Paper · Oct 12, 2023
Rotational symmetry of ancient solutions to fully nonlinear curvature flows
As on arXiv
Abstract
We address the classification of ancient solutions to fully nonlinear curvature flows for hypersurfaces. Under natural conditions on the speed of motion we classify ancient solutions which are convex, noncollapsing, uniformly two-convex and noncompact. There are exactly two possibilities – every such solution is either a self-similarly shrinking cylinder, or else is a rotationally symmetric translating soliton. For a large class of flows this yields a complete classification of the blow-up limits that can arise at a singularity of a solution which is compact, embedded and two-convex.
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications
Topics Rotational symmetry & warped products, Self-shrinkers & translators, Other curvature flows · Shared terms “fully nonlinear”, “symmetric translating”, “translating solitons”
-
Ancient pancake solutions to fully nonlinear curvature flows
Topics Ancient solutions, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”
-
Ancient solutions of Ricci flow with Type I curvature growth
Topics (Non)collapsing, Ancient solutions, Boundaries & non-compact · Same author S. Lynch
-
Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture
Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Shared terms “complete classification”, “translating solitons”, “rotationally symmetric”
-
Classification of ancient noncollapsed flows in
Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Shared terms “rotationally symmetric”
-
Finite-Shift Inverse-Concavity and Exterior Noncollapsing for Fully Nonlinear Curvature Flows of Nonconvex Hypersurfaces
Topics (Non)collapsing, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”, “noncollapsing”
-
Rotational symmetry of solutions of mean curvature flow coming out of a double cone II
Topics Rotational symmetry & warped products, Singularities · Shared terms “rotationally symmetric”
-
Non-homothetic convex ancient solutions for flows by high powers of curvature
Topics Rotational symmetry & warped products, Ancient solutions, Other curvature flows · Shared terms “rotationally symmetric”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 12B.4Compactness and Universal Estimates for κ-Solutions
Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.
- 12B.3The Structure of κ-Solutions
Necks and caps, compactness, and universal estimates.
- 9B.3Collapsing and Noncollapsing
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.