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 figuring out of algebraic curve concept and the geometry of surfaces at the foundation of a quaternionic-valued functionality thought. The ebook deals an hassle-free advent to the topic yet takes the reader to particularly complicated themes. Willmore surfaces within the foursphere, their Bäcklund and Darboux transforms are lined, and a brand new evidence of the category of Willmore spheres is given.

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3. E(S) harmonic submanifold = E inner The energy 7. M is End(V) I S2 Fnd(EV) I XS -SX}, I YS SY}. JY E End(fff) IX (indefinite) < defined = fm := functional < dS A *dS >. with respect to variations of S are called M to Z. S is harmonic if and only if the Z-tangential component of dS vanishes: (d This condition is equivalent to * any dS)T of the F. 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.

HIE E is is the called a is equivalent to the fact 0 E F(E), projection. 1) do) and hence induces a holomorphic structure F(E) into itself, complex line bundle E. line bundle A complex line bundle E C H induces a quaternionic of H maps the on L=EH=EE)Ej CH. to the structure of a JE admits a unique extension complex structure bundle (L, J), namely right-multiplication complex quaternionic by (-i) on line bundle (L, J) C H induces a a complex quaternionic Ej. Conversely, line bundle complex The E:= Definition lift 10.

HP1 with deriva- L C H in curve Then there exist complex structures J such that We want to extend J and j = to S E a jj complex structure F(End(H)) such that SL this Q case". e. 4) dSL c L. of S is'clear: The existence bundle L' Since L' Identify not unique, L. is H = 0, R can be Q: interpreted I((S 4 + 1(SdS 4 Q+ V) E F(L), SIL complementary Ji SIP := j. := S + R is kerR, and We compute If some if and RS + SR Note that for and define 7r, RH c L c whence R2 L E) L' = It is easy to see that S is not unique.

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