## 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“.

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

Photon surfaces are timelike, totally umbilic hypersurfaces of Lorentzian spacetimes. In the first part of this paper, we locally characterize all possible photon surfaces in a class of static, spherically symmetric spacetimes that includes Schwarzschild, Reissner--Nordström, Schwarzschild-anti de Sitter, etc., in n+1dimensions. In the second part, we prove that any static, vacuum, "asymptotically isotropic" n+1-dimensional spacetime that possesses what we call an "equipotential" and "outward directed" photon surface is isometric to the Schwarzschild spacetime of the same (necessarily positive) mass, using a uniqueness result by the first named author.

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

Link to preprint version | |

Link to published version |

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

We establish a one-to-one correspondence between Finsler structures on the \(2\)-sphere with constant curvature \(1\) and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $$\mathbb{CP}(a_1,a_2)\to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)$$ of weighted projective spaces provide examples of Finsler \(2\)-spheres of constant curvature and all geodesics closed.

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

This article studies the bundle of Weyl structures associated to a parabolic geometry. Given a parabolic geometry of any type on a smooth manifold \(M\), this is a natural bundle \(A\to M\), whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on \(A\), which induces an almost bi-Lagrangian structure on \(A\) and a compatible linear connection on \(TA\). In the first part of the article, we prove that all elements of the theory of Weyl structures can be interpreted in terms of natural geometric operations on \(A\). In the second part of the article, we turn our point of view around and use the relation to parabolic geometries and Weyl structures to study the intrinsic geometry on \(A\). This geometry is rather exotic outside of the class of torsion free parabolic geometries associated to a \(|1|\)-grading (i.e. AHS structures), to which we restrict for the rest of the article. We prove that the split-signature metric provided by the almost bi-Lagrangian structure is always Einstein. We proceed to study Weyl structures via the submanifold geometry of the image of the corresponding section in \(A\). For Weyl structures satisfying appropriate non-degeneracy conditions, we derive a universal formula for the second fundamental form of this image. In the last part of the article, we show that for locally flat projective structures, this has close relations to solutions of a projectively invariant Monge-Ampère equation and thus to properly convex projective structures. Analogs for other AHS structures are indicated at the end of the article.

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

We construct a slant product on the analytic structure group of Higson and Roe and the K-theory of the stable Higson corona of Emerson and Meyer, which is the domain of the co-assembly map. In fact, we obtain such products simultaneously on the entire Higson-Roe sequence. The existence of these products implies injectivity results for external product maps on each term of the Higson-Roe sequence. Our results apply in particular to taking products with aspherical manifolds whose fundamental groups admit coarse embeddings into Hilbert space. To conceptualize the general class of manifolds where this method applies, we introduce the notion of Higson-essentialness. A complete spin-c manifold is Higson-essential if its fundamental class is detected by the stable Higson corona via the co-assembly map. We prove that coarsely hypereuclidean manifolds are Higson-essential. Finally, we draw conclusions for positive scalar curvature metrics on product spaces, in particular on non-compact manifolds. We also construct suitable equivariant versions of these slant products and discuss related problems of exactness and amenability of the stable Higson corona.

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

In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prove that the entropies are monotone along the singular Ricci de Turck flow. We employ these entropies to show that in the singular setting, Ricci solitons are gradient and that steady or expanding Ricci solitons are Einstein.

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry**23**Spectral geometry, index theory and geometric flows on singular spaces

We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this paper seems to be new even in $\mathbb{R}^n$.

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

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.

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

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.

**Related project(s):****15**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact 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 show \(R\)-sectoriality for the fractional powers of possibly non-invertible \(R\)-sectorial operators. Applications concern existence, uniqueness and maximal \(L^{q}\)-regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided and its relation to the local geometry is established. Our method extends the freezing-of-coefficients method to the case of non-local operators that are expressed as linear combinations of terms in the form of a product of a function and a fractional power of a local operator.

Pages | 27 |

Link to preprint version |

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

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