## 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 prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains. Our class of domains contains the class of domains with isolated oscillating conical singularities. Our results thus generalize the classical results of Kondratiev on domains with conical singularities. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier results on manifolds with boundary and bounded geometry.

Journal | Comptes Rendus Mathématique Sér. I 357 487-493 (2019) |

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

Let M be a smooth manifold with (smooth) boundary ∂M and bounded geometry and ∂<sub>D</sub>M ⊂ ∂M be an open and closed subset. We prove the well-posedness of the mixed Robin boundary value problem Pu = f in M, u = 0 on ∂<sub>D</sub>M, ∂<sup>P</sup><sub>ν</sub> u + bu = 0 on ∂M \ ∂<sub>D</sub>M under the following assumptions. First, we assume that P satisfies the strong Legendre condition (which reduces to the uniformly strong ellipticity condition in the scalar case) and that it has totally bounded coefficients (that is, that the coefficients of P and all their derivatives are bounded). Let ∂<sub>R</sub>M ⊂ ∂M \ ∂<sub>D</sub>M be the set where b≠ 0.

Journal | Rev. Roumaine Math. Pures Appl. 64 85-111 (2019) |

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

Let M be a manifold with boundary and bounded geometry. We assume that M has "finite width", that is, that the distance from any point to the boundary is bounded uniformly. Under this assumption, we prove that the Poincaré inequality for vector valued functions holds on M. We also prove a general regularity result for uniformly strongly elliptic equations and systems on general manifolds with boundary and bounded geometry. By combining the Poincaré inequality with the regularity result, we obtain-as in the classical case-that uniformly strongly elliptic equations and systems are well-posed on M in Hadamard's sense between the usual Sobolev spaces associated to the metric. We also provide variants of these results that apply to suitable mixed Dirichlet-Neumann boundary conditions. We also indicate applications to boundary value problems on singular domains.

Journal | Mathematische Nachrichten |

Publisher | Wiley |

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

We compute the *p*-central and exponent-*p* series of all right angled Artin groups, and compute the dimensions of their subquotients. We also describe their associated Lie algebras, and relate them to the cohomology ring of the group as well as to a partially commuting polynomial ring and power series ring. We finally show how the growth series of these various objects are related to each other.

**Related project(s):****18**Analytic L2-invariants of non-positively curved spaces

Let *N*⊂*M* be a submanifold embedding of spin manifolds of some codimension *k*≥1. A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that *M* does not admit a metric of positive scalar curvature if *k*=2 and the Dirac operator of *N* has non-trivial index, provided that suitable conditions are satisfied. In the cases *k*=1 and *k*=2, Zeidler and Kubota, respectively, established more systematic results: There exists a transfer KO∗(C∗*π*1*M*)→KO∗−*k*(C∗*π*1*N*) which maps the index class of *M* to the index class of *N*. The main goal of this article is to construct analogous transfer maps *E*∗(B*π*1*M*)→*E*∗−*k*(B*π*1*N*) for different generalized homology theories *E* and suitable submanifold embeddings. The design criterion is that it is compatible with the transfer *E*∗(*M*)→*E*∗−*k*(*N*) induced by the inclusion *N*⊂*M* for a chosen orientation on the normal bundle. Under varying restrictions on homotopy groups and the normal bundle, we construct transfers in the following cases in particular: In ordinary homology, it works for all codimensions. This slightly generalizes a result of Engel and simplifies his proof. In complex K-homology, we achieve it for *k*≤3. For *k*≤2, we have a transfer on the equivariant KO-homology of the classifying space for proper actions.

**Related project(s):****9**Diffeomorphisms and the topology of positive scalar curvature

Let Γ be a finitely generated discrete group and let *M*˜ be a Galois Γ-covering of a smooth compact manifold *M*. Let *u*:*X*→*B*Γ be the associated classifying map. Finally, let SΓ∗(*M*˜) be the analytic structure group, a K-theory group appearing in the Higson-Roe exact sequence ⋯→SΓ∗(*M*˜)→*K*∗(*M*)→*K*∗(*C*∗Γ)→⋯. Under suitable assumptions on Γ we construct two pairings, first between SΓ∗(*M*˜) and the delocalized part of the cyclic cohomology of CΓ, and secondly between SΓ∗(*M*˜) and the relative cohomology *H*∗(*M*→*B*Γ). Both are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class *ρ*(*D*˜)∈SΓ∗(*M*˜) of an invertible Γ-equivariant Dirac type operator on *M*˜. Regarding the first pairing we establish in fact a more general result, valid without additional assumptions on Γ: indeed, we prove that it is possible to map the Higson-Roe analytic surgery sequence to the long exact sequence in noncommutative de Rham homology ⋯−→*j*∗*H*∗−1(AΓ)→*ι**H**d**e**l*∗−1(AΓ)→*δ**H**e*∗(AΓ)−→*j*∗⋯ with AΓ a dense homomorphically closed subalgebra of *C*∗*r*Γ and *H**d**e**l*∗(AΓ) and *H**e*∗(AΓ) denoting versions of the delocalized homology and the homology localized at the identity element, respectively.

**Related project(s):****9**Diffeomorphisms and the topology of positive scalar curvature

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick, Xie.

**Related project(s):****9**Diffeomorphisms and the topology of positive scalar curvature

Using standard results from higher (secondary) index theory, we prove that the positive scalar curvature bordism groups of a cartesian product GxZ are infinite in dimension 4n if n>0 G a group with non-trivial torsion. We construct representatives of each of these classes which are connected and with fundamental group GxZ. We get the same result in dimension 4n+2 (n>0) if G is finite and contains an element which is not conjugate to its inverse. This generalizes the main result of Kazaras, Ruberman, Saveliev, "On positive scalar curvature cobordism and the conformal Laplacian on end-periodic manifolds" to arbitrary even dimensions and arbitrary groups with torsion.

**Related project(s):****9**Diffeomorphisms and the topology of positive scalar curvature

We show that the moduli space of metrics of nonnegative sectional curvature on every homotopy RP^5 has infinitely many path components. We also show that in each dimension 4k+1 there are at least 2^{2k} homotopy RP^{4k+1}s of pairwise distinct oriented diffeomorphism type for which the moduli space of metrics of positive Ricci curvature has infinitely many path components. Examples of closed manifolds with finite fundamental group with these properties were known before only in dimensions 4k+3≥7.

Journal | to appear in Transactions of the AMS |

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

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index, and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractiblegeometric realizations.

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

In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.

**Related project(s):****1**Hitchin components for orbifolds

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 a work of Barbot et al. and disproves Conjecture 8.11 of Barbot-Mérigot [Groups Geom. Dyn. 6 (2012), pp. 441-483]. 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 (Ph.D. Thesis) for Coxeter groups built on Danciger-Guéritaud-Kassel's 2017 work 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.

Journal | Transactions of the American Mathematical Society |

Volume | 372 |

Pages | 153-186 |

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**Related project(s):****1**Hitchin components for orbifolds

We study the Thurston–Parreau boundary both of the Hitchinand of the maximal character varieties and determine therein an open setof discontinuity for the action of the mapping class group. This result isobtained as consequence of a canonical decomposition of a geodesic cur-rent on a surface of finite type arising from a topological decompositionof the surface along special geodesics. We show that each componenteither is associated to a measured lamination or has positive systole. Fora current with positive systole, we show that the intersection function onthe set of closed curves is bi-Lipschitz equivalent to the length functionwith respect to a hyperbolic metric.

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

In this paper we investigate the Hausdorff dimension of limitsetsof Anosov representations. In this context we revisit and extend the frameworkof hyperconvex representations and establish a convergence property for them,analogue to a differentiability property. As an applicationof this convergence,we prove that the Hausdorff dimension of the limit set of a hyperconvex rep-resentation is equal to a suitably chosen critical exponent. In the appendix, incollaboration with M. Bridgeman, we extend a classical result on the Hessianof the Hausdorff dimension on purely imaginary directions.

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

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 |

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**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