Plateau’s problem for integral currents in locally non-compact metric spaces
نویسنده
چکیده
The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak -compactness theorem for integral currents in dual spaces of separable Banach spaces. Our theorems generalize results of Ambrosio–Kirchheim, Lang, the author, and recent results of Ambrosio–Schmidt.
منابع مشابه
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 s...
متن کاملInstitute for Mathematical Physics Currents in Metric Spaces Currents in Metric Spaces
We develop a theory of currents in metric spaces which extends the classical theory of Federer{Fleming in euclidean spaces and in Riemannian manifolds. The main idea, suggested in 20, 21], is to replace the duality with diierential forms with the duality with (k + 1)-ples (f; 1; : : : ; k) of Lipschitz functions, where k is the dimension of the current. We show, by a metric proof which is new e...
متن کاملCurrents in Metric Spaces
We develop a theory of currents in metric spaces which extends the classical theory of Federer–Fleming in euclidean spaces and in Riemannian manifolds. The main idea, suggested in [20, 21], is to replace the duality with differential forms with the duality with (k+ 1)-ples (f, π1, . . . , πk) of Lipschitz functions, where k is the dimension of the current. We show, by a metric proof which is ne...
متن کاملIntegration of Hölder Forms and Currents in Snowflake Spaces
M f dg1 ∧ · · · ∧ dgn in case the functions f, g1, . . . , gn are merely Hölder continuous of a certain order by extending the construction of the Riemann-Stieltjes integral to higher dimensions. More generally, we show that for α ∈ ( n n+1 , 1] the n-dimensional locally normal currents in a locally compact metric space (X, d) represent a subspace of the n-dimensional currents in (X, d). On the...
متن کاملFlat Convergence for Integral Currents in Metric Spaces
It is well known that in compact local Lipschitz neighborhood retracts in Rn flat convergence for Euclidean integer rectifiable currents amounts just to weak convergence. The purpose of the present paper is to extend this result to integral currents in complete metric spaces admitting a local cone type inequality. This includes for example all Banach spaces and complete CAT(κ)-spaces, κ ∈ R. Th...
متن کامل