## 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 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

In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied.

Journal | Calc. Var. Part. Differ. Eq. |

Publisher | Springer |

Volume | 58 |

Pages | 75 |

Link to preprint version | |

Link to published version |

**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 consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold *M*×R, where *M* is asymptotically flat. If the initial hypersurface *F*0⊂*M*×R is uniformly spacelike and asymptotic to *M*×{*s*} for some *s*∈R at infinity, we show that the mean curvature flow starting at *F*0 exists for all times and converges uniformly to *M*×{*s*} as *t*→∞.

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry**29**Curvature flows without singularities**30**Nonlinear evolution equations on singular manifolds**31**Solutions to Ricci flow whose scalar curvature is bounded in Lp.

We consider the Cahn-Hilliard equation on manifolds with conical singularities. For appropriate initial data we show that the solution exists in the maximal \(L^q\)-regularity space for all times and becomes instantaneously smooth in space and time, where the maximal \(L^q\)-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we provide precise information concerning the asymptotic behavior of the solution close to the conical tips in terms of the local geometry.

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

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the canonical Obata connection on the Hilbert scheme of cohomologically stable twisted rational curves of degree *d* in the ℙ3∖ℙ1 is flat for any *d*≥3.

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

We investigate the geometry of the Kodaira moduli space* M* of sections of a twistor projection, the normal bundle of which is allowed to jump. In particular, we identify the natural assumptions which guarantee that the Obata connection of the hypercomplex part of *M* extends to a logarithmic connection on *M*.

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

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 extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of the pressure metric, cyclic Higgs bundles, and the deformation theory of real projective structures on 3-manifolds.

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

In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds can not be established.

Journal | to appear in Proceedings of the American Mathematical Society |

Link to preprint version |

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

In this note we prove an optimal volume growth condition for stochastic completeness of graphs under very mild assumptions. This is realized by proving a uniqueness class criterion for the heat equation which is an analogue to a corresponding result of Grigor'yan on manifolds. This uniqueness class criterion is shown to hold for graphs that we call globally local, i.e., graphs where we control the jump size far outside. The transfer from general graphs to globally local graphs is then carried out via so called refinements.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

In this expository paper we answer two fundamental questions concerning discrete magnetic Schrödinger operator associated with weighted graphs. We discuss when formal expressions of such operators give rise to self-adjoint operators, i.e., when they have self-adjoint restrictions. If such self-adjoint restrictions exist, we explore when they are unique.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

We study pairs of Dirichlet forms related by an intertwining order

isomorphisms between the associated \(L^2\)-spaces. We consider the

measurable, the topological and the geometric setting respectively.

In the measurable setting, we deal with arbitrary (irreducible)

Dirichlet forms and show that any intertwining order isomorphism is

necessarily unitary (up to a constant). In the topological setting

we deal with quasi-regular forms and show that any intertwining

order isomorphism induces a quasi-homeomorphism between the

underlying spaces. In the geometric setting we deal with both

regular Dirichlet forms as well as resistance forms and essentially

show that the geometry defined by these forms is preserved by

intertwining order isomorphisms. In particular, we prove in the

strongly local regular case that intertwining order isomorphisms

induce isometries with respect to the intrinsic metrics between the

underlying spaces under fairly mild assumptions. This applies to a

wide variety of metric measure spaces including

\(\mathrm{RCD}(K,N)\)-spaces, complete weighted Riemannian manifolds

and complete quantum graphs. In the non-local regular case our

results cover in particular graphs as well as fractional Laplacians

as arising in the treatment of \(\alpha\)-stable Lévy processes. For

resistance forms we show that intertwining order isomorphisms are

isometries with respect to the resistance metrics.

Our results can can be understood as saying that diffusion always

determines the Hilbert space, and -- under natural compatibility

assumptions -- the topology and the geometry respectively. As special

instances they cover earlier results for manifolds and graphs.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces