Regularity of area minimizing currents mod p
Camillo De Lellis, Jonas Hirsch, Andrea Marchese, Salvatore Stuvard
Abstract
Abstract We establish a first general partial regularity theorem for area minimizing currents $${\mathrm{mod}}(p)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>mod</mml:mi><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> , for every p , in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an m -dimensional area minimizing current $${\mathrm{mod}}(p)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>mod</mml:mi><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> cannot be larger than $$m-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> . Additionally, we show that, when p is odd, the interior singular set is $$(m-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> -rectifiable with locally finite $$(m-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> -dimensional measure.