## 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“.

On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator $\mathrm{D}_{\mathcal B}$ in $\mathrm{L}^{2}$ depends Riesz continuously on $\mathrm{L}^{\infty}$ perturbations of local boundary conditions ${\mathcal B}$. The Lipschitz bound for the map ${\mathcal B} \to {\mathrm{D}}_{\mathcal B}(1 + {\mathrm{D}}_{\mathcal B}^2)^{-\frac{1}{2}}$ depends on Lipschitz smoothness and ellipticity of ${\mathcal B}$ and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions.

Journal | Communications in Partial Differential Equations |

Publisher | Taylor and Francis |

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

Let **M**(n,D) be the space of closed n-dimensional Riemannian manifolds (M,g) with diam(M)≤D and |secM|≤1. In this paper we consider sequences (Mi,gi) in **M**(n,D) converging in the Gromov–Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient vol(BMir(x))/injMi(x) can be uniformly bounded from below by a positive constant C(n, r, Y) for all points x∈Mi. On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient vol(BMir(x))/injMi(x) uniformly from below for all x∈Mi. As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in **M**(n,D) with C≤vol(M)/inj(M).

Journal | Journal of Geometric Analysis |

Publisher | Springer |

Volume | 28, no. 3 |

Pages | 2707-2724 |

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

We investigate the problem of calculating the Fredholm index of a geometric Dirac operator subject to local (e.g. Dirichlet and Neumann) and non-local (APS) boundary conditions posed on the strata of a manifold with corners. The boundary strata of the manifold with corners can intersect in higher codimension. To calculate the index we introduce a glueing construction and a corresponding Lie groupoid. We describe the Dirac operator subject to mixed boundary conditions via an equivariant family of Dirac operators on the fibers of the Lie groupoid. Using a heat kernel method with rescaling we derive a general index formula of the Atiyah-Singer type.

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

We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus at least 3 without punctures is Sp_2*g*(2), thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz's results on C-linear representations of mapping class groups to projective representations over any field.

Journal | To appear in Groups Geom. Dyn. |

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

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal \(L^q\)-regularity space for all times and is instantaneously smooth in space and time, where the maximal \(L^q\)-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.

Journal | Comm. Partial Differential Equations 43, no 10, 1456-1484 (2018) |

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

We construct a codimension 3completely non-holonomic subbundle on the Gromoll–Meyer exotic 7-sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method can be applied to construct a codimension 3 completely non-holonomic subbundle on the standard 7-sphere (or more general on a (4*n *+3)-dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides a new and simple proof for the standard sphere *S*7.

Journal | Appl. Anal. 96 (2017), 2390–2407. |

Link to preprint version |

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

In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.

**Related project(s):****4**Secondary invariants for foliations**11**Topological and equivariant rigidity in the presence of lower curvature bounds

We decompose locally in time maximal \(L^{q}\)-regular solutions of abstract quasilinear parabolic equations as a sum of a smooth term and an arbitrary small−with respect to the maximal \(L^{q}\)-regularity space norm−remainder. In view of this observation, we next consider the porous medium equation and the Swift-Hohenberg equation on manifolds with conical singularities. We write locally in time each solution as a sum of three terms, namely a term that near the singularity is expressed as a linear combination of complex powers and logarithmic integer powers of the singular variable, a term that decays to zero close to the singularity faster than each of the non-constant summands of the previous term and a remainder that can be chosen arbitrary small with respect e.g. to the \(C^{0}\)-norm. The powers in the first term are time independent and determined explicitly by the local geometry around the singularity, e.g. by the spectrum of the boundary Laplacian in the situation of straight conical tips. The case of the above two problems on closed manifolds is also considered and local space asymptotics for the solutions are provided.

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

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds and construct, in particular, the first classes of manifolds for which these spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist closed (respectively, open) manifolds for which the moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. An analogous statement holds for spaces of non-negative Ricci curvature metrics in every dimension at least eleven (respectively, twelve).

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

Let M be a Milnor sphere or, more generally, the total space of a linear S^3-bundle over S^4 with H^4(M;Q) = 0. We show that the moduli space of metrics of nonnegative sectional curvature on M has infinitely many path components. The same holds true for the moduli space of m etrics of positive Ricci curvature on M.

Journal | preprint arXiv |

Pages | 11 pages |

Link to preprint version |

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

We study immersed tori in 3-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes (0,b) with $b\rightarrow 1$ the homogenous tori are known to be unique constrained Willmore minimizers (up to invariance). In this paper we generalize the result and determine constrained Willmore minimizers in non-rectangular conformal classes (a,b). In a first step we explicitly construct a 2-dimensional family of putative minimizers parametrized by their conformal class (a,b). For $b\rightarrow 1$, b≠1 fixed, this family is then shown to minimize for $a\rightarrow 0^+$. Difficulties arise from the fact that these minimizers are non-degenerate for a≠0 but smoothly converge to the degenerate homogenous tori as a→0. As a byproduct of our arguments, we show that the minimal Willmore energy ω(a,b) is real analytic and concave in a∈(0, a^b) for some b>0 and fixed $b \rightarrow 1$,

**Related project(s):****16**Minimizer of the Willmore energy with prescribed rectangular conformal class

We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are H\"older continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.

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

In this paper we prove that any smooth surfaces can be locally isometrically embedded into as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.

Journal | Annales de l'Institut Henri Poincare (C) Non Linear Analysis |

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**Related project(s):****22**Willmore functional and Lagrangian surfaces

We prove that if an ALE Ricci-flat manifold (*M*,*g*) is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to *g* exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to *g*. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable.

**Related project(s):****21**Stability and instability of Einstein manifolds with prescribed asymptotic geometry

We prove a gluing theorem for solutions \((A_0, \Phi_0)\) of Hitchin's self-duality equations with logarithmic singularities on a rank-\(2\) vector bundle over a noded Riemann surface \( \Sigma\) representing a boundary point of Teichmüller moduli space. We show that every nearby smooth Riemann surface \( \Sigma_1\) carries a smooth solution \((A_1, \Phi_1)\) of the self-duality equations, which may be viewed as a desingularization of \((A_0, \Phi_0)\).

Journal | Adv. Math. |

Publisher | Elsevier |

Volume | 322 |

Pages | 637-681 |

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

We prove that the Teichmüller space of negatively curved metrics on a hyperbolic manifold *M* has nontrivial *i*-th rational homotopy groups for some *i* > dim* M*. Moreover, some elements of infinite order in the i-th homotopy group of *B*Diff(*M*) can be represented by bundles over a sphere with fiberwise negatively curved metrics.

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

We use classical results in smoothing theory to extract information about the rational homotopy groups of the space of negatively curved metrics on a high dimensional manifold. It is also shown that smooth M-bundles over spheres equipped with fiberwise negatively curved metrics, represent elements of finite order in the homotopy groups of the classifying space for smooth M-bundles, provided the dimension of M is large enough.

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

We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic problems is achieved using the theory of Fourier integral operators on Lie groupoids. We show that our calculus is closed under composition and furnishes a so-called noncommutative completion of the given embedding. A representation of the algebra is defined and the continuity of the operators in the algebra on suitable Sobolev spaces is established.

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

The canonical map from the \(\mathbb{Z}/2\)-equivariant Lazard ring to the \(\mathbb{Z}/2\)-equivariant complex bordism ring is an isomorphism.

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

We study the Kazdan-Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.

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