OAI.SharpIntegralFillings.coefficient_optimal

The theorem states that for every integer n ≥ 2 there exists an integral n-current T in Euclidean space ℝ^{n+1}, in the metric-current sense (a multilinear functional on a bounded Lipschitz function and n Lipschitz functions satisfying linearity, continuity, locality and finite-mass axioms, whose mass is the least total mass of a finite controlling measure, and which is a countable sum of integer-multiplicity bi-Lipschitz chart pieces with disjoint images and summable masses, with an integral boundary as well), such that T has compact support, is a cycle (its boundary is zero), and has strictly positive mass, and such that every compactly supported integral (n+1)-current S in ℝ^{n+1} whose boundary equals T satisfies mass(S) ≥ c_n · mass(T)^{(n+1)/n}. Here the filling coefficient is c_n = 1/((n+1)·σ_n^{1/n}), where σ_n = (n+1)·ω_{n+1} and ω_{n+1} is the Lebesgue volume of the unit ball in ℝ^{n+1}. This asserts that the filling inequality with this constant is attained: some nonzero compactly supported integral cycle has no filling of smaller mass than the bound, so the constant cannot be improved. The theorem is admitted in the source without proof.