TY - GEN

T1 - Minimal surfaces under constrained Willmore transformation

AU - Quintino, Áurea C.

N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - The class of constrained Willmore (CW) surfaces in space-forms constitutes a Möbius invariant class of surfaces with strong links to the theory of integrable systems, with a spectral deformation [8], defined by the action of a loop of flat metric connections, and Bäcklund transformations [9], defined by a dressing action by simple factors. Constant mean curvature (CMC) surfaces in 3-dimensional space-forms are [25] examples of CW surfaces, characterized by the existence of some polynomial conserved quantity [21, 22, 24]. Both CW spectral deformation and CW Bäcklund transformation preserve [21, 22, 24] the existence of such a conserved quantity, defining, in particular, transformations within the class of CMC surfaces in 3-dimensional space-forms, with, furthermore [21, 22, 24], preservation of both the space-form and the mean curvature, in the latter case. A classical result by Thomsen [28] characterizes, on the other hand, isothermic Willmore surfaces in 3-space as minimal surfaces in some 3-dimensional space-form. CW transformation preserves [8, 9] the class of Willmore surfaces, as well as the isothermic condition, in the particular case of spectral deformation [8]. We define, in this way, a CW spectral deformation and CW Bäcklund transformations of minimal surfaces in 3-dimensional space-forms into new ones, with preservation of the space-form in the latter case. This paper is dedicated to a reader-friendly overview of the topic.

AB - The class of constrained Willmore (CW) surfaces in space-forms constitutes a Möbius invariant class of surfaces with strong links to the theory of integrable systems, with a spectral deformation [8], defined by the action of a loop of flat metric connections, and Bäcklund transformations [9], defined by a dressing action by simple factors. Constant mean curvature (CMC) surfaces in 3-dimensional space-forms are [25] examples of CW surfaces, characterized by the existence of some polynomial conserved quantity [21, 22, 24]. Both CW spectral deformation and CW Bäcklund transformation preserve [21, 22, 24] the existence of such a conserved quantity, defining, in particular, transformations within the class of CMC surfaces in 3-dimensional space-forms, with, furthermore [21, 22, 24], preservation of both the space-form and the mean curvature, in the latter case. A classical result by Thomsen [28] characterizes, on the other hand, isothermic Willmore surfaces in 3-space as minimal surfaces in some 3-dimensional space-form. CW transformation preserves [8, 9] the class of Willmore surfaces, as well as the isothermic condition, in the particular case of spectral deformation [8]. We define, in this way, a CW spectral deformation and CW Bäcklund transformations of minimal surfaces in 3-dimensional space-forms into new ones, with preservation of the space-form in the latter case. This paper is dedicated to a reader-friendly overview of the topic.

KW - Bäcklund transformations

KW - Constant mean curvature surfaces

KW - Constrained Willmore surfaces

KW - Isothermic surfaces

KW - Minimal surfaces

KW - Polynomial conserved quantities

KW - Spectral deformation

KW - Willmore energy

UR - http://www.scopus.com/inward/record.url?scp=85111154644&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-68541-6_13

DO - 10.1007/978-3-030-68541-6_13

M3 - Conference contribution

AN - SCOPUS:85111154644

SN - 9783030685409

T3 - Springer Proceedings in Mathematics and Statistics

SP - 229

EP - 245

BT - Minimal Surfaces

A2 - Hoffmann, Tim

A2 - Kilian, Martin

A2 - Leschke, Katrin

A2 - Martin, Francisco

PB - Springer

T2 - Workshop Series of Minimal Surfaces: Integrable Systems and Visualisation, 2016-19

Y2 - 27 March 2017 through 29 March 2017

ER -