The c-Isoperimetric Mass of Currents and the c-Plateau Problem

Abstract

We introduce and study the c-isoperimetric mass Mc(T):=M(T)+cM(∂T)κ for integer multiplicity n-rectifiable currents T, where M is the usual mass and \(\kappa= \frac{n}{n-1}\) is the exponent as in the isoperimetric inequality. We study the c-Plateau problem: Given Γ an integer multiplicity (n−1)-rectifiable current without boundary, find if possible Tc minimizing c-isoperimetric mass amongst all T integer multiplicity n-rectifiable currents with boundary of the form ∂T=Σ+Γ where Σ, the free boundary, has support disjoint from Γ. We resolve the c-Plateau problem for Γ an (n−1)-dimensional sphere, and for Γ a square in the plane. For this, we develop variational tools, such as computing the first and second variation with respect to Mc. Consequently, we conclude in Theorem 8.2 that there are no solutions Tc to the c-Plateau problem with nonempty free boundary Σc a smooth constant mean curvature (n−1)-dimensional submanifold, near which Tc corresponds to a smooth n-dimensional submanifold-with-boundary.