2022/12/06 by Keaton Naff
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · doi:10.1515/crelle-2022-0068
crossref issued 2022/12/06 · crossref published 2022/12/06 · crossref published-online 2022/12/06 · openalex publication_date 2022/12/06 · crossref created 2022/12/07 · crossref deposited 2022/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30 · crossref indexed 2026/07/31
Abstract We consider ancient solutions to the mean curvature flow in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> ℝn+1 ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> n≥ 3 ) that are weakly convex, uniformly two-convex, and satisfy two pointwise derivative estimates <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo></m:mo> <m:mi>A</m:mi> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:msub> <m:mi>γ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>H</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> |∇ A|≤γ1| H|2 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:msup> <m:mo>∇</m:mo> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mi>A</m:mi> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:msub> <m:mi>γ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>H</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> |∇2A|≤γ2| H|3 . We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact 𝑛-dimensional ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>5</m:mn> </m:mrow> </m:math> n≥ 5 ) solutions to the mean curvature flow in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> ℝN that satisfy the pinching condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>A</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo><</m:mo> <m:mrow> <m:mi>c</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>H</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> | A|2<c| H|2 for a suitable constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>c</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>c</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> c=c(n) . We conclude that any blow-up model at the first singular time must be a codimension