Simply-connected Nonpositive Curved Surfaces in R3

where |B| is the length of second fundamental form. |B|2 = 4|H|2 − 2K, where H is the mean curvature and K is the Gauss curvature. The curvature of minimal surfaces is nonpositive. Then we exam what is still hold for some minimal surface theorems if extending the minimal condition H ≡ 0 to the surfaces with K ≤ 0 and (1). In 2001, F. Xavier [18] has the following theorem: Theorem [Xavier]. Let M ⊂ R3 be a complete simply-connected embedded minimal surface whose Gaussian curvature is bounded from below. If there is a plane whose intersection with M is transversal and connected then M is a plane or a helicoid. Meeks and Rosenberg [12] has the following theorem: Theorem [Meeks and Rosenberg]. A properly embedded simply-connected minimal surface in R3 is either a plane or a helicoid. The minimal surfaces can be divided into two classes. One is with finite total curvature(FTC) and the other is with infinite total curvature. For the case of minimal surfaces with FTC, ∫ |B|2 = −2 ∫ K < ∞, so condition (1) is satisfied. In this paper, we shall prove the following theorem: Theorem 1. A complete simply-connected embedded C2-surface M in R3 with K ≤ 0 and (1) is a plane.