Mean Convex Neighborhood Conjecture Resolved for Cylindrical Flows
Published:Dec 31, 2025 00:12
•1 min read
•ArXiv
Analysis
This paper provides a complete classification of ancient, asymptotically cylindrical mean curvature flows, resolving the Mean Convex Neighborhood Conjecture. The results have implications for understanding the behavior of these flows near singularities, offering a deeper understanding of geometric evolution equations. The paper's independence from prior work and self-contained nature make it a significant contribution to the field.
Key Takeaways
- •Resolves the Mean Convex Neighborhood Conjecture for mean curvature flows with cylindrical singularities.
- •Provides a complete classification of ancient, asymptotically cylindrical flows.
- •Establishes a canonical neighborhood theorem near cylindrical singularities.
- •Offers a new proof of the existence of flying wing solitons.
Reference
“The paper proves that any ancient, asymptotically cylindrical flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons.”