## 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“.

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

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

The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [2016] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, à la Mantoulidis and Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of an ad-hoc notion of charged Barnik mass for suitable charged minimal Bartnik data.

**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 |

Link to preprint version | |

Link to published version |

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

In this paper, we study the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle $2\pi\beta$ along the divisor, then for any $\beta'$ sufficiently close to $\beta$, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle $2\pi\beta'$ along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence before. As corollaries, we give parabolic proofs of Donaldson's openness theorem and his existence conjecture for the conical Kähler-Einstein metrics with positive Ricci curvatures.

**Related project(s):****31**Solutions to Ricci flow whose scalar curvature is bounded in Lp.

In this paper, we study the limiting flow of conical Kähler-Ricci flows when the cone angles tend to $0$. We prove the existence and uniqueness of this limiting flow with cusp singularity on compact Kähler manifold $M$ which carries a smooth hypersurface $D$ such that the twisted canonical bundle $K_M+D$ is ample. Furthermore, we prove that this limiting flow converge to a unique cusp Kähler-Einstein metric.

Journal | Annali di Matematica Pura ed Applicata (1923 -) |

Link to preprint version |

**Related project(s):****31**Solutions to Ricci flow whose scalar curvature is bounded in Lp.

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.

Journal | Comm. Math. Phys. |

Publisher | Springer |

Volume | 367, no. 1 |

Pages | 151-191 |

Link to preprint version | |

Link to published version |

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

We prove that the computation of the Fredholm index for fully elliptic pseudodifferential operators on Lie manifolds can be reduced to the computation of the index of Dirac operators perturbed by smoothing operators. To this end we adapt to our framework ideas coming from Baum-Douglas geometric K-homology and in particular we introduce a notion of geometric cycles that can be classified into a variant of the famous geometric K-homology groups, for the specific situation here. We also define comparison maps between this geometric K-homology theory and relative K-theory.

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

Moduli spaces of stable parabolic bundles of parabolic degree \(0\) over the Riemann sphere are stratified according to the Harder-Narasimhan filtration of underlying vector bundles. Over a Zariski open subset \(\mathscr{N}_{0}\) of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function \(\mathscr{S}\) is defined as the regularized critical value of the non-compact Wess-Zumino-Novikov-Witten action functional. The definition of \(\mathscr{S}\) depends on a suitable notion of parabolic bundle 'uniformization map' following from the Mehta-Seshadri and Birkhoff-Grothendieck theorems. It is shown that \(-\mathscr{S}\) is a primitive for a (1,0)-form \(\vartheta\) on \(\mathscr{N}_{0}\) associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that \(-\mathscr{S}\) is a Kähler potential for \((\Omega-\Omega_{\mathrm{T}})|_{\mathscr{N}_{0}}\), where \(\Omega\) is the Narasimhan-Atiyah-Bott Kähler form in \(\mathscr{N}\) and \(\Omega_{\mathrm{T}}\) is a certain linear combination of tautological \((1,1)\)-forms associated with the marked points. These results provide an explicit relation between the cohomology class \([\Omega]\) and tautological classes, which holds globally over certain open chambers of parabolic weights where \(\mathscr{N}_{0} = \mathscr{N}\).

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

Motivated by the work of Leznov-Mostovoy, we classify the linear deformations of standard \(2n\)-dimensional phase space that preserve the obvious symplectic \(\mathfrak{o}(n)\)-symmetry. As a consequence, we describe standard phase space, as well as \(T^{*}S^{n}\) and \(T^{*}\mathbb{H}^{n}\) with their standard symplectic forms, as degenerations of a 3-dimensional family of coadjoint orbits, which in a generic regime are identified with the Grassmannian of oriented 2-planes in \(\mathbb{R}^{n+2}\).

Journal | Journal of Geometric Mechanics |

Publisher | American Institute of Mathematical Sciences |

Volume | 11(1) |

Pages | 45-58 |

Link to preprint version | |

Link to published version |

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