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Minimal surfaces in metric spaces

The main goals of the project are as follows:

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

## Publications

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):
24Minimal surfaces in metric spaces

We prove that a minimal disc in a CAT(0) space is a local embedding away from a finite set of "branch points". On the way we establish several basic properties of minimal surfaces: monotonicity of area densities, density bounds, limit theorems and the existence of tangent maps.

As an application, we prove Fary-Milnor's theorem in the CAT(0) setting.

 Link to preprint version

Related project(s):
24Minimal 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.

 Link to preprint version

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

We prove an analog of Schoen-Yau univalentness theorem for saddle maps between discs.

 Link to preprint version Link to published version

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

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. Volume Online First Link to preprint version

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

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## Team Members

Paul Creutz
Doctoral student
Universität Köln
paul.creutz(at)ish.de