## Prof. Dr. Alexander Lytchak

### Project leader

Universität Köln

E-mail: alytchak(at)math.uni-koeln.de

Telephone: +49 221 470-5709

Homepage: http://www.mi.uni-koeln.de/~alytchak/

## Project

**24**Minimal surfaces in metric spaces
**66**Minimal surfaces in metric spaces II

## Publications within SPP2026

We discuss solutions of several questions concerning the geometry of conformal planes.

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

We analyze weak convergence on CAT(0) spaces and the existence and properties of corresponding weak topologies.

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

We show that cyclic products of projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bačák. Namely, we construct an example of convex subsets in a Hadamard space such that the corresponding cyclic product of projections is not asymptotically regular.

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

We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold (M,g). For example, if a convex set in (M,g) is bounded by a smooth hypersurface, then it is strictly convex.

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

We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

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

We prove that a topological 4-manifold of globally non-positive curvature is homeomorphic to Euclidean space.

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

We show that closed subsets with vanishing first homology in two-dimensional spaces inherit the upper curvature bound from their ambient spaces and discuss topological applications.

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

We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a C^{1,1}-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an *α*-Hölder continuous Lorentzian metric admit a C^{1,*α*/4}-parametrization.

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

We investigate the geometric and topological structure of equidistant decompositions of Riemannian manifolds.

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

We construct short retractions of a CAT(1) space to its small convex subsets. This construction provides an alternative geometric description of an analytic tool introduced by Wilfrid Kendall.

Our construction uses a tractrix flow which can be defined as a gradient flow for a family of functions of certain type. In an appendix we prove a general existence result for gradient flows of time-dependent locally Lipschitz semiconcave functions, which is of independent interest.

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

We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.

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

We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.

Journal | Trans. Amer. Math. Soc. |

Link to preprint version | |

Link to published version |

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

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. |

Link to preprint version | |

Link to published version |

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

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces

Journal | Geom. Topol. |

Volume | 22 |

Pages | 591-644 |

Link to preprint version |

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

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds.

Journal | Acta Math. |

Volume | 221 |

Pages | 159-202 |

Link to preprint version | |

Link to published version |

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

We study geometric and topological properties of locally compact, geodesically complete spaces with an upper curvature bound. We control the size of singular subsets, discuss homotopical and measure-theoretic stratifications and regularity of the metric structure on a large part.

Journal | Geom. Funct. Anal. |

Volume | To appear |

Link to preprint version |

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

We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.

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