| In this paper, we give explicit constructions and formulations for harmonic maps from R1,1 into classical real semisimple Lie groups by using Darboux transformation. We also discuss pluriharmonic maps from complex manifoldsinto symmetric spaces and Willmore surfaces in Sn. By converting geometric conditions satisfied by these maps into integrable systems, and using the the-ory of integrable systems, we give explicit constructions for pluriharmonic maps from complex manifolds into symmetric spaces and the Willmore surfaces in Sn respectively. Finally, we classify hypersurfaces in Sn+1 with three distinct prin-ciple curvatures and zero Mobius form using the theory of Mobius geometry. The paper consists of four chapters. We arrange it as follows:In chapter one, we give some explicit constructions of harmonic maps from Lorentz space R1,1 to classical semisimple Lie groups SL(N,R),SU(p,q) Sp(p,q), SO(p,q) by using Darboux transformation, i.e., we may obtain a new parallel transport from a known one via purely algebraic algorithm. Thus we obtain a new harmonic map.Let G be semisimple matrix group with Lie algebra g, T a fixed maximal abelian subalgebra of g, T the orthogonal complement of T with respect to the Killing form <,> of g, and a, b 6 T a fixed regular elements. If (x;t) = A(x,t)dx + B(x,t)dt is a flat,g-valued connection 1-form, there exists a unique : R2 --. G such that (0,0) = e(e is a unit of G). Such $ is called the parallel transport of Let be a smooth map.Terng has proved there is a bijective correspondence between the space of harmonicmaps with and the space of solution of the -1-flow equation Ut = [a, Qb,-1(w)](cf.[Tl]),where G . XQ denotes the Adjoint G-orbit at X0. Let v = Qb,-1(u) E g. Since the -1-flow equation had Lax pair, i.e.,where u T. In this chapter, we construct a Darboux matrix D(x, t, A) such that satisfiesWe construct a Darboux matrix for each groups G in the following.Theorem 1.2.2. Let G = SL(N,R), A = diag(, where Let hp be a row solution of (1.1-4) with H = is N x N nondegenerated matrix. If S = H H is a real matrix at one point, then u ,, and a new parallel transport is obtained from a known parallel transport via the following transformationThe. corresponding new harmonic map 0 : .R1,1 SL(N,R) is given by the followingwithwhere G ?XQ denotes the Adjoint G?orbit at XQ.If we choose a nonzero complex number such that(2) Choose row vectors , such that are of rank k and rank N ?k matrices at one point respectively, where (3) At one point, satisfyThen we haveTheorem 1.2.5. Let G ?SU(p, q). Choose and h1, h2, ???,hN as (1), (2), (3). Set H = , and S = H-1AH. Then a new parallel transport is obtained from a known parallel transport via the transformation (1.2.10). The corresponding new harmonic map SU(p,q) is given by (1.2.11) with (1.2.12).If we choose a nonzero complex number p, such that(2)' Choose lp(p = 1, 2, ???, 27V) such that 1 a 2N) are of rank f, rank 2N ?k matrices at a point respectively, where (3)' At one point, satisfywhere Then we haveTheorem 1.2.7. Lef G = Sp(N) be a symplectic group. and Then a new parallel transport is obtained from a known parallel transport. via transformation (1.2.10). The corresponding new harmonic map Sp(N) is given by (1.2.11) with (1.2.12).Remark. The method of the above Darboux transformation can be used in (p, q)-type symplectic group Sp(p,q). But we need rectify (3)' (cf. remark 1.2.3).Theorem 1.2.8. Let G = SO(p, q)(N = p + q is even). Choose ???, (k = N/2) and h1, h2, ...... , hN as (1), (2), (3). Set H = S = H-1kH is real at one point. Then a new parallel transport is obtained from a known parallel transport via transformation (1.2.10). The corresponding new harmonic map R1.1 -->SO(p,q) is given by (1.2.11) with (1.2.12).In chapter two, we prove that there is a bijective correspondence between the space of pluricharmonic maps from complex manifolds into symmetry spaces and the space of extended lifts modulo ga... |
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