In terms of the almost complex affine connection and moving unitary frames, all totally rael minimal immersions from R-2 into the nearly Kahler S-6 axe determine explicitly. Moreover, the complete flat almost complex curves in the nearly Kahler S-6 are determined completely.
By using Darboux transformations, the authors give the explicit construction for local iso-metric immersions of space forms Mn(c) into space forms M2n-1(c + ε2) via purely algebraicalgorithm.
The Ribaucour transformations for flat Lagrangian submanifolds in Cn and CPn via loop group actions are given. As a consequence, the authors obtain a family of new flat Lagrangian submanifolds from a given one via a purely algebraic algorithm. At the same time, it is shown that such Ribaucour transformation always comes with a permutability formula.
By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1＋1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n＋1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1＋1 are considered.
The first and second variation formulas of the energy functional for a nondegenerate map between Finsler manifolds is derived. As an application, some nonexistence theorems of nonconstant stable harmonic maps from a Finsler manifold to a Riemannian manifold are given.
In this paper,we reformulate the Euler-Lagrange equations of Willmore surfaces in S^n as the flatness of a family of certain loop algebra-valued 1-forms.Therefore we can give the Weierstrass type representation of conformal Willmore surfaces.We also discuss the relations between conformal Willmore surfaces in S^n and minimal surfaces in constant curvature spaces S^n,R^n,H^n,and prove that some special Willmore surfaces can be derived from minimal surfaces in S^n,R^n,H^n.
