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

We study the asymptotics of the natural \(L^2\) metric on the Hitchin moduli space with group \(G=SU(2)\). Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore, is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric. We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. Very recent work by Dumas and Neitzke indicates that the convergence rate for the metric is exponential, at least in certain directions.

 

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

We study immersed tori in 3-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes (0,b) with $b\rightarrow 1$ the homogenous tori are known to be unique constrained Willmore minimizers (up to invariance). In this paper we generalize the result and determine constrained Willmore minimizers in non-rectangular conformal classes (a,b). In a first step we explicitly construct a 2-dimensional family of putative minimizers parametrized by their conformal class (a,b). For $b\rightarrow 1$, b≠1 fixed, this family is then shown to minimize for $a\rightarrow 0^+$. Difficulties arise from the fact that these minimizers are non-degenerate for a≠0 but smoothly converge to the degenerate homogenous tori as a→0. As a byproduct of our arguments, we show that the minimal Willmore energy ω(a,b) is real analytic and concave in a∈(0, a^b) for some b>0 and fixed $b \rightarrow 1$, 

 

Related project(s):
16Minimizer of the Willmore energy with prescribed rectangular conformal class

On a compact globally hyperbolic Lorentzian spin manifold with smooth spacelike Cauchy boundary the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah-Patodi-Singer boundary conditions are imposed. In this paper we investigate to what extent these boundary conditions can be replaced by more general ones and how the index then changes. There are some differences to the classical case of the elliptic Dirac operator on a Riemannian manifold with boundary.

 

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

We decompose locally in time maximal \(L^{q}\)-regular solutions of abstract quasilinear parabolic equations as a sum of a smooth term and an arbitrary small−with respect to the maximal \(L^{q}\)-regularity space norm−remainder. In view of this observation, we next consider the porous medium equation and the Swift-Hohenberg equation on manifolds with conical singularities. We write locally in time each solution as a sum of three terms, namely a term that near the singularity is expressed as a linear combination of complex powers and logarithmic integer powers of the singular variable, a term that decays to zero close to the singularity faster than each of the non-constant summands of the previous term and a remainder that can be chosen arbitrary small with respect e.g. to the \(C^{0}\)-norm. The powers in the first term are time independent and determined explicitly by the local geometry around the singularity, e.g. by the spectrum of the boundary Laplacian in the situation of straight conical tips. The case of the above two problems on closed manifolds is also considered and local space asymptotics for the solutions are provided.

 

Related project(s):
30Nonlinear evolution equations on singular manifolds

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.

 

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

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${\mathcal B} \to {\mathrm{D}}_{\mathcal B}(1 + {\mathrm{D}}_{\mathcal B}^2)^{-\frac{1}{2}}$ depends on Lipschitz smoothness and ellipticity of ${\mathcal B}$ and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.

 

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

In this paper we prove that any smooth surfaces can be locally isometrically embedded into as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.

 

JournalAnnales de l'Institut Henri Poincare (C) Non Linear Analysis
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Related project(s):
22Willmore functional and Lagrangian surfaces

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 [9A12] and disproves Conjecture 8.11 of Barbot-Mérigot [BM12]. 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 [Mou88] for Coxeter groups built on Danciger-Guéritaud-Kassel [DGK17] 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.

 

Related project(s):
1Hitchin components for orbifolds

We prove that if an ALE Ricci-flat manifold (M,g) is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to g. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable.

 

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible \(\sigma\)-compact space in which the corona sits as a \(\mathbb{Z}\)-set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space \(BG\), then our constructions yield a \(\mathbb{Z}\)-structure for the group.

Related project(s):
10Duality and the coarse assembly map

We prove a gluing theorem for solutions \((A_0, \Phi_0)\)  of Hitchin's self-duality equations with logarithmic singularities on a rank-\(2\) vector bundle over a noded Riemann surface \( \Sigma\) representing a boundary point of Teichmüller moduli space. We show that every nearby smooth Riemann surface \( \Sigma_1\) carries a smooth solution \((A_1, \Phi_1)\) of the self-duality equations, which may be viewed as a desingularization of \((A_0, \Phi_0)\).

JournalAdv. Math.
PublisherElsevier
Volume322
Pages637-681
Link to preprint version
Link to published version

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

We prove that the Teichmüller space of negatively curved metrics on a hyperbolic manifold M has nontrivial i-th rational homotopy groups for some i > dim M. Moreover, some elements of infinite order in the i-th homotopy group of BDiff(M) can be represented by bundles over a sphere with fiberwise negatively curved metrics.

 

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

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in the homotopy groups of the classifying space for smooth M-bundles, provided the dimension of M is large enough.

 

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

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic problems is achieved using the theory of Fourier integral operators on Lie groupoids. We show that our calculus is closed under composition and furnishes a so-called noncommutative completion of the given embedding. A representation of the algebra is defined and the continuity of the operators in the algebra on suitable Sobolev spaces is established.

 

 

Related project(s):
3Geometric operators on a class of manifolds with bounded geometry

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal \(L^q\)-regularity space for all times and is instantaneously smooth in space and time, where the maximal \(L^q\)-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.

 

Related project(s):
30Nonlinear evolution equations on singular manifolds

The canonical map from the \(\mathbb{Z}/2\)-equivariant Lazard ring to the \(\mathbb{Z}/2\)-equivariant complex bordism ring is an isomorphism. 

 

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

We study the Kazdan-Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.

 

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

We prove that the Atiyah-Singer Dirac operator ${\mathrm D}_{\mathrm g}$ in ${\mathrm L}^2$ depends Riesz continuously on ${\mathrm L}^{\infty}$ perturbations of complete metrics ${\mathrm g}$ on a smooth manifold. The Lipschitz bound for the map ${\mathrm g} \to {\mathrm D}_{\mathrm g}(1 + {\mathrm D}_{\mathrm g}^2)^{-\frac{1}{2}}$ depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\'on's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.

 

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

In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces \( \Sigma_R\) converging for \(R\searrow 0\) to a surface \( \Sigma_0\) with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration \(\mathcal M_d \to \Sigma_R\) gives rise to a graph-continuous Fredholm family, the index of it being stable when passing to the limit. We also report on similarities and differences between properties of the Hitchin fibration in this degeneration and in the limit of large Higgs fields as studied in Mazzeo et al. (Duke Math. J. 165(12):2227–2271, 2016).

 

JournalAbh. Math. Semin. Univ. Hambg.
PublisherSpringer Berlin Heidelberg
Volume86
Pages189--201
Link to preprint version
Link to published version

Related project(s):
32Asymptotic geometry of the Higgs bundle moduli space

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the η-invariant of the Cauchy hypersurfaces.

 

JournalCommun. Math. Phys.
PublisherSpringer
Volume347
Pages703-721
Link to preprint version
Link to published version

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