Surfaces Isometric Embedding Of A Number Of Relevant Issues

In the first part of the present paper the infinitesimal nonrigidity of a class of convex surfaces with planar boundary, is given. This result shows that if the image of the Gauss map of an evolution convex surface with planar boundary covers some hemisphere, such a surface maybe be of infinitesimal nonrigidity for the isometric infinitesimal deformation of planar boundary. Meanwhile we present a kind of convex surfaces which are of infinitesimally nonrigidity and global nonrigidity. Second, the analyticity of solution to a class of degenerate elliptic equation is obtained. and some application in a geometric problem is presented. Finally, we discuss the realization of Alexandroff's positive annule into R~3 and attain some elementary results.