Willmore surfaces in spheres via loop groups. III: On minimal surfaces in space forms. (English) Zbl 1368.53010

Wang, Peng

Tohoku Math. J. (2) 69, No. 1, 141-160 (2017).

This article under review is the third paper of a series of papers concerning the global geometry of Willmore surfaces in terms of loop group theory. In the first part of the paper, the forms of the normalized potentials of strongly conformally harmonic maps containing a non-zero constant real vector is derived. In the last part of the paper, expressing the normalized potentials by some strictly upper triangular matrix-valued 1-forms, the author gives by direct computations a concrete description of all Willmore surfaces corresponding to the first type of nilpotent Lie subalgebras investigated in the second article of the series [the author, “Willmore surfaces in spheres via loop groups. II: A coarse classification of Willmore two-spheres by potentials”, Preprint, arxiv:1412.6737].
For Part IV, see [the author, Chin. Ann. Math., Ser. B 42, No. 3, 383–408 (2021; Zbl 1473.53086)].

Reviewer: Gabriel Eduard Vilcu (Ploieşti)

MSC:

53A30	Conformal differential geometry (MSC2010)
58E20	Harmonic maps, etc.
53C43	Differential geometric aspects of harmonic maps
53C35	Differential geometry of symmetric spaces

Keywords:

Willmore surfaces; normalized potential; minimal surfaces; Iwasawa decompositions

Citations:

Zbl 1473.53086