Communications in Analysis and Geometry

Volume 28 (2020)

Number 1

Area minimizing discs in locally non-compact metric spaces

Pages: 89 – 112



Chang-Yu Guo (Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, China)

Stefan Wenger (Department of Mathematics, University of Fribourg, Switzerland)


We solve the classical problem of Plateau in every metric space which is $1$‑complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual Banach spaces, some non-dual Banach spaces such as $L^1$, all Hadamard spaces, and many more. Our results generalize corresponding results of Lytchak and the second author from the setting of proper metric spaces to that of locally non-compact ones.We furthermore solve the Dirichlet problem in the same class of spaces. The main new ingredient in our proofs is a suitable generalization of the Rellich–Kondrachov compactness theorem, from which we deduce a result about ultra-limits of sequences of Sobolev maps.

Research partially supported by Swiss National Science Foundation Grants 153599 and 165848.

Received 17 May 2017

Accepted 7 November 2017

Published 12 March 2020