Area minimizing surfaces in homotopy classes in metric spaces

Abstract

We introduce and study a notion of relative 1 1 -homotopy type for Sobolev maps from a surface to a metric space spanning a given collection of Jordan curves. We use this to establish the existence and local Hölder regularity of area minimizing surfaces in a given relative 1 1 -homotopy class in proper geodesic metric spaces admitting a local quadratic isoperimetric inequality. If the underlying space has trivial second homotopy group then relatively 1 1 -homotopic maps are relatively homotopic. We also obtain an analog for closed surfaces in a given 1 1 -homotopy class. Our theorems generalize and strengthen results of Lemaire [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9 (1982), pp. 91–103], Jost [J. Reine Angew. Math. 359 (1985), pp. 37-54], Schoen–Yau [Ann. of Math. (2) 110 (1979), pp. 127–142], and Sacks–Uhlenbeck [Trans. Amer. Math. Soc. 271 (1982), pp. 639–652].