On biminimal submanifolds in nonpositively curved manifolds. (English) Zbl 1295.53065
Differ. Geom. Appl. 35, 1-8 (2014); corrigendum ibid. 47, 256-259 (2016).
Summary: Biminimal immersions are critical points of the bienergy for normal variations with fixed energy, that is critical points of the functional \(E_2(\cdot)+\lambda E(\cdot)\), \(\lambda\in\mathbb R\), for normal variations. A submanifold is called a biminimal submanifold if it is a biminimal isometric immersion. In this note we prove that positive (that is \(\lambda>0\)) complete biminimal submanifolds in nonpositively curved manifolds are minimal. Furthermore for nonpositive (that is \(\lambda\leq 0\)) complete biminimal submanifolds in negative space forms we get this result under certain proper conditions. Our result is sharp by the examples from [E. Loubeau and S. Montaldo, Proc. Edinb. Math. Soc., II. Ser. 51, No. 2, 421–437 (2008; Zbl 1144.58010)]. These results inspire us to make two conjectures on biminimal submanifolds.
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53C24 | Rigidity results |
| 53C40 | Global submanifolds |
Citations:
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