Tohoku Mathematical Journal

Differentiable pinching theorems for submanifolds via Ricci flow

Fei Huang, Hongwei Xu, and Entao Zhao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Two differentiable pinching theorems are verified via the Ricci flow and stable currents. We first prove a differentiable sphere theorem for positively pinched submanifolds in a space form. Moreover, we obtain a differentiable sphere theorem for submanifolds in the sphere $\mathbb{S}^{n+p}$ under extrinsic restriction.

Article information

Source
Tohoku Math. J. (2), Volume 67, Number 4 (2015), 531-540.

Dates
First available in Project Euclid: 22 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1450798071

Digital Object Identifier
doi:10.2748/tmj/1450798071

Mathematical Reviews number (MathSciNet)
MR3436540

Zentralblatt MATH identifier
1334.53025

Subjects
Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 53C40: Global submanifolds [See also 53B25]

Keywords
Submanifolds differentiable sphere theorem Ricci flow stable current curvature pinching

Citation

Xu, Hongwei; Huang, Fei; Zhao, Entao. Differentiable pinching theorems for submanifolds via Ricci flow. Tohoku Math. J. (2) 67 (2015), no. 4, 531--540. doi:10.2748/tmj/1450798071. https://projecteuclid.org/euclid.tmj/1450798071


Export citation

References

  • M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990), 429–445.
  • M. Berger, Les variétés Riemanniennes (1/4)-pincées, Ann. Scuola Norm. Sup. Pisa 14 (1960), 161–170.
  • M. Berger, Sur quelques variétés riemanniennes suffisamment pincées, Bull. Soc. Math. France 88 (1960), 51–71.
  • R. Bott, Lectures on Morse Theory, old or new, Bull. Amer. Math. Soc. 7 (1982), 331–358.
  • S. Brendle, A general convergence result for the Ricci flow in higher dimensions, Duke Math. J. 145 (2008), 585–601.
  • S. Brendle and R. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc. 22 (2009), 287–307.
  • H. Gauchman, Minimal submanifolds of a sphere with bounded second fundamental form, Tran. Amer. Math. Soc. 292 (1986), 779–791.
  • K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. 106 (1977), 201–211.
  • R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255–306.
  • W. Klingenberg, Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung, Comment. Math. Helv. 35 (1961), 47–54.
  • H. B. Lawson and J. Simons, On stable currents and their application to global problems in real and complex geometry, Ann. of Math. 98 (1973), 427–450.
  • P. F. Leung, On the topology of a compact submanifold of a sphere with bounded second fundamental form, Manus. Math. 79 (1993), 183–185.
  • M. Micallef and J. D. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. 127 (1988), 199–227.
  • Y. Otsu, K. Shiohama and T. Yamaguchi, A new version of differentiable sphere theorem, Invent. Math. 98 (1989), 219–228.
  • G. Perelman, Manifolds of positive Ricci curvature with almost maximal volume, J. Amer. Math. Soc. 7 (1994), 299–305.
  • H. E. Rauch, A contribution to differential geometry in the large, Ann. of Math. 54 (1951), 38–55.
  • K. Shiohama, Sphere theorems, Handbook of Differential Geometry, Vol. 1, F.J.E. Dillen and L.C.A. Verstraelen (eds.), Elsevier Science B.V., Amsterdam, 2000.
  • K. Shiohama and H. W. Xu, The topological sphere theorem for complete submanifolds, Compositio Math. 107 (1997), 221–232.
  • H. W. Xu and W. Fang, Geometric and topological rigidity of complete submanifolds, preprint, 2001.
  • H. W. Xu and J. R. Gu, An optimal differentiable sphere theorem for complete manifolds, Math. Res. Lett. 17 (2010), 1111–1124.
  • H. W. Xu, W. Fang and F. Xiang, A generalization of Gauchman's rigidity theorem, Pacific J. Math. 228 (2006), 185–199.
  • H. W. Xu, F. Huang and F. Xiang, An extrinsic rigidity theorem for submanifolds with parallel mean curvature in a sphere, Kodai Math. J. 34 (2011), 85–104.
  • H. W. Xu and G. X. Yang, Topological sphere theorems for submanifolds with positive curvature, preprint, 2008.
  • H. W. Xu and E. T. Zhao, Topological and differentiable sphere theorems for complete submanifolds, Comm. Anal. Geom. 17 (2009), 565–585.
  • T. Yamaguchi, Lipschitz convergence of manifolds of positive Ricci curvature with large volume, Math. Ann. 284 (1989), 423–436.