Lipschitz Approximation for General Almost Area Minimizing Currents

Abstract We prove a Lipschitz approximation with superlinear error terms for integral currents $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -minimizing the area functional, where $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> is a modulus of continuity satisfying a Dini condition. We also present an almost monotonicity result for the mass ratio of these general almost area minimzing integral currents.