On the curvature bounded sphere problem in ℝ³

2025/06/28 by Qiu, Hongda
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2507.06245

openalex publication_date 2025/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that if a topological sphere smoothly embedded into ℝ³ with normal curvatures absolutely bounded by 1 is contained in an open ball of radius 2, then the region it bounds must contain a unit ball. This result suggests a potential direction for a problem formulated by D.Burago and A.Petrunin asking whether a topological sphere smoothly embedded in ℝ³ with normal curvatures absolutely bounded by 1 encloses a volume of at least (4)/(3)π. The appendix presents an example illustrating an alternative aspect for this problem.

Citations

• Extremal problems for surfaces of prescribed topological type (1)

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