## 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“.

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):****28**Rigidity, 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):****15**Spaces 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):****15**Spaces 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.

Journal | Journal de Mathématiques Pures et Appliquées. (9) |

Volume | 126 |

Pages | 109--143 |

Link to preprint version | |

Link to published version |

**Related project(s):****19**Boundaries, 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):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact 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):****31**Solutions 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.

Journal | Amer. J. Math. |

Publisher | John Hopkins Univ. Press |

Volume | 141 (5) |

Pages | 1421-1455 |

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Link to published version |

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

Let *X* be a compact manifold, *G* a Lie group, *P*→*X* a principal *G*-bundle, and *B_**P* the infinite-dimensional moduli space of connections on *P* modulo gauge. For a real elliptic operator *E* we previously studied orientations on the real determinant line bundle over *B_**P*. These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson.

Here we consider complex elliptic operators *F* and introduce the idea of spin structures, square roots of the complex determinant line bundle of *F*. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures. Our main result identifies spin structures on *X* with orientations on *X*×*S*1. Thus, if *P*→*X* and *Q*→*X*×*S*1 are principal *G*-bundles with *Q*|*X*×{1}≅*P*, we relate spin structures on (*B_**P*,*F*) to orientations on (*B_**Q*,*E*) for a certain class of operators *F* on *X* and *E* on *X*×*S*1.

Combined with arXiv:1811.02405, we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups *G*=*U*(*m*),*S**U*(*m*). In a sequel we will apply this to define canonical orientation data for all Calabi-Yau 3-folds *X* over the complex numbers, as in Kontsevich-Soibelman arXiv:0811.2435, solving a long-standing problem in Donaldson-Thomas theory.

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

We investigate Bartnik's static metric extension conjecture under the additional assumption of axisymmetry of both the given Bartnik data and the desired static extensions. To do so, we suggest a geometric flow approach, coupled to the Weyl-Papapetrou formalism for axisymmetric static solutions to the Einstein vacuum equations. The elliptic Weyl-Papapetrou system becomes a free boundary value problem in our approach. We study this new flow and the coupled flow--free boundary value problem numerically and find axisymmetric static extensions for axisymmetric Bartnik data in many situations, including near round spheres in spatial Schwarzschild of positive mass.

Journal | Pure and Applied Mathematics Quaterly |

Link to preprint version |

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

We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.

We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We provide examples which are conveniently treated by our methods.

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

We study the set of trapped photons of a subcritical (a<M) Kerr spacetime as a subset of the phase space. First, we present an explicit proof that the photons of constant Boyer--Lindquist coordinate radius are the only photons in the Kerr exterior region that are trapped in the sense that they stay away both from the horizon and from spacelike infinity. We then proceed to identify the set of trapped photons as a subset of the (co-)tangent bundle of the subcritical Kerr spacetime. We give a new proof showing that this set is a smooth 5-dimensional submanifold of the (co-)tangent bundle with topology SO(3)×R2 using results about the classification of 3-manifolds and of Seifert fiber spaces. Both results are covered by the rigorous analysis of Dyatlov [5]; however, the methods we use are very different and shed new light on the results and possible applications.

Journal | General Relativity and Gravitation |

Publisher | Springer |

Link to preprint version |

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

For every finite graph Γ, we define a simplicial complex associated to the outer automorphism group of the RAAG A_Γ. These complexes are defined as coset complexes of parabolic subgroups of Out^0(A_Γ) and interpolate between Tits buildings and free factor complexes. We show that each of these complexes is homotopy Cohen-Macaulay and in particular homotopy equivalent to a wedge of d-spheres. The dimension d can be read off from the defining graph Γ and is determined by the rank of a certain Coxeter subgroup of Out^0(A_Γ). In order to show this, we refine the decomposition sequence for Out^0(A_Γ) established by Day-Wade, generalise a result of Brown concerning the behaviour of coset posets under short exact sequences and determine the homotopy type of relative free factor complexes associated to Fouxe-Rabinovitch groups.

**Related project(s):****8**Parabolics and invariants

We prove the K- and L-theoretic Farrell-Jones Conjecture with coefficients in an additive category for every normally poly-free group, in particular for even Artin groups of FC-type, and for all groups of the form A⋊Z where A is a right-angled Artin group. Our proof relies on the work of Bestvina-Fujiwara-Wigglesworth on the Farrell-Jones Conjecture for free-by-cyclic groups.

**Related project(s):****8**Parabolics and invariants

Realizations of differential operators subject to differential boundary conditions on manifolds with conical singularities are shown to have a bounded \(H_{\infty}\)-calculus in appropriate \(L_{p}\)-Sobolev spaces provided suitable conditions of parameter-ellipticity are satisfied. Applications concern the Dirichlet and Neumann Laplacian and the porous medium equation.

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

We prove an affine analog of Scharlau's reduction theorem for spherical buildings. To be a bit more precise let *X* be a euclidean building with spherical building ∂*X* at infinity. Then there exists a euclidean building *X*¯ such that *X* splits as a product of *X*¯ with some euclidean *k*-space such that ∂*X*¯ is the thick reduction of ∂*X* in the sense of Scharlau. \newline In addition we prove a converse statement saying that an embedding of a thick spherical building at infinity extends to an embedding of the euclidean building having the extended spherical building as its boundary.

**Related project(s):****20**Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings

We strengthen a result of Hanke-Schick about the strong Novikov conjecture for low degree cohomology by showing that their non-vanishing result for the maximal group C*-algebra even holds for the reduced group C*-algebra. To achieve this we provide a Fell absorption principle for certain exotic crossed product functors.

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

We construct examples of fibered three-manifolds with fibered faces all of whose monodromies extend to a handlebody.

**Related project(s):****8**Parabolics and invariants

Mantoulidis and Schoen developed a novel technique to handcraft asymptotically flat extensions of Riemannian manifolds (Σ≅S2,g), with g satisfying λ1=λ1(−Δg+K(g))>0, where λ1 is the first eigenvalue of the operator −Δg+K(g) and K(g) is the Gaussian curvature of g, with control on the ADM mass of the extension. Remarkably, this procedure allowed them to compute the Bartnik mass in this so-called minimal case; the Bartnik mass is a notion of quasi-local mass in General Relativity which is very challenging to compute. In this survey, we describe the Mantoulidis-Schoen construction, its impact and influence in subsequent research related to Bartnik mass estimates when the minimality assumption is dropped, and its adaptation to other settings of interest in General Relativity.

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

We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian 2-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties. In particular, we prove that they admit a dominated splitting and we identify special cases in which the flows are Anosov. In the latter case, we study when they admit an invariant measure in the Lebesgue class and the regularity of the weak foliations.

Journal | Mathematische Annalen |

Publisher | Springer |

Volume | 373 |

Pages | 553--580 |

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Link to published version |

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