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 -