Dynamics of convex mean curvature flow

Abstract There is an extensive and growing body of work analyzing convex ancient solutions to mean curvature flow (MCF), or equivalently of rescaled mean curvature flow (RMCF). The goal of this paper is to complement the existing literature, which analyzes ancient solutions one at a time, by considering the space 𝑋 of all convex hypersurfaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>⊂</m:mo> <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:mrow> </m:math> M\subset\mathbb{R}^{n+1} , regard RMCF as a semiflow on this space, and study the dynamics of this semiflow. To this end, we first extend the well-known existence and uniqueness of solutions to MCF with smooth compact convex initial data to include the case of arbitrary noncompact and nonsmooth initial convex hypersurfaces. We identify a suitable weak topology with good compactness properties on the space 𝑋 of convex hypersurfaces and show that RMCF defines a continuous local semiflow on 𝑋 whose fixed points are the shrinking cylinder solitons <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>S</m:mi> <m:mi>k</m:mi> </m:msup> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msup> </m:mrow> </m:math> S^{k}\times\mathbb{R}^{n-k} , and for which the Huisken energy is a Lyapunov function. Ancient solutions to MCF are then complete orbits of the RMCF semiflow on 𝑋. We consider the set of all hypersurfaces that lie on an ancient solution that, in backward time, is asymptotic to one of the shrinking cylinder solitons and prove various topological properties of this set. We show that this space is a path connected, compact subset of 𝑋, and considering only point symmetric hypersurfaces, that it is topologically trivial in the sense of Čech cohomology. We also prove that the space of all convex ancient solutions with point symmetry 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> \mathbb{R}^{n+1} is homeomorphic to an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (n-1) -dimensional simplex, in the case when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n=2 or <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 , and conjecture that it holds true for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n\geq 2 .