Ancient mean curvature flows that converge rapidly to a compact self-shrinker at time minus infinity must have the same codimension as the shrinker, and if the convergence is super-exponential the flow equals the shrinker.
Ancient gradient flows of elliptic functionals and Morse index
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinking equatorial spheres. In the mean curvature flow case in S^3, we classify ancient flows with more relaxed area bounds: they are steady or shrinking equators or Clifford tori. In the embedded curve shortening case in S^2, we completely classify ancient flows of bounded length: they are steady or shrinking circles.
fields
math.DG 1years
2019 1verdicts
CONDITIONAL 1representative citing papers
citing papers explorer
-
Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$
Ancient mean curvature flows that converge rapidly to a compact self-shrinker at time minus infinity must have the same codimension as the shrinker, and if the convergence is super-exponential the flow equals the shrinker.