On the Darboux equation.
On the Darboux equation.
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关于达布方程。
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发表时间：
2001
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通讯作者：
Zbigniew Blocki
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作者：
Zbigniew Blocki
给定一个度量张量为g的黎曼曲面M，我们计算了度量g−Du⊗Du的高斯曲率，其中u是M上的光滑函数。导言。在他著名的专著[2]中，达布等人考虑了在R3中嵌入抽象黎曼曲面的问题。如果g是M上的度量张量，则在M上寻找三个函数u，v，w，使得g=Du⊗Du+dv⊗dv+dw⊗dw。在本地，其中两个，比如v和w，必须满足g̃：=dv⊗dv+dw⊗dw>；0。因此，对于新度规g̃的高斯曲率K̃，我们有(0.1)K̃=0，这实际上是一个仅针对第一分量u的方程，称为达布方程。繁琐的计算(见[3])表明，用现代术语来说，这个方程等价于(0.2)M(U)=K(1−|∇u|)，其中M是Monge-Ampère算子，K是高斯曲率(相对于原始度规g)。本文的目的是给出K̃的精确公式，特别是证明(0.1)和(0.2)是等价的。也就是说，我们将证明以下结果。2000年数学学科分类。53C42。KBN Grant 7 T07A 003 16.
Given a Riemannian surface M with a metric tensor g, we compute the Gauss curvature of a metric g− du⊗ du, where u is a smooth function on M . Introduction. In his celebrated monograph [2], Darboux, among other things, considered the problem of embedding abstract Riemannian surfaces in R3. If g is a metric tensor on M then one looks for three functions u, v, w on M , such that g = du⊗ du + dv ⊗ dv + dw ⊗ dw. Locally, two of them, say v and w, must satisfy g̃ := dv ⊗ dv + dw ⊗ dw > 0. For the Gauss curvature K̃ of the new metric g̃ we thus have (0.1) K̃ = 0, which is in fact an equation just for the first component u, known as the Darboux equation. Tedious calculations (see e.g. [3]) show that this equation, is, in modern terms, equivalent to (0.2) M(u) = K(1− |∇u|), where M is the Monge-Ampère operator and K the Gauss curvature (with respect to the original metric g). The aim of this note is to give the precise formula for K̃, which will in particular show that (0.1) and (0.2) are equivalent. Namely we shall prove the following result. 2000 Mathematics Subject Classification. 53C42. Supported by KBN Grant 7 T07A 003 16.