The main goals of the project are as follows:

- Solve the asymptotic Plateau problem in nonpositively, respectively negatively curved spaces / groups. Analyze solutions and find applications.
- Study conformal changes of spaces with upper curvature bounds and find a new approach to Reshetnyak's theory of surfaces with upper curvature bounds.
- Investigate the analytical, geometrical and topological properties of geodesic surfaces with quadratic isoperimetric inequalities and study the compactification of the class of such surfaces.
- Find optimal isoperimetric inequalities for curves in normed spaces and applications to regularity of minimal surfaces.
- Construct minimal surfaces in metric spaces by topological methods and / or min-max techniques.

## Publications

We describe the Gromov-Hausdorff closure of the class of length spaces being homeomorphic to a fixed closed surface.

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

This is the second in a two part series of papers concerning Morse quasiflats - higher dimensional analogs of Morse quasigeodesics. Our focus here is on their asymptotic structure. In metric spaces with convex geodesic bicombings, we prove asymptotic conicality, uniqueness of tangent cones at infinity and Euclidean volume growth rigidity for Morse quasiflats. Moreover, we provide some immediate consequences.

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

This is the first in a series of papers concerned with Morse quasiflats, which are a generalization of Morse quasigeodesics to arbitrary dimension. In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry invariant; we also give a variety of examples. The second paper proves that Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries; it also gives some first applications.

**Related project(s):****24**Minimal surfaces in metric spaces**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 prove that any length metric space homeomorphic to a surface may be decomposed into non-overlapping convex triangles of arbitrarily small diameter. This generalizes a previous result of Alexandrov--Zalgaller for surfaces of bounded curvature.

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

We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution of Plateau's problem in metric spaces by Lytchak and Wenger as generalizations of minimal surfaces.

Journal | Geom. Dedicata |

Volume | 210 |

Pages | 151-164 |

Link to preprint version | |

Link to published version |

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

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 find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak--Wenger, which satisfies a related maximality condition.

Journal | Calc. Var. Partial Differential Equations |

Volume | 59 |

Link to preprint version | |

Link to published version |

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

Assume you are given a finite configuration $\Gamma$ of disjoint rectifiable Jordan curves in $\mathbb{R}^n$. The Plateau-Douglas problem asks whether there exists a minimizer of area among all compact surfaces of genus at most $p$ which span $\Gamma$. While the solution to this problem is well-known, the classical approaches break down if one allows for singular configurations $\Gamma$ where the curves are potentially non-disjoint or self-intersecting. Our main result solves the Plateau-Douglas problem for such potentially singular configurations. Moreover, our proof works not only in $\mathbb{R}^n$ but in general proper metric spaces. Thus we are also able to extend previously known existence results of Jürgen Jost as well as of the second author together with Stefan Wenger for regular configurations. In particular, existence is new for disjoint configurations of Jordan curves in general complete Riemannian manifolds. A minimal surface of fixed genus $p$ bounding a given configuration $\Gamma$ need not always exist, even in the most regular settings. Concerning this problem, we also generalize the approach for singular configurations via minimal sequences satisfying conditions of cohesion and adhesion to the setting of metric spaces.

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

We study the structure of the branch set of solutions to Plateau's problem in metric spaces satisfying a quadratic isoperimetric inequality. In our first result, we give examples of spaces with isoperimetric constant arbitrarily close to the Euclidean isoperimetric constant $(4\pi)^{−1}$ for which solutions have large branch set. This complements recent results of Lytchak--Wenger and Stadler stating, respectively, that any space with Euclidean isoperimetric constant is a CAT(0) space and solutions to Plateau's problem in a CAT(0) space have only isolated branch points. We also show that any planar cell-like set can appear as the branch set of a solution to Plateau's problem. These results answer two questions posed by Lytchak and Wenger. Moreover, we investigate several related questions about energy-minimizing parametrizations of metric disks: when such a map is quasisymmetric, when its branch set is empty, and when it is unique up to a conformal diffeomorphism.

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

Our main result gives an improved bound on the filling areas of closed curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu's classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds.

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

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 | Trans. Amer. Math. Soc. |

Volume | 373 |

Pages | 1577-1596 |

Link to preprint version | |

Link to published version |

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

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

## Team Members

** Paul Creutz**

Doctoral student

Universität Köln

paul.creutz(at)ish.de

**Prof. Dr. Alexander Lytchak**

Project leader

Karlsruher Institut für Technologie

alexander.lytchak(at)kit.edu

**Dr. Artem Nepechiy**

Researcher

Karlsruher Institut für Technologie

artem.nepechiy(at)kit.edu

**Dr. Stephan Stadler**

Project leader

Max Planck Instutute for Mathematics Bonn

stadler(at)mpim-bonn.mpg.de