A Survey on Classical Minimal Surface Theory

2012 ◽  
Author(s):  
William Meeks ◽  
Joaquín Pérez
Keyword(s):  
Minimal Surface ◽  
Surface Theory

scholarly journals A generalization of a completeness lemma in minimal surface theory

2014 ◽  
Vol 37 (2) ◽  
pp. 506-517
Author(s):  
Yûsuke Okuyama ◽  
Katsutoshi Yamanoi
Keyword(s):  
Minimal Surface ◽  
Surface Theory

scholarly journals An analogue of minimal surface theory in $\operatorname {SL}(n,\mathbf C)/\operatorname {SU}(n)$

2001 ◽  
Vol 354 (4) ◽  
pp. 1299-1325 ◽  
Author(s):  
M. Kokubu ◽  
M. Takahashi ◽  
M. Umehara ◽  
K. Yamada
Keyword(s):  
Minimal Surface ◽  
Surface Theory

scholarly journals Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

2017 ◽  
Vol 28 (09) ◽  
pp. 1740004 ◽  
Author(s):  
Antonio Alarcón ◽  
Finnur Lárusson
Keyword(s):  
Riemann Surface ◽  
Convex Hull ◽  
Minimal Surface ◽  
Cohomology Class ◽  
Holomorphic Maps ◽  
De Rham Cohomology ◽  
Surface Theory ◽  
Open Riemann Surface ◽  
De Rham ◽  
Smooth Locus

Let [Formula: see text] be a connected open Riemann surface. Let [Formula: see text] be an Oka domain in the smooth locus of an analytic subvariety of [Formula: see text], [Formula: see text], such that the convex hull of [Formula: see text] is all of [Formula: see text]. Let [Formula: see text] be the space of nondegenerate holomorphic maps [Formula: see text]. Take a holomorphic 1-form [Formula: see text] on [Formula: see text], not identically zero, and let [Formula: see text] send a map [Formula: see text] to the cohomology class of [Formula: see text]. Our main theorem states that [Formula: see text] is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on [Formula: see text] can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.

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