Quantitative long range curvature estimate for mean curvature flow

Abstract We prove that 𝛼-noncollapsed ancient mean curvature flow satisfies a quantitative curvature estimate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mi>C</m:mi> <m:mo>⁢</m:mo> <m:mi>H</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> H(y,t)\leq CH(x,t)(H(x,t)\lvert x-y\rvert+1)^{2} for any pair of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> </m:math> x,y . In other words, the rescaled curvature grows at most quadratically in terms of the rescaled extrinsic distance.