Paper · Oct 19, 2026
Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow
As on arXiv
Abstract
We develop a pinching method for mean curvature flow in higher codimension. Motivated by pinching constructions for Ricci flow, the method uses a nonlinear deformation of the second fundamental form. We introduce two tensorial curvature conditions for closed immersions in Euclidean space which extend beyond the scope of classical scalar pinching results. In dimensions at least ten, we apply the method to prove convergence to a round point under the first condition and a canonical neighborhood theorem under the second.
Comments: 41 pages, comments are welcome!
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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
- 11A.7Maximum Principles under Ricci Flow
Scalar and tensor maximum principles, and preserved curvature conditions.
- 6A.9Reaction–Diffusion: Fronts, Patterns and Pinching
Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.
- 11A.10Hamilton–Ivey Pinching
Why high curvature in dimension 3 is almost positive.