Pinching Problems of Minimal Submanifolds in a Product Space

Article

Abstract

Let \(\mathbb {M}^{m_{1}}(c)\times \mathbb {R}^{m_{2}}\) be a Riemannian product of a space form \(\mathbb {M}^{m_{1}}(c)\) of constant sectional curvature c and a Euclidean space \(\mathbb {R}^{m_{2}}\). Let M n (n ≥ 2) be an n-dimensional immersed connected submanifold in \(\mathbb {M}^{m_{1}}(c)\times \mathbb {R}^{m_{2}}\). We firstly derive the compatible equations for immersion of M n into \(\mathbb {M}^{m_{1}}(c)\times \mathbb {R}^{m_{2}}\). Then, we derive a Simons’ type equation on the squared length of the second fundamental form of M n . When M n is compact and minimal in \(\mathbb {S}^{m_{1}}(c)\times \mathbb {R}^{m_{2}}\), we prove a series of pinching theorems on the Ricci curvature, the squared length, and the squared maximum norm of the second fundamental form of M n .

Keywords

Pinching problems Submanifolds Simons’ type equation Product space form 

Mathematics Subject Classification (2010)

53C40 53C42 

Notes

Acknowledgements

The authors would like to express their sincere thanks and gratefulness to the referees for their precious help and guidance on the original manuscript. We believe that their comments and suggestions have increased the accuracy and quality of this paper.

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Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.The Institute of MathematicsDalian University of TechnologyDalianChina
  2. 2.College of SciencesJiujiang UniversityJiujiangChina

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