Complete surfaces inE3 with constant mean curvature
Complete surfaces inE3 with constant mean curvature
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E3 中具有恒定平均曲率的完整曲面
DOI：
10.1007/bf02566886
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发表时间：
1966
影响因子：
0.9
通讯作者：
R. Osserman
中科院分区：
文献类型：
--
作者：
T. Klotz;R. Osserman
本文讨论了E3中具有常平均曲率H的完备曲面.我们的主要结果是：一个具有H-cr且其上高斯曲率K不变号的完全浸入曲面一定是球面或直圆柱面，而在紧致曲面中只有球面具有常数H。H. HOPF（[7]，p. 241）对于亏格为零的曲面。AD ALEXANDROV（[2]，或[8]，第7章）在曲面没有自交，即嵌入E3的假设下，给出了任意亏格曲面的一个证明.虽然下面所述的定理是在E a中用常数H表征所有完整曲面的方向上迈出的一步，但它并没有给出关于紧致情况的新信息。
In this paper we discuss complete surfaces in E 3 which have constant mean curvature H. Our main result states that a complete, immersed surface with H-cr on which the Gaussian curvature K does not change sign must be a sphere or a right circular cylinder.It has long been conjectured that among compact surfaces only the sphere has constant H. This conjecture was proved by H. HOPF ([7], p. 241) for surfaces of genus zero. A proof for surfaces of arbitrary genus was given by AD ALEXANDROV ([2], or [8], Chapter 7) under the assumption that the surfaces have no self intersections, ie, that they are embedded in E 3. While the theorem stated below is a step in the direction of characterizing all complete surfaces in E a with constant H, it does not give new information about the compact case.