On the thread problem for minimal surfaces

Abstract Calibrations are a possible tool to validate the minimality of a certain candidate. They have been introduced in the context of minimal surfaces and adapted to the case of the Steiner problem in several variants. Our goal is to compare the different notions of calibrations for the Steiner problem and for planar minimal partitions that are already present in the literature. The paper is then complemented with remarks on the convexification of the problem, on nonexistence of calibrations and on calibrations in families.

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Minimal Surfaces from a Complex Analytic Viewpoint

10.1007/978-3-030-69056-4 ◽

2021 ◽

Author(s):

Antonio Alarcón ◽

Franc Forstnerič ◽

Francisco J. López

Keyword(s):

Minimal Surfaces

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The Adjunction Inequality for Weyl-Harmonic Maps

Complex Manifolds ◽

10.1515/coma-2020-0007 ◽

2020 ◽

Vol 7 (1) ◽

pp. 129-140

Author(s):

Robert Ream

Keyword(s):

Minimal Surfaces ◽

Twistor Space ◽

Harmonic Maps ◽

Holomorphic Curves ◽

Weyl Connection ◽

Almost Kähler ◽

Conformal Manifolds

AbstractIn this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality\chi \left( {{T_f}\sum } \right) + \chi \left( {{N_f}\sum } \right) \le \pm {c_1}\left( {f*{T^{\left( {1,0} \right)}}M} \right).The ±J-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well as their extension to almost-Kähler 4-manifolds by Chen-Tian, Ville, and Ma.

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