Energy and area minimizers in metric spaces

Abstract

We show that in the setting of proper metric spaces one obtains a solution of the classical two-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area (in the sense of convex geometry) has been chosen appropriately. We prove the quasi-convexity of this new definition of area. Under the assumption of a quadratic isoperimetric inequality we establish regularity results for energy minimizers and improve Hoelder exponents of some area-minimizing discs.

Regularity of harmonic discs in spaces with quadratic isoperimetric inequality

“…Note that the class of quasi-harmonic maps remains unchanged if the Reshetnyak energy is replaced for example by Korevaar-Schoen's energy mentioned above or by any other definition of energy in the sense of [25]. Harmonic maps are particular examples of 1-quasi-harmonic maps.…”

“…Much more refined methods than the one used in our proofs would be needed in order to obtain optimal regularity results. In [24] and [25] we obtained optimal Hölder exponents for maps which are parametrized quasi-conformally in the sense of [24] and minimize area rather than energy among all maps with the same trace. Note that such maps u are quasi-harmonic since every v ∈ W 1,2 (Ω ′ ) with tr(v) = tr(u| Ω ′ ) satisfies…”

Intrinsic structure of minimal discs in metric spaces

“…Indeed, a choice of a definition of area µ is equivalent to a choice of a Jacobian J µ : S 2 → [0, ∞) which satisfies natural transformation and monotonicity conditions, cf. [LW16c], Section 2.3.…”

“…Other prominent examples are the Busemann definition H 2 , the Holmes-Thompson definition µ ht , Gromov's mass *definition m * . We refer to [APT04], [Iva08] for a thorough discussion of these examples and of the whole subject and to [LW16c] for a detailed description of the corresponding Jacobians.…”

“…Thus q(µ) equals to the infimum of the quotient J µ (s)/J µ i (s) taken over all norms s. In particular, cf. [LW16c],…”