume25 (2021)
Pages497-546
Link to preprint version
Link to published version

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

The unit conjecture, commonly attributed to Kaplansky, predicts that if \(K\) is a field and \(G\) is a torsion-free group then the only units of the group ring \(K[G]\) are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.

 

Related project(s):
48Profinite and RFRS groups

For a countable group G we construct a small, idempotent complete, symmetric monoidal, stable ∞-category KK^G_sep whose homotopy category recovers the triangulated equivariant Kasparov category of separable G-C*-algebras, and exhibit its universal property. Likewise, we consider an associated presentably symmetric monoidal, stable ∞-category KK^G which receives a symmetric monoidal functor kk^G from possibly non-separable G-C*-algebras and discuss its universal property. In addition to the symmetric monoidal structures, we construct various change-of-group functors relating these KK-categories for varying G. We use this to define and establish key properties of a (spectrum valued) equivariant, locally finite K-homology theory on proper and locally compact G-topological spaces, allowing for coefficients in arbitrary G-C*-algebras. Finally, we extend the functor kk^G from G-C*-algebras to G-C*-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps.

 

Related project(s):
45Macroscopic invariants of manifolds

We prove that any length metric space homeomorphic to a surface may be decomposed into non-overlapping convex triangles of arbitrarily small diameter. This generalizes a previous result of Alexandrov--Zalgaller for surfaces of bounded curvature.

 

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

Based on  Morse theory for the energy functional on path spaces we develop a deformation theory for mapping spaces of  spheres into orthogonal groups. This is used to show that these mapping spaces are weakly homotopy equivalent, in a stable range, to mapping spaces associated to orthogonal Clifford representations. Given an oriented Euclidean bundle $V \to X$ of rank divisible by four over a finite complex $X$ we derive a stable decomposition result for vector bundles over  the sphere bundle $\mathbb{S}( \mathbb{R} \oplus V)$  in terms of vector bundles and Clifford module bundles over $X$. After passing to topological K-theory these results imply classical Bott-Thom isomorphism theorems.

 

 

 

JournalSão Paulo Journal of Mathematical Sciences (to appear)
Link to preprint version

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 investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution of Plateau's problem in metric spaces by Lytchak and Wenger as generalizations of minimal surfaces.

 

JournalGeom. Dedicata
Volume210
Pages151-164
Link to preprint version
Link to published version

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


We show that a strict, nearly Kähler 6-manifold with either second or third Betti number nonzero is linearly unstable with respect to the \(\nu\)-entropy of Perelman and hence is dynamically unstable for the Ricci flow.

 

 

JournalAnn. Global Anal. Geom. 57 no. 1, 15-22 (2020)
Link to preprint version

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

We introduce a new invariant of finitely generated groups, the ambiguity function, and we prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate \(\lim \limits_{n \rightarrow \infty} \sqrt[n]{\vert {B^{X}_H(n)} \vert}\) of a subgroup \(H\) of a finitely generated acylindrically hyperbolic group $G$ exists with respect to every finite generating set \(X\) of \(G\), if \(H\) contains a loxodromic element of \(G\). Further we prove that the relative exponential growth rate of every finitely generated subgroup \(H\) of a right-angled Artin group \(A_{\Gamma}\) exists with respect to every finite generating set of \(A_{\Gamma}\).

 

Related project(s):
8Parabolics and invariants

A Calderón projector for an elliptic operator $P$ on a manifold with boundary $X$ is a projection from general boundary data to the set of boundary data of solutions $u$ of $Pu=0$. Seeley proved in 1966 that for compact $X$ and for $P$ uniformly elliptic up to the boundary there is a Calder\'on projector which is a pseudodifferential operator on $\partial X$. We generalize this result to the setting of fibred cusp operators, a class of elliptic operators on certain non-compact manifolds having a special fibred structure at infinity.

 

This applies, for example, to the Laplacian on certain locally symmetric spaces or on particular singular spaces, such as a domain with cusp singularity or the complement of two touching smooth strictly convex domains in Euclidean space. Our main technical tool is the $\phi$-pseudodifferential calculus introduced by Mazzeo and Melrose.

 

In our presentation we provide a setting that may  be useful for doing analogous constructions for other types of singularities.

 

Related project(s):
13Analysis on spaces with fibred cusps49Analysis on spaces with fibred cusps II

We study the low energy resolvent of the Hodge Laplacian on a manifold  equipped with a fibred boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibred boundary (aka $\phi$-) pseudodifferential operator when the resolvent parameter tends to zero.

 

This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibres are points. The new feature in the case of non-trivial fibres is that the resolvent has different asymptotic behavior on the subspace of forms that are fibrewise harmonic  and on its orthogonal complement. To deal with this, we introduce an appropriate 'split' pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on $\phi$-manifolds.

Related project(s):
13Analysis on spaces with fibred cusps23Spectral geometry, index theory and geometric flows on singular spaces49Analysis on spaces with fibred cusps II

We define a functional \({\cal J}(h)\) for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that \({\cal J}(h)\) is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We derive a formula relating \({\cal J}(h)\) and another functional \({\cal I}(h)\), closely related to the Yang-Mills-Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of \({\cal J}(h)\), which is expressed as a certain \(L^{2}\)-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of \({\cal J}(h)\) if and only if the corresponding Hitchin-Simpson mean curvature is parallel with respect to the Hitchin-Simpson connection.

 

JournalInternational Journal of Geometric Methods in Modern Physics
PublisherWorld Scientific
Volume17(13)
Lineart. no. 2050200
Link to preprint version
Link to published version

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space69Wall-crossing and hyperkähler geometry of moduli spaces

We construct smooth bundles with base and fiber products of two spheres whose total spaces have nonvanishing A-hat-genus. We then use these bundles to locate nontrivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all Spin manifolds of dimension 6 or at least 10, which admit such a metric and are a connected sum of some manifold and \(S^n\times S^n\) or \(S^n\times S^{n+1}\), respectively. We also construct manifolds M whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order that lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of M.

 

JournalInternational Mathematics Research Notices
Link to preprint version
Link to published version

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 Manifolds

We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.

 

Related project(s):
36Cohomogeneity, curvature, cohomology

In each dimension $4k+1\geq 9$, we exhibit infinite families of closed manifolds with fundamental group $\mathbb Z_2$ for which the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with this property were known before only in dimension $5$ and dimensions $4k+3\geq 7$.

 

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 show that a properly convex projective structure \(\mathfrak{p}\) on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if \(\mathfrak{p}\) is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that \(\mathfrak{p}\) admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this non-linear PDE into a transport equation, we obtain our result by applying methods from geometric inverse problems. In particular, we use an extension of a remarkable \(L^2\)-energy identity known as Pestov's identity to prove a vanishing theorem for the relevant transport equation.

 

JournalAnalysis & PDE
PublisherMathematical Sciences Publishers
Volume13
Pages1073--1097
Link to preprint version
Link to published version

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

Let M be a simply connected spin manifold of dimension at least six which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on M has non-trivial higher homotopy groups.

Moreover, denote by \mathcal{M}^+_0(M) the moduli space of positive scalar cuvature metrics on M associated to the group of orientation-preserving diffeomorphisms of M. We show that if M belongs to a certain class of manifolds which includes (2n−2)-connected (4n−2)-dimensional manifolds, then the fundamental group of \mathcal{M}^+_0(M) is non-trivial.

 

JournalInt. Math. Res. Not. IMRN
Link to preprint version
Link to published version

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

The main result of this paper is that when M_0, M_1 are two simply connected spin manifolds of the same dimension d≥5 which both admit a metric of positive scalar curvature, the spaces \mathcal{M}^+(M0) and \mathcal{M}^+(M_1) of such metrics are homotopy equivalent. This supersedes a previous result of Chernysh and Walsh which gives the same conclusion when M_0 and M_1 are also spin cobordant.

We also prove an analogous result for simply connected manifolds which do not admit a spin structure; we need to assume that d≠8 in that case.

 

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

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of \(\mathbb{R}\) or \(\mathbb{C}\). We prove an adelic version of superrigidity which implies that two such commensurability classes define the same profinite commensurability class if and only if the algebraic groups are adelically isomorphic. We discuss noteworthy consequences on profinite rigidity questions.

 

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

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