Partial Regularity for Mass-minimizing Currents in Hilbert Spaces

Recently, the theory of currents and the existence theory for Plateau’s problem have been extended to the case of finite-dimensional currents in infinitedimensional manifolds or even metric spaces; see [5] (and also [7, 39] for the most recent developments). In this paper, in the case when the ambient space is Hilbert, we provide the first partial regularity result, in a dense open set of the support, for n-dimensional integral currents which locally minimize the mass. Our proof follows with minor variants [34], implementing Lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space.