Some Results On Harmonic Maps Of Surfaces And Geometry Of Submanifolds

This paper consists two charpters.In Charpter 1,which includes six sections,we consider some properties for harmonic maps from surfaces into complex Grassrnann manifolds.In Charpter 2, which includes four sections,we study some topics in geometry of submanifolds.Charpter 1 Some Results on Harmonic Maps from Surfaces into Complex Grassmann ManifoldsThis charpter includes six sections.We obtain some results for harmonic maps from surfaces into complex Grassmann manifolds.First in 2 we establish the generalized Frenet formulae for harmonic maps from surfaces into complex Grassmann manifolds,which generalizes the corresponding Frenet formulae for holomor-phic maps.Then in 3 we discuss the curvature pinching properties for harmonic maps and generalize the related results for pseudo-holomorphic curves into general harmonic maps.It is well-known that starting from a harmonic map , we can construct the following harmonic sequence:Suppose that the harmonic map satisfies the following condition:when 0 j k0, rank( j) = m0 = m0 ; and when 1 t p,ifthen rank( j) = mt ; here p is a natural number.Define a function c : [0, ] - R byWe denote by and K the Kaehler angle and the Gaussian curvature of M of the metric induced by , respectively. Our main results are the following.Theorem 1.3 Let M be a compact Riemannian surface, : M - G(m,n) he a harmonic map described above.If K c( ), then K = c( ) , and = [ ],where : M - G(l,n)is holomorphic which generates the Frenet harmonic sequence; and : M - G(m - l,n) is anti-holomorphic, I = rauk( ' ).Theorem 1.5 Let : M - G(m, n) be a harmonic map from a compact Riemannian surface into G(m,n) which generates the following harmonic sequence:If K J 4/[m.(2j(r - j) 4- r)] , then K ] = 4/[m(2j(r - j] + r)],and (p is holomorphic; if K j 4/[m(2j(r - j) + r)] .and det Ai 0, i = 0, ... , j -1, then the same conclusion holds.Theorem 1.6 Let : S2 - G(m,n) be a harmonic map. If for some p 2,rank( j) = m, |det Aj | 0, j = 0, ... ,p - 2 (rasp. rank( -j) = m, | det A_j | then K equals to the end-values of above formula and is a pseudo-holomorphic curve.For a given harmonic map .it is natural to ask that when the harmonic map is '-irreducible or "-irreducible, and when is isotropic?If is non-isotropic,how to calaulate its isotropy order?These are all the elementary topics for harmonic maps. The purpose of 4 is to deal with these questions.For known results,when m = 1,i.e.,for harmonic maps of M into G(1, n) = CPn-1,Peng and Jiao provides a method to determine the isotropy order ord( )(see [PJ]). Put (j) = 0 1 j, then (j) is a smooth map of M into G(j + 1,n). Peng and Jiao prove that if is a non-isotropic and linearly full harmonic map, and there is a natural number k such that (0) = , (1), ..., (k-1) are harmonic,but (k) is not harmonic, then ord() = k.Obviously,this method is not practicable,because it is far more difficult to determine whether (j) is harmonic than to determine whether 0 is orthogonal to j, while it is also difficult to determine whether is non-isotropic and linearly full. In 4 we obtain an explicit formula to calculate ord( ) which is suitable for any harmonic map of M into G(m, n). Moreover, the formula depends on (p only and is independent of 0). On the other hand, we also provide a criterion to decide when is '-irreducible or "-irreducible.Our main results are the following.Theorem 1.7 Let M be a connected iemannian surface,and : M -CPn-l = G(1,n) U(n) be a harmonic map. thenhere we formally write -1AZ = 0.Theorem 1.9 Let M be a connected Riemannian surface, and (p : M -G(m, n) be a harmonic map. Then (f is 0'-irreducible if and only if rank(Az ) = m;while is "-irreducible if and only if rank( Az) = m.Theorem 1.10 Let M be a connected Biemannian surface, and : M -G(m, n) be a harmonic map. thenIn 5 we discuss the degeneration of the harmonic sequences induced by harmonic maps,i.e., we want to determine when the harmonic sequences must contain d...