Area Minimizing Discs in Metric Spaces

Abstract

Abstract. We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.

Improvements of upper curvature bounds

“…In particular, Λ (Γ, B) is non-empty. If B is compact, the existence of an energy minimizer in Λ(Γ, B) is proved in [LW17]. The same classical argument, extended in [LW17] to proper metric spaces also works in the present non-compact case as follows.…”

“…By now there exists a well established theory of Sobolev maps with values in metric spaces, [HKST15]. We will follow [LW17] and restrict our revision to the special case needed in this paper.…”

“…A general solution of the classical Plateau's problem has been provided in [LW17] for proper metric spaces and in [GW17] for complete CAT(0) spaces. We need to discuss an appropriate extension to non-proper CAT(κ) spaces with κ > 0.…”

Majorization by hemispheres and quadratic isoperimetric constants

“…In this section we discuss how our results can be applied to the work of Alexander Lytchak and Stefan Wenger in [LW17a] and [LW17b] to obtain regularity results for minimal discs.…”

Conformal deformations of CAT(0) spaces

“…(b) The existence of a minimal integral k-current filling any prescribed boundary in U; see [10] and [6,Theorem 3.3]. (c) The existence of a conformally parametrized disc u : D → U of minimal area for a given boundary curve γ, which is a Jordan curve of finite length in U; see [11] and [7, Theorem 1.2]. (d) For any Radon measure µ on U there exists a center of mass x ∈ U for the measure µ [2,12].…”