By Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit, Ulrich Pinkall (auth.)
The conformal geometry of surfaces lately built by means of the authors results in a unified realizing of algebraic curve idea and the geometry of surfaces at the foundation of a quaternionic-valued functionality concept. The ebook bargains an uncomplicated advent to the topic yet takes the reader to relatively complex themes. Willmore surfaces within the foursphere, their Bäcklund and Darboux transforms are coated, and a brand new evidence of the class of Willmore spheres is given.
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Extra resources for Conformal Geometry of Surfaces in S 4 and Quaternions
Given vature vector of is the have, mean immersed an f at fact, the same sphere. in M E x mean holomorphic the L curve mean cur- is determined of sphere curvature Geometry Sx, by Sx. On the other hand, S" Example 17. 12) simplify the coordinate expressions for the Hopf fields, which we now write as follows Proposition 12. 12) and we can w w = w 0 dR + R dH + H * dfH * 0) G-1, dR) G-', + R dH * dH + * 1H(NdN - 2 We Proof. 15) H * dN. *dN). 14) rewrite dH + R = dN 0 * -2dH + IH dN (dN * 2 - - N * dN) H - 1H* (dN+N*dN) w.
3 The Willmore Condition in Affine Coordinates We use the notations of the previous Proposition 12, and in addition abbre- viate v, = dR+R*dR. Note that V Proposition 13. -dR +. *dRR = The Willmore -dR -- integrand A A *A > = 16 For f : M -4 R, this is * dR = -v. given by 1 1 < R - JRdR - *dR12 is the classical = 4 (IHI2 - K integrand 1 < A A *A >= 4 (Ih 12 - K)Idfl2. - K-L)JdfJ2. 3 The Willmore Condition in Affine Coordinates 45 Proof. < A A *A > 8 traceR(-A' 1 4 Now see We Proposition now 1V) 4 Re( - (*A)) = IV12 = 2 16 4 express the and * A = JdR + R * dR12 16 jRdR - *dR12.
DN). 14) rewrite dH + R = dN 0 * -2dH + IH dN (dN * 2 - - N * dN) H - 1H* (dN+N*dN) w. * dN 2 2 But H(NdN - *dN). 3 The Willmore Condition in Affine Coordinates We use the notations of the previous Proposition 12, and in addition abbre- viate v, = dR+R*dR. Note that V Proposition 13. -dR +. *dRR = The Willmore -dR -- integrand A A *A > = 16 For f : M -4 R, this is * dR = -v. given by 1 1 < R - JRdR - *dR12 is the classical = 4 (IHI2 - K integrand 1 < A A *A >= 4 (Ih 12 - K)Idfl2. - K-L)JdfJ2. 3 The Willmore Condition in Affine Coordinates 45 Proof.