On the mean curvature function for compact surfaces
On the mean curvature function for compact surfaces
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DOI：
10.4310/jdg/1214436095
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发表时间：
1981
影响因子：
2.5
通讯作者：
H. Lawson, Jr.;R. Tribuzy
中科院分区：
文献类型：
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作者：
H. Lawson, Jr.;R. Tribuzy
R 中的任何曲面都由其第一和第二基本形式全等确定，这是一个经典事实。我们将在本文中证明，紧致曲面本质上是由第一个基本形式和第二个基本形式的迹（即度量和平均曲率函数）决定的。这种现象唯一可能的例外发生在平均曲率恒定的情况下。当然，霍普夫长期以来的猜想是，唯一这样的（紧）表面是圆球体。我们的主要结果的明确陈述如下。用 M(c) 表示恒定截面曲率 c 的完整单连接 3 流形。定理。令 Σ 为配备黎曼度量的紧致曲面，并令 H: Σ -^ R 为平滑函数。如果 H 不是常数，则最多存在两个几何上不同的 Σ 到 M(c) 中的平均曲率 H 的等距浸没。备注。 1. 如果两个浸没没有 M(c) 的等距差异（即同余），则称它们在几何上不同。 2. 上述定理可以立即应用于不可定向的表面。这里，函数 H: Σ7 —» R 必须被 2 片可定向覆盖表面 π: Σ -> Σ 上的函数 H: Σ -»R 替换，其性质为 H(a(ρ)) = -H(p)，其中 a: Σ ̂ >Σ 是覆盖 T 的甲板变换。 3. 上述结果代表了第二作者[5]的博士论文中证明的定理对大于一的属的推广。第一作者坚持认为证明的难点部分和主要思想源于此。
It is a classical fact that any surface in R is determined up to congruences by its first and second fundamental forms. We shall prove in this article that compact surfaces are essentially determined by the first fundamental form and only the trace of the second, that is, by the metric and the mean curvature function. The only possible exception to this phenomenon occurs in the case of constant mean curvature. Of course, it is a long-standing conjecture of Hopf that the only such (compact) surfaces are the round spheres. An explicit statement of our main result is as follows. Denote by M(c) the complete simply-connected 3-manifold of constant sectional curvature c. Theorem. Let Σ be a compact oriented surface equipped with a riemannian metric, and let H: Σ -^ R be a smooth function. If H is not constant, then there exist at most two geometrically distinct isometric immersions of Σ into M(c) with mean curvature H. Remarks. 1. Two immersions are said to be geometrically distinct if they do not differ by an isometry of M(c), i.e., by a congruence. 2. The theorem above can be immediately applied to nonorientable surfaces. Here the function H: Σ7 —» R must be replaced by a function H: Σ -»R on the 2-sheeted orientable covering surface π: Σ -> Σ with the property that H(a(ρ)) = -H(p) where a: Σ ̂ >Σ is the deck transformation of the covering TΓ. 3. The result above represents a generalization to genus greater than one, of a theorem proved in the doctoral dissertation of the second author [5]. The first author insists on stating that the hard part of the proof and the principal ideas originated there.