The second variation of area of minimal surfaces in four-manifolds

Micallef, Mario; Wolfson, Jon

doi:10.1007/bf01444887

Cited by 51 publications

(31 citation statements)

References 20 publications

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“…In this section, we recall the geometry of the normal bundle ν of in M (see [4]) and fix our notation. We choose a local orthonormal frames {e 1 , e 2 , e 3 , e 4 } in M such that {e 1 , e 2 } is a local orthonormal frames of and {e 3 , e 4 } is a local orthonormal frames of the normal bundle ν of in M. We shall use the following conventions on the range of the indices,…”

Section: Preliminariesmentioning

confidence: 99%

The Second Variation of the Functional $$L$$ L of Symplectic Critical Surfaces in Kähler Surfaces

Han

2014

Commun. Math. Stat.

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Let M be a complete Kähler surface and be a symplectic surface which is smoothly immersed in M. Let α be the Kähler angle of in M. In the previous paper Han and Li (JEMS 12:505-527, 2010) 2010, we study the symplectic critical surfaces, which are critical points of the functional L = 1 cos α dμ in the class of symplectic surfaces. In this paper, we calculate the second variation of the functional L and derive some consequences. In particular, we show that, if the scalar curvature of M is positive, is a stable symplectic critical surface with cos α ≥ δ > 0, whose normal bundle admits a holomorphic section X ∈ L 2 ( ), then is holomorphic. We construct symplectic critical surfaces in C 2 . We also prove a Liouville theorem for symplectic critical surfaces in C 2 .

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Section: Preliminariesmentioning

confidence: 99%

The Second Variation of the Functional $$L$$ L of Symplectic Critical Surfaces in Kähler Surfaces

Han

2014

Commun. Math. Stat.

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“…See [12] and [13] for minimal surfaces in hyperkähler manifolds, in particular, in K3 surfaces with the hyperkähler metric, for example.…”

Section: Applications To Surfaces In Hyperkähler Manifoldsmentioning

confidence: 99%

Surfaces in four-dimensional hyperkähler manifolds whose twistor lifts are harmonic sections

Hasegawa

2011

Proc. Amer. Math. Soc.

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Abstract. We determine surfaces of genus zero in self-dual Einstein manifolds whose twistor lifts are harmonic sections. We apply our main theorem to the case of four-dimensional hyperkähler manifolds. As a corollary, we prove that a surface of genus zero in four-dimensional Euclidean space is twistor holomorphic if its twistor lift is a harmonic section. In particular, if the mean curvature vector field is parallel with respect to the normal connection, then the surface is totally umbilic. Thus, our main theorem is a generalization of Hopf's theorem for a constant mean curvature surface of genus zero in threedimensional Euclidean space. Moreover, we can also see that a Lagrangian surface of genus zero in the complex Euclidean plane with conformal Maslov form is the Whitney sphere.

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“…Main tools to prove Theorem 1.2 are (a) an algorithm to compute the Morse index and the nullity which is given by the first author [3 ,4 ] (see § 2 and the previous paper [6 ]), (b) periods of the Abelian differentials of the second kind, that is, meromorphic differentials with zero residues on a minimal surface. The situation is quite different from the genus 3 case and we solve the difficulties by Micallef and Wolfson's technique [11 ].…”

Section: Introductionmentioning

confidence: 99%

On Hyperelliptic Minimal Surfaces With Even Genus

Ejiri¹,

Shoda²

2015

Current Developments in Differential Geometry and Its Related Fields

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In this paper, we announce our results for compact oriented minimal surfaces in flat 4-tori which are given in our preprint [ 7 ]. We consider (i) the Morse index of a one-parameter family, (ii) the existence of non-holomorphic hyperelliptic minimal surfaces with even genus. Key tools are established by the first author [ 3 , 4 ], and we continue the work related to our previous papers [ 5 , 6 ] for the higher genus and codimensional case. Our key tools can be applied for the higher genus and codimensional case, but the situation is different from that of the previous paper. It is crucial to solve such difficulties, and we give a partial answer for it. We will give a clear explanation about the long story.

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The second variation of area of minimal surfaces in four-manifolds

Cited by 51 publications

References 20 publications

The Second Variation of the Functional $$L$$ L of Symplectic Critical Surfaces in Kähler Surfaces

The Second Variation of the Functional $$L$$ L of Symplectic Critical Surfaces in Kähler Surfaces

Surfaces in four-dimensional hyperkähler manifolds whose twistor lifts are harmonic sections

On Hyperelliptic Minimal Surfaces With Even Genus

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