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A Bernstein Theorem for Minimal Maps with Small Second Fundamental Form

  • Felix LubbeEmail author
Article

Abstract

We consider minimal maps \(f:M\rightarrow N\) between Riemannian manifolds \((M,\mathrm {g}_M)\) and \((N,\mathrm {g}_N)\), where M is compact and where the sectional curvatures satisfy \(\sec _N\le \sigma \le \sec _M\) for some \(\sigma >0\). Under certain assumptions on the differential of the map and the second fundamental form of the graph \(\varGamma (f)\) of f, we show that f is either the constant map or a totally geodesic isometric immersion.

Keywords

Minimal maps Bernstein theorem higher codimension 

Mathematics Subject Classification

Primary 53C42 53C40 58J05 

Notes

Acknowledgements

I am grateful to Andreas Savas-Halilaj for valuable discussions. This research was initiated while I was supported by the Research Training Group 1463 of the DFG at the Leibniz Universität Hannover.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HamburgHamburgGermany

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