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Submanifolds with harmonic mean curvature vector field in contact 3-manifolds. (English) Zbl 1076.53065

The paper under review concerns curves and surfaces in 3-dimensional contact manifolds whose mean curvature vector field \(H\) is in the kernel of certain elliptic differential operators. The study of such submanifolds is inspired by a conjecture given in [B.-Y. Chen, Soochow J. Math. 22, 117–337 (1996; Zbl 0867.53001)], namely, harmonicity of the mean curvature vector field implies harmonicity of the immersion. In the first part of the paper the harmonicity of mean curvature vector fields of curves and surfaces in 3-dimensional Sasakian space forms is studied. Several results for the 3-dimensional sphere \(S^3\) are generalized to 3-dimensional Sasakian space forms.
Two of the main results are the following:
Theorem 2.1: A Hopf cylinder \(S_{\overline{\gamma}}\) in a 3-dimensional regular Sasaki manifold satisfies \(\Delta H=\lambda H\) if and only if \(\overline{\gamma}\) is a geodesic \((\lambda =0)\) or a Riemannian circle \((\lambda\neq 0)\). If \(\lambda \neq 0\), then \(\lambda\geq 2\).
Theorem 2.2: A Hopf cylinder \(S_{\overline{\gamma}}\) satisfies \(\Delta^{\bot} H=0\) (normal-harmonicity) if and only if \(\overline{\gamma}\) is one of the following: (1) a geodesic, (2) a Riemannian circle, (3) a Cornu spiral.
One result given in the second part of the paper is the existence of nonminimal biharmonic Hopf cylinders in Sasakian space forms of holomorphic sectional curvature greater than 1. In this part, the author also classifies Legendre curves and Hopf cylinders in Sasakian space forms which are biharmonic in this sense of J. Eells, jun. and J. H. Sampson.

MSC:

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Citations:

Zbl 0867.53001
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