## 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 immersed surfaces in R3 which are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane orthogonally along the boundary, and when the boundary is contained in a line. In both cases we derive weak forms of the resulting free boundary conditions and prove regularity by reflection.

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

For the Willmore flow of spheres in R^n with small energy, we prove stability estimates for the barycenter, the quadratic moment, and in case n=3 also for the enclosed volume and averaged mean curvature. As applications, we give a new proof for a quasi-rigidity estimate due to De Lellis and Müller, also for an inequality by Röger and Schätzle for the isoperimetric deficit.

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

Let M be a compact Riemannian manifold which does not admit any immersed surface which is totally geodesic. We prove that then any completely immersed surface in M has area bounded in terms of the L^2 norm of the second fundamental form.

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

A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely solvable up to gauge solutions, given initial data on a spacelike Cauchy hypersurface. The solution map is an isomorphism between initial data (modulo gauge producing initial data) and solutions (modulo gauge solutions). In the first part of this work, we show that the solution map is actually an isomorphism of locally convex topological vector spaces. This implies that the equivalence class of solutions depends continuously on the equivalence class of initial data. We may therefore conclude well-posedness of the Cauchy problem. In the second part, we show that the linearised constraint equations can always be solved on a closed manifold with vanishing scalar curvature. This generalises the classical notion of TT-tensors on flat space used to produce models of gravitational waves. All our results are proven for smooth and distributional initial data of arbitrary real Sobolev regularity.

Journal | Annales Henri Poincaré |

Publisher | Springer International Publishing |

Volume | 20 |

Pages | 3849–3888 |

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Link to published version |

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

We prove that any compact Cauchy horizon with constant non-zero surface gravity in a smooth vacuum spacetime is a Killing horizon. The novelty here is that the Killing vector field is shown to exist on both sides of the horizon. This generalises classical results by Moncrief and Isenberg, by dropping the assumption that the metric is analytic. In previous work by Rácz and the author, the Killing vector field was constructed on the globally hyperbolic side of the horizon. In this paper, we prove a new unique continuation theorem for wave equations through smooth compact lightlike (characteristic) hypersurfaces which allows us to extend the Killing vector field beyond the horizon. The main ingredient in the proof of this theorem is a novel Carleman type estimate. Using a well-known construction, our result applies in particular to smooth stationary asymptotically flat vacuum black hole spacetimes with event horizons with constant non-zero surface gravity. As a special case, we therefore recover Hawking's local rigidity theorem for such black holes, which was recently proven by Alexakis-Ionescu-Klainerman using a different Carleman type estimate.

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

An action of a compact Lie group is called equivariantly formal, if the Leray--Serre spectral sequence of its Borel fibration degenerates at the E_2-term. This term is as prominent as it is restrictive.

In this article, also motivated by the lack of junction between the notion of equivariant formality and the concept of formality of spaces (surging from rational homotopy theory) we suggest two new variations of equivariant formality: "MOD-formal actions" and "actions of formal core".

We investigate and characterize these new terms in many different ways involving various tools from rational homotopy theory, Hirsch--Brown models, A∞-algebras, etc., and, in particular, we provide different applications ranging from actions on symplectic manifolds and rationally elliptic spaces to manifolds of non-negative sectional curvature.

A major motivation for the new definitions was that an almost free action of a torus Tn↷X possessing any of the two new properties satisfies the toral rank conjecture, i.e. dimH∗(X;Q)≥2n. This generalizes and proves the toral rank conjecture for actions with formal orbit spaces.

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

We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization.

Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory.

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

We show that the space introduced by Vasy in order to construct a pseudodifferential calculus adapted to the N-body problem can be obtained as the primitive ideal spectrum of one of the N-body algebras considered by Georgescu. In the process, we provide an alternative description of the iterated blow-up space of a manifold with corners with respect to a clean semilattice of adapted submanifolds (i.e. p-submanifolds). Since our constructions and proofs rely heavily on manifolds with corners and their submanifolds, we found it necessary to clarify the various notions of submanifolds of a manifold with corners.

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

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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Link to published version |

**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 |

Link to preprint version |

**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