## 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“.

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 |

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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

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 |

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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

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 prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane. In the course of the proof we obtain new characterizations of Berwald metrics in terms of the so-called linear parallel transport.

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

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits of Riemannian manifolds and deduce a sphere theorem.

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

We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.

Journal | J. Eur. Math. Soc. |

Volume | Online first article |

Link to preprint version |

**Related project(s):****24**Minimal surfaces in metric spaces**66**Minimal surfaces in metric spaces II

We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.

Journal | Math. Ann. |

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

A surface which does not admit a length nonincreasing deformation is called metric minimizing. We show that metric minimizing surfaces in CAT(0) spaces are locally CAT(0) with respect to their intrinsic metric.

Journal | Geom. Topol. |

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