Paper · Sep 28, 2026
Counterexamples to preservation of flat normal bundles under mean curvature flow
As on arXiv
Abstract
We construct an embedded torus and an entire graph in whose normal bundles are initially flat but lose this property instantaneously under mean curvature flow. We also give an example showing that the parallel principal normal condition is not preserved under mean curvature flow, even though the normal bundle remains flat along the flow.
Comments: 23 pages. Revised Section 5 and made minor improvements throughout the manuscript
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.
-
Uniqueness of entire graphs evolving by Mean Curvature flow
Topics Mean curvature flow · Shared terms “entire graph”
-
Codimension two mean curvature flow of entire graphs
Topics Mean curvature flow · Shared terms “entire graph”, “preserved”
-
Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow
Same author Keita Kunikawa
-
Almost splitting and quantitative stratification for super Ricci flow
Same author Keita Kunikawa
-
Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy
Topics Mean curvature flow
-
Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecture
Topics Mean curvature flow · Shared terms “counterexamples”
-
Singularity formation along the line bundle mean curvature flow
Topics Mean curvature flow
-
Genus Increase in Closed Mean Curvature Flows
Topics Mean curvature flow · Shared terms “counterexamples”
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
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.