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Download Conformal Geometry of Surfaces in Quaternions by Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz PDF

By Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit, Ulrich Pinkall

The conformal geometry of surfaces lately constructed through the authors ends up in a unified realizing of algebraic curve concept and the geometry of surfaces at the foundation of a quaternionic-valued functionality idea. The e-book bargains an simple creation to the topic yet takes the reader to really complex themes. Willmore surfaces within the foursphere, their Bäcklund and Darboux transforms are lined, and a brand new facts of the type of Willmore spheres is given.

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Its is In par- *JEJ)O) + ALIE AL by determines *Sdo) - + do). AL. following the curves of holomorphic Willmore vanishing in with immersed second space, ellipse. curvature and Cp3 are ex- functional. Immersions conformally the E G Ej, 0- + = implies Super-Conformal 0. = (6E Sdo) 1. 2 = = (-i)-eigenspace 2 AL *d(JO))i). *dS),o 4 Comparison + L in in the *d(Jo)) + 2 + J(dO + the component = A0 *d(JO)) *d(Jo)) + irEALO + R4, into fundamental centered The surface the at is called the image of form (a is mean a double curvature super-conformal tangential cover vector, if this of) the ellipse circle.

39 - 46, 2002 © Springer-Verlag Berlin Heidelberg 2002 * + df (Y) Ndf (Y)dR(X) + *df (Y)dR(X). 40 Proposition The 8. 'Rdf Proof. mean (*dR 2 By definition + 'H vector curvature dfR RdR), 2 II trace 2 is given by (7-3) NdN). 5) of the trace, jdfJ2 4'H Geometry Conformal and Affine 7 Metric = *dfdR = -df (*dR dN * - df RdR) + df - + * (*dN + but (*dN If follows + NdN)df = *dNdf = -df A - dN * dR = df -dN A = -df (*dR df = -d(Ndf) RdR). + that 27ildfI2 = -df (*dR RdR), + and 2 ldfTf Similarly for Proposition and let K' dR + (*dR = RdR)Tf + N.

E. : LNM 1772, pp. 31 - 38, 2002 © Springer-Verlag Berlin Heidelberg 2002 = 0. 2) = 0, (6-3) = 0. 4) fact, d(S Proof. St be Let a Q * = 4d * of S in variation fm d d Wt- E(S) the 4d = S(d dS)T * Z with = (Sd variational * dS)T. (6-5) field vector Y. 10) formula < dS A *dY > =< < dY A *dS > + < dS A *dY traceR(AB) and dS(-dY) *dS - traceR(BA), = we dY >=< dY A *dS > * >. get . Thus d Wt- E(S) 2 = Therefore f' < dY A *dS >= -2 JM S is harmonic For the other 0 only if and first equivalences, = d = (d d(S') * * with *Q - 8d * * dS >= -2 * fm < Yd*dS >.

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