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Complex points of minimal surfaces in almost Kähler manifolds

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Abstract

If (N, ω, J, g) is an almost Kähler manifold andM is a branched minimal immersion which is not aJ-holomorphic curve, we show that the complex tangents are isolated and that each has a negative index, which extends the results in the Kähler case by S. S. Chern and J. Wolfson [2] and S. Webster [7] to almost Kähler manifolds. As an application, we get lower estimates for the genus of embedded minimal surfaces in almost Kähler manifolds. The proofs of these results are based on the well-known Cartan’s moving frame methods as in [2, 7]. In our case, we must compute the torsion of the almost complex structures and find a useful representation of torsion. Finally, we prove that the minimal surfaces in complex projective plane with any almost complex structure is aJ-holomorphic curve if it is homologous to the complex line.

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Supported in parts by the Chinese Educational Committee’s Foundation for Returned Scholar and NSF C-19501024 of P.R. China

Postscript: This paper was written in the August of 1995 and presented at the 85th year’s birthday of Tsinghua University in the April of 1996 and US-CHINA differential equation’s meeting in June of 1996 in Hang Zhou of China.

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Ma, R. Complex points of minimal surfaces in almost Kähler manifolds. Manuscripta Math 95, 159–168 (1998). https://doi.org/10.1007/BF02678022

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  • DOI: https://doi.org/10.1007/BF02678022

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