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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

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

 

JournalGeom. Topol.
Volume22
Pages591-644
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.

 

JournalActa Math.
Volume221
Pages159-202
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.

 

JournalMath. Ann.
VolumeOnline 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.

 

JournalGeom. Funct. Anal.
VolumeTo 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

Dr. Stephan Stadler
Project leader
Ludwig-Maximilians-Universität München
stadler(at)math.lmu.de

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