Contemporary Mathematics Volume 646, 2015 Plateau Problems in Metric Spaces and Related Homology and Cohomology Theories Robert M. Hardt Abstract. The classical Plateau problem, studied by T. Rado (1930) and J. Douglas (1931), involved area-minimizing parameterized surfaces in R3. Mass-minimizing objects of dimensions greater than two in Rn were first treated around 1960 in separate works by E. De Giorgi, H. Federer, W. Flem- ing, and E. Reifenberg. Important developments were made by B. White (1999) for chains with very general normed coefficient groups and by L. Am- brosio and B. Kirchheim (2000) for currents in a metric space. Here we de- scribe work with T. De Pauw (2012) generalizing these and treating general chains in a complete metric space that have coefficients in a general com- plete normed Abelian group. We mention some applications for special metric spaces, groups, or norms. Homology theories using chains of finite mass and rectifiable chains reflect the metric, as well as topological, properties of the spaces. For chains with coefficients in (R,∣⋅∣) and compact metric spaces satis- fying a linearly isoperimetric condition, there is a continuous duality between the chain homology and a cohomology theory based on variationally topolo- gized linear cochains. We recall this and other results from recent work with T. De Pauw and W. Pfeffer. 1. Introduction 1.1. A Motivating Problem from Topology. A basic general question from Algebraic Topology is the following: Given an m 1 chain B in a space X with ∂B = 0, does there exist an m chain A in X with ∂A = B? This may have various answers depending on the definiton of chain or boundary. In any case, we note here that either answer, YES or NO, will give an interesting variational problem whenever X has a metric and each chain has a well-defined mass. A situation for which the answer is YES is the example where X = R3 and B is the singular chain corresponding to a oriented embedded smooth closed curve. For instance, one suitable A that has boundary B may be obtained by simply forming the cone over B from the origin, as illustrated below. 2010 Mathematics Subject Classification. Primary 49Q15, 28A75 Secondary 55N35. The author was supported in part by NSF grant DMS1207702. ©2015 American Mathematical Society 1
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