## 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 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.

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**30**Nonlinear evolution equations on singular manifolds

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 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

The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model time-symmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant. Following the ideas of Mantoulidis and Schoen [2016], of Miao and Xie [2016], and of joint work of Miao and the authors [2017], we construct asymptotically hyperbolic extensions of minimal and constant mean curvature (CMC) Bartnik data while controlling the total mass of the extensions. We establish that for minimal surfaces satisfying a stability condition, the Bartnik mass is bounded above by the conjectured lower bound coming from the asymptotically hyperbolic Riemannian Penrose inequality. We also obtain estimates for such a hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature.

Journal | J. Geom. Phys. |

Publisher | Elsevier |

Volume | 132 |

Pages | 338--357 |

Link to preprint version | |

Link to published version |

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

We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a covariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau. We show that a unique STCMC-foliation exists near infinity of any asymptotically Euclidean initial data set with non-vanishing energy which allows for the definition of a new notion of total center of mass for isolated systems. This STCMC-center of mass transforms equivariantly under the asymptotic Poincaré group of the ambient spacetime and in particular evolves under the Einstein evolution equations like a point particle in Special Relativity. The new definition also remedies subtle deficiencies in the CMC-approach to defining the total center of mass suggested by Huisken and Yau which were described by Cederbaum and Nerz.

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

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

We define a functional \({\cal J}(h)\) for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that \({\cal J}(h)\) is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We derive a formula relating \({\cal J}(h)\) and another functional \({\cal I}(h)\), closely related to the Yang-Mills-Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of \({\cal J}(h)\), which is expressed as a certain \(L^{2}\)-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of \({\cal J}(h)\) if and only if the corresponding Hitchin-Simpson mean curvature is parallel with respect to the Hitchin-Simpson connection.

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

For a smooth manifold \(M\), possibly with boundary and corners, and a Lie group \(G\), we consider a suitable description of gauge fields in terms of parallel transport, as groupoid homomorphisms from a certain path groupoid in \(M\) to \(G\). Using a cotriangulation \(\mathscr{C}\) of \(M\), and collections of finite-dimensional families of paths relative to \(\mathscr{C}\), we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains further local topological information sufficient to reconstruct a principal \(G\)-bundle on \(M\) up to equivalence. The space of ELG fields of a given pair \((M,\mathscr{C})\) is a covering for the space of fields in Lattice Gauge Theory, whose connected components parametrize equivalence classes of principal \(G\)-bundles on \(M\). We give a criterion to determine when ELG fields over different cotriangulations define equivalent bundles.

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