scholarly journals Uniqueness of minimal surfaces, Jacobi fields,and flat structures

2020 ◽  
Vol 148 (7) ◽  
pp. 3059-3071
Author(s):  
Hojoo Lee
Keyword(s):  
Minimal Surfaces ◽  
Jacobi Fields ◽  
Flat Structures

scholarly journals A Björling representation for Jacobi fields on minimal surfaces and soap film instabilities

2020 ◽  
Vol 476 (2238) ◽  
pp. 20190903
Author(s):  
Gareth P. Alexander ◽  
Thomas Machon
Keyword(s):  
Minimal Surfaces ◽  
General Framework ◽  
Soap Film ◽  
Jacobi Field ◽  
Soap Films ◽  
Jacobi Fields ◽  
One Dimensional ◽  
Stable Domain ◽  
Transverse Gradient ◽  
Geometric Nature

We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localized if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schrödinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasize that the geometric nature of the Björling representation allows direct connection with instabilities observed in soap films.

scholarly journals On different notions of calibrations for minimal partitions and minimal networks in ℝ2

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Marcello Carioni ◽  
Alessandra Pluda
Keyword(s):  
Minimal Surfaces ◽  
Steiner Problem ◽  
Minimal Networks

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.

Minimal Surfaces from a Complex Analytic Viewpoint

2021 ◽  
Author(s):  
Antonio Alarcón ◽  
Franc Forstnerič ◽  
Francisco J. López
Keyword(s):  
Minimal Surfaces

scholarly journals The Adjunction Inequality for Weyl-Harmonic Maps

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.

scholarly journals Structural Models Based on Minimal Surfaces and Geodesics

2021 ◽  
Author(s):  
Francisco Gonzalez-Quintial ◽  
Andres Martin-Pastor
Keyword(s):  
Minimal Surfaces ◽  
Structural Models
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