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Immersions with small normal curvature

Otis Chodosh, Chao Li

TL;DR

This work analyzes Gromov's minimal curvature invariant for immersions into unit balls, ultimately determining $\\mathcal{C}_N(S^n\\times S^1)=\\sqrt{\\tfrac{3}{2}}$ for large $N$ via a Veronese-tensor construction and a Petrunin-type obstruction. It proves a differentiable sphere theorem for immersions with small normal curvature by showing conformal positivity of sectional or isotropic curvature, then invoking Brendle–Schoen-type classifications. The paper also extends the Veronese framework through tensor products to generate further low-curvature immersions for products of spheres and tori, and establishes existence results for minimizers under favorable bounds. AI-assisted exploration informs the construction and curvature condition reasoning, while rigorous geometric analysis ensures sharpness and topological consequences. Overall, the results connect minimal normal curvature to deep curvature-positivity phenomena, yielding sharp bounds and rigidity for certain product manifolds.

Abstract

We study Gromov's problem concerning minimal normal curvature immersions in the unit ball. In particular, we determine the minimal possible value of the normal curvature of an $S^n\times S^1$. We also prove a differentiable sphere theorem and an existence result for minimizers in this context.

Immersions with small normal curvature

TL;DR

This work analyzes Gromov's minimal curvature invariant for immersions into unit balls, ultimately determining for large via a Veronese-tensor construction and a Petrunin-type obstruction. It proves a differentiable sphere theorem for immersions with small normal curvature by showing conformal positivity of sectional or isotropic curvature, then invoking Brendle–Schoen-type classifications. The paper also extends the Veronese framework through tensor products to generate further low-curvature immersions for products of spheres and tori, and establishes existence results for minimizers under favorable bounds. AI-assisted exploration informs the construction and curvature condition reasoning, while rigorous geometric analysis ensures sharpness and topological consequences. Overall, the results connect minimal normal curvature to deep curvature-positivity phenomena, yielding sharp bounds and rigidity for certain product manifolds.

Abstract

We study Gromov's problem concerning minimal normal curvature immersions in the unit ball. In particular, we determine the minimal possible value of the normal curvature of an . We also prove a differentiable sphere theorem and an existence result for minimizers in this context.
Paper Structure (19 sections, 13 theorems, 64 equations)

This paper contains 19 sections, 13 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. The minimal curvature invariant of Sn x S1
  3. Existence of minimizers
  4. A differentiable sphere theorem for immersions with small normal curvatures
  5. General Veronese tensor products
  6. On the use of AI
  7. Acknowledgements
  8. Preliminaries
  9. The Gauss equation
  10. Conformal change of curvature
  11. Immersions with small normal curvature
  12. Higher Veronese immersions
  13. Tensor products of Veronese immersions
  14. Low curvature examples
  15. Obstructions to low curvature maps
  16. ...and 4 more sections

Key Result

Theorem 1

For $n\geq 2$ we have $\mathcal{C}_{N}(S^n \times S^1) = \sqrt{\frac{3}{2}}$ for $N \geq 2n+4$.

Theorems & Definitions (27)

  • Theorem 1
  • Proposition 2
  • Remark 3
  • Corollary 4
  • Theorem 5
  • Proposition 6
  • Remark 7
  • Proposition 8
  • proof
  • Proposition 9
  • ...and 17 more