## 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 describe the natural geometry of Hilbert schemes of curves in projective spaces.

**Related project(s):****7**Asymptotic geometry of moduli spaces of curves

In this paper, we study the long-time behavior of modified Calabi flow to study the existence of generalized Kähler-Ricci soliton. We first give a new expression of the modified $K$-energy and prove its convexity along weak geodesics. Then we extend this functional to some finite energy spaces. After that, we study the long-time behavior of modified Calabi flow.

Journal | The Journal of Geometric Analysis |

Link to preprint version |

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

We develop a categorical index calculus for elliptic symbol families. The categorified index problems we consider are a secondary version of the traditional problem of expressing the index class in K-theory in terms of differential-topological data. They include orientation problems for moduli spaces as well as similar problems for skew-adjoint and self-adjoint operators. The main result of this paper is an excision principle which allows the comparison of categorified index problems on different manifolds. Excision is a powerful technique for actually solving the orientation problem; applications appear in the companion papers arXiv:1811.01096, arXiv:1811.02405, and arXiv:1811.09658.

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

Pseudo H-type Lie groups \(G_{r,s}\) of signature (r,s) are defined via a module action of the Clifford algebra \(C\ell_{r,s}\) on a vector space V≅\(\mathbb{R}^{2n}\). They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let \(\mathcal{N}_{r,s}\) denote the Lie algebra corresponding to \(G_{r,s}\). A choice of left-invariant vector fields [\(X_1, \ldots, X_{2n}\)] which generate a complement of the center of \(\mathcal{N}_{r,s}\) gives rise to a second order operator

\(\Delta_{r,s}:=\big{(}X_1^2+ \ldots + X_n^2\big{)}- \big{(}X_{n+1}^2+ \ldots +X_{2n}^2 \big{)}\)

which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of \(\Delta_{r,s}\) in the case r=0, s>0 and study their properties. In the case of r>0 we prove that \(\Delta_{r,s}\) admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of \(\Delta_{r,s}\) and the existence of a fundamental solution in the space of Schwartz distributions.

**Related project(s):****6**Spectral Analysis of Sub-Riemannian Structures

We prove that the sign of the Euler characteristic of arithmetic groups with CSP is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F. Our methods imply similar results for L2-torsion as well as a strong profiniteness statement for Novikov--Shubin invariants.

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

Let *N* be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold *M*. In this paper, we study the extent to which *N* admits as much symmetry as *M*. Our main results are examples of *N* that exhibit two extremes of behavior. On the one hand, we find *N* with maximal symmetry, i.e. Isom(*M*) acts on *N* by isometries with respect to some negatively curved metric on *N*. For these examples, Isom(*M*) can be made arbitrarily large. On the other hand, we find *N* with little symmetry, i.e. no subgroup of Isom(*M*) of "small" index acts by diffeomorphisms of *N*. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky-Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group.

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

In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This survey is succinct rather than comprehensive, and its aim is to inspire geometers and analysts alike to study these methods so that they can be adapted and potentially applied more widely.

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

We construct a Baum--Connes assembly map localised at the unit element of a discrete group $\Gamma$.

This morphism, called $\mu_\tau$, is defined in $KK$-theory with coefficients in $\mathbb{R}$ by means of the action of the projection $[\tau]\in KK_\mathbb{R}^\Gamma(\mathbb{C},\mathbb{C})$ canonically associated to the group trace of $\Gamma$. The right hand side of $\mu_\tau$ is functorial with respect to the group $\Gamma$.

We show that the corresponding $\tau$-Baum--Connes conjecture is weaker then the classical one but still implies the strong Novikov conjecture.

**Related project(s):****4**Secondary invariants for foliations

We show that a properly convex projective structure \(\mathfrak{p}\) on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if \(\mathfrak{p}\) is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that \(\mathfrak{p}\) admits a compatible Weyl connection if and only if a certain holomorphic curve exists. Turning this non-linear PDE into a transport equation, we obtain our result by applying methods from geometric inverse problems. In particular, we use an extension of a remarkable \(L^2\)-energy identity known as Pestov's identity to prove a vanishing theorem for the relevant transport equation.

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

We construct $\eta$- and $\rho$-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah–Patodi–Singer index theorem in this setting, as well as its higher generalisation. Applications concern the classification of positive scalar curvature metrics on closed spin manifolds. We also investigate the properties of these twisted invariants for the signature operator and the relation to the higher invariants.

Journal | Math. Proc. Camb. Philos. Soc. |

Publisher | Cambridge University Press |

Volume | August 2018 |

Link to preprint version | |

Link to published version |

**Related project(s):****4**Secondary invariants for foliations

We show that a finitely generated residually finite rationally solvable (or RFRS) group *G* is virtually fibred, in the sense that it admits a virtual surjection to $\mathbb Z$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of *G* vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of 3-manifolds.

Journal | to appear in J. Amer. Math. Soc |

Link to preprint version |

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

We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of Sikorav.

We offer several applications: we reprove Thurston's theorem on the existence of a polytope controlling the BNS invariants of a 3-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincare duality groups of type F in dimension 3 and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl.

We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the *L*2-torsion polytope of Friedl-Lueck is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl-Lueck-Tillmann.

Journal | to appear in Invent. Math. |

Link to preprint version |

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

Let $X$ be a Banach space or more generally a complete metric space admitting a conical geodesic bicombing. We prove that every closed $L$-Lipschitz curve $\gamma:S^1\rightarrow X$ may be extended to an $L$-Lipschitz map defined on the hemisphere $f:H^2\rightarrow X$. This implies that $X$ satisfies a quadratic isoperimetric inequality (for curves) with constant $\frac{1}{2\pi}$. We discuss how this fact controls the regularity of minimal discs in Finsler manifolds when applied to the work of Alexander Lytchak and Stefan Wenger.

Journal | Transactions of the American Mathematical Society |

Link to preprint version |

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

Let (Mi,gi)i∈N be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower dimensional Riemannian manifold (B,h) in the Gromov-Hausdorff topology. Lott showed that the spectrum converges to the spectrum of a certain first order elliptic differential operator D on B. In this article we give an explicit description of D. We conclude that D is self-adjoint and characterize the special case where D is the Dirac operator on B.

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

For a closed, connected direct product Riemannian manifold (M,g)=(M1×⋯×Ml,g1⊕⋯⊕gl), we define its multiconformal class [[g]] as the totality {f12g1⊕⋯⊕fl2gl} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a function fi2>0 on the total space M. A multiconformal class [[g]] contains not only all warped product type deformations of g but also the whole conformal class [g~] of every g~∈[[g]]. In this article, we prove that [[g]] carries a metric of positive scalar curvature if and only if the conformal class of some factor (Mi,gi) does, under the technical assumption dimMi≥2. We also show that, even in the case where every factor (Mi,gi) has positive scalar curvature, [[g]] carries a metric of scalar curvature constantly equal to −1 and with arbitrarily large volume, provided l≥2 and dimM≥3. In this case, such negative scalar curvature metrics within [[g]] for l=2 cannot be of any warped product type.

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

We study the behavior of the spectrum of the Dirac operator together with a symmetric W1,∞-potential on a collapsing sequence of spin manifolds with bounded sectional curvature and diameter losing one dimension in the limit. If there is an induced spin or pin− structure on the limit space N, then there are eigenvalues that converge to the spectrum of a first order differential operator D on N together with a symmetric W1,∞-potential. In the case of an orientable limit space N, D is the spin Dirac operator DN on N if the dimension of the limit space is even and if the dimension of the limit space is odd, then D=DN⊕−DN.

Journal | Manuscripta Mathematica |

Publisher | Springer |

Pages | 1-24 |

Link to preprint version | |

Link to published version |

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

We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross-section we show well posedness of the short time solution in the \(L^q\)-setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asymptotic expansion of the evolving metric close to the boundary in terms of the initial local geometry. Due to the blow up of the scalar curvature close to the singularities we use maximal \(L^q\)-regularity theory for conically degenerate operators.

Journal | J. Evol. Equ. |

Link to preprint version |

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

We consider pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators, which are defined on the product manifold $M_1 \times M_2$ for closed manifolds $M_1$ and $M_2$. We prove a topological index theorem for Fredholm operators of tensor product type. To this end we construct a suitable double deformation groupoid and prove a Poincaré duality type result in relative $K$-theory.

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

We extend two known existence results to simply connected manifolds with

positive sectional curvature: we show that there exist pairs of simply

connected positively-curved manifolds that are tangentially homotopy equivalent

but not homeomorphic, and we deduce that an open manifold may admit a pair of

non-homeomorphic simply connected and positively-curved souls. Examples of such

pairs are given by explicit pairs of Eschenburg spaces. To deduce the second

statement from the first, we extend our earlier work on the stable converse

soul question and show that it has a positive answer for a class of spaces that

includes all Eschenburg spaces.

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Link to preprint version |

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