Inspired by the Gromov-Hausdorff distance, we define a new notion called the intrinsic flat distance between oriented m dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all common metric spaces. This is made rigorous by applying Ambrosio-Kirchheim's extension of Federer-Fleming's notion of integral currents to arbitrary metric spaces.We prove the intrinsic flat distance between two compact oriented Riemannian manifolds is zero iff they have an orientation preserving isometry between them. Using the theory of Ambrosio-Kirchheim, we study converging sequences of manifolds and their limits, which are in a class of metric spaces that we call integral current spaces. We describe the properties of such spaces including the fact that they are countably H m rectifiable spaces and present numerous examples. Contents

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.

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We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces. 7 2. Preliminaries 2.1. Basic notation. The following notation will be used throughout the paper. The Euclidean norm of a vector v ∈ R n is denoted by |v|.We denote the open unit disc in R 2 by D. A domain will always mean an open, bounded, connected subset of R 2 .Metric spaces appearing in this paper will be assumed complete. A metric space is called proper if its closed bounded subsets are compact. We will denote distances in a metric space X by d or d X . Let X = (X, d) be a metric space. The open ball in X of radius r and center x 0 ∈ X is denoted by B(x 0 , r) = B X (x 0 , r) = {x ∈ X : d(x 0 , x) < r}.

In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in terms of topology of the supports which ensure that no cancellation occurs and that the support of the weak limit agrees with the Hausdorff limit of the supports. We use our results to prove countable H m -rectifiability of the Gromov-Hausdorff limit of sequences of Lipschitz manifolds M n all of which are λ-linearly locally contractible up to some scale r 0 . In the Appendix, we show that the Gromov-Hausdorff limit need not be countably H m -rectifiable if the M n have a common local geometric contractibility function which is only concave (and not linear). We also relate our results to work of Cheeger-Colding on the limits of noncollapsing sequences of manifolds with nonnegative Ricci curvature. Mathematics Subject Classification (2000) 53C23 · 49Q15