Existence Of H-~ 2 ¡Á R And S ~ 2 ¡Á The R Complete Regular Gaussian Curvature Surface

In this paper,we mainly study the existence of complete surfaces with constant Gauss curvature in H²Ã—R and S²Ã—R.In section 1,we introduce some knowledge about this subject.In section 2,we prove that in H²Ã—R there exist complete revolution surfaces with constant Gauss curvature≥-1;in S²Ã—R there exist complete revolution surfaces with constant Gauss curvature＞1 or = 0.In section 3,we prove that there is no complete surface with constant Gauss curvature＜-1 in H²Ã—R and S²Ã—R.In section 4,we mainly study the existence of complete surfaces with constant Gauss curvature∈(0,1) or[-1,0) in S²Ã—R.we prove that in S²Ã—R there is no complete surface with constant Gauss curvature∈(0,1);in the case of Gauss curvature∈[-1,0),there is no complete surface with constant Gauss curvature which Gauss curvature and height function's module satisfy some relationship;there is no compact surface with constant Gauss curvature∈[-1,0);there is no complete surface with constant Gauss curvature＜(2-5^1/2)/4.