THE VALUE DISTRIBUTION OF GAUSS MAPS OF IMMERSED HARMONIC SURFACES WITH RAMIFICATION

Motivated by the result of Chen-Liu-Ru[1],we investigate the value distribution properties for the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(n) with ramification,which can be seen as a generalization of the results in the case of the minimal surfaces.In addition,we give an estimate of the Gauss curvature for the K-quasiconfomal harmonic surfaces whose generalized Gauss map is ramified over a set of hyperplanes.

RIGIDITY RESULTS FOR SELF-SHRINKING SURFACES IN R^(4)

In this paper,we give some rigidity results for complete self-shrinking surfaces properly immersed in R^(4) under some assumptions regarding their Gauss images.More precisely,we prove that this has to be a plane,provided that the images of either Gauss map projection lies in an open hemisphere or S^(2)(1/2–√)∖S^(-1)+(1/2–√).We also give the classification of complete self-shrinking surfaces properly immersed in R^(4) provided that the images of Gauss map projection lies in some closed hemispheres.As an application of the above results,we give a new proof for the result of Zhou.Moreover,we establish a Bernstein-type theorem.

On Twistor Gauss Maps of Surfaces in 4-Spheres

In this paper we study surfaces in S^4 and their twistor Gauss maps.Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given.We find many examples of nonisotropic harmonic maps from a surface to(?)P^3.

The Gauss maps of Demoulin surfaces with conformal coordinates

Demoulin surfaces in the real projective 3-space are investigated. Our result enables us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian 6-symmetric space.

Laguerre Geoinetry of Surfaces in R^3

Let f ： M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L（f） = f（H2 - K）/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R^3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R^3. And we give a classification theorem of surfaces in R^3 with vanishing Laguerre form.

Laguerre Minimal Surfaces in ■~3

Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space R^3. We show that any Laguerre minimal surface in R^3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in R^3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space R0^3.

Minimal Lagrangian submanifolds of the complex hyperquadric

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angle functions encoding the geometry of the Lagrangian submanifold at hand.We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface.We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions,respectively all but one,coincide.

Moment maps and isoparametric hypersurfaces of OT-FKM type

Associated with a Clifford system on R^(2 l),a Spin(m+1)action is induced on R^(2 l).An isoparametric hypersurface N in S^(2 l-1)of OT-FKM(Ozeki,Takeuchi,Ferns,Karcher and Miinzner)type is invariant under this action,and so is the Cartan-Munzner polynomial F(x).This action is extended to a Hamiltonian action on C^(2 l).We give a new description of F(x)by the moment mapμ:C2 l→t^(*),where t≌o(m+1)is the Lie algebra of Spin(m+1).It also induces a Hamiltonian action on CP^(2 l-1).We consider the Gauss map g of N into the complex hyperquadric Q_(2 l-2)(C)■CP^(2 l-1),and show that g(N)lies in the zero level set of the moment map restricted to Q_(2 l-2)(C).