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NEW EXAMPLES OF WILLMORE TORI IN S4 A. GOUBERMAN, K. LESCHKE
 

Summary: NEW EXAMPLES OF WILLMORE TORI IN S4
A. GOUBERMAN, K. LESCHKE
Abstract. Using the (generalized) Darboux transformation in the case of the Clifford
torus, we construct for all Pythagorean triples (p, q, n) Z3
a CP3
­family of Willmore
tori in S4
with Willmore energy 2n2
.
1. Introduction
Classical geometers like Bianchi, Darboux and Bšacklund used local transformations to ob-
tain new examples of a particular class of surfaces out of simple known ones by geometric
constructions. For instance, the Darboux transformation was classically [6] defined for
isothermic surfaces, that is surfaces which allow a conformal curvature line parametriza-
tion: two immersions f and f form a Darboux pair if there exists a sphere congruence
which envelopes both surfaces f and f . In this case, both f and f are isothermic.
In modern days, the Darboux transformation is used to study global properties of surfaces:
relaxing the enveloping condition one obtains a (generalized) Darboux transformation for
conformal immersions f : M S4 of a Riemann surface into the 4­sphere. The existence
of a Riemann surface worth of global solutions is, at least in the case when M = T2 is a

  

Source: Applebaum, David - Department of Probability and Statistics, University of Sheffield

 

Collections: Mathematics