Area minimizing surfaces of bounded genus in metric spaces

Abstract

The Plateau-Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disctype surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

Morrey's $\varepsilon$-conformality lemma in metric spaces

“…The metric g can be chosen to have constant sectional curvature −1, 0, or 1 and such that the boundary of M is geodesic. Theorem 1.2 and Proposition 1.1 yield the following corollary which strengthens [7,Theorem 4.2].…”

“…Unlike in Theorem 1.6 we do not obtain a good parametrization. This is in contrast to the situation in our paper [7], where we solved the Plateau-Douglas problem under the Douglas condition. Notice that minimizers with respect to the Reshetnyak energy need not minimize the Hausdorff area in general, see [18,Proposition 11.6].…”

“…In the setting of Euclidean space or, more generally, Riemannian manifolds the problem has been solved under two different conditions of non-degeneracy: the so-called Douglas condition considered in [5], [11], as well as Courant's condition of cohesion used in [25], [4], [26]. Assuming the Douglas condition we have recently solved the Plateau-Douglas problem in any proper metric space admitting a local quadratic isoperimetric inequality in [7]. In the present paper we solve this problem using Courant's condition of cohesion in any proper metric space.…”

“…The Riemannian metric g can be chosen in such a way that (M, g) has constant curvature −1, 0 or 1 and that M has geodesic boundary. If the metric space X admits a local quadratic isoperimetric inequality for curves then the map u in Theorem 1.6 has a unique representative which is locally Hölder continuous in the interior of M and is continuous up to the boundary, see [7,Theorem 1.4].…”

“…so the sequence (u n ) is also an E 2 + -energy minimizing sequence in Λ(M, Γ, X) by (5.1). Since (u n ) satisfies the condition of cohesion, [7,Theorem 8.2] implies the existence of some u ∈ Λ(M, Γ, X) and of a Riemannian metric g such that E 2 + (u, g) = e(M, Γ, X) and u is infinitesimally isotropic with respect to g. Moreover, the Riemannian metric g can be chosen in such a way that (M, g) has constant curvature −1, 0, 1 and that ∂M is geodesic. Proposition 5.1 shows that u is µ i -area minimizing in Λ(M, Γ, X).…”