On the existence of integral currents with prescribed mean curvature vector

Abstract

Given an integralm-currentT ₀ in ℝ^m+k and a tensorH of typ (m, 1) on ℝ^m+k with values orthogonal to each of its arguments we prove the existence of an integralm-currentT with boundary ∂T=∂T ₀ having prescribed mean curvature vectorH, i. e.$T = \underline{\underline \tau } (M,\theta ,\xi )$ is a solution of

for all vectorfieldsX: ℝ^m+k → ℝ^m+k with spt(X)∩spt(∂T)=Ø. It turns out that we can solve the above equation assuming

$$\left| H \right|< \gamma _m^{ - 1} 2^{ - 1/m} m(m + 1)^{ - 1 - 1/m} M(\tilde T)^{ - 1/m} ,$$

where γ _m denotes the constant of Almgren’s Isoperimetric Theorem and$\tilde T$ is an integralm-current minimizing mass for the boundary ∂T ₀.