Liouville equation and spherical convex polytopes
Liouville equation and spherical convex polytopes
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DOI：
10.1090/s0002-9939-1992-1137227-5
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发表时间：
1992-04
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影响因子：
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通讯作者：
F. Luo;G. Tian
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文献类型：
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作者：
F. Luo;G. Tian
研究复平面上具有给定奇性的Liouville方程Au = -e2 u，得到了解存在的一个充分必要条件.证明是基于连续性方法和唯一性定理。假设M是一个有n个点V1，...，Vn被移除并且ai，.，an E（0，27 r）是n个给定的数。我们感兴趣的是寻找一个凸多面体，其边界P在S3中具有n个顶点，使得P {V1，...，Vn}是Riemann表面M，Vi处的锥角是ai。我们的结果是下面的定理1。设M是n-穿孔黎曼球面（n > 3），ai是（0，27）中的n个数。则在S3中存在一个具有n个顶点的凸多面体，其边界P满足（a）P { V1，...，Vn }与M共形等价，（B）Vi处的锥角为ai，当且仅当n（1）Zai> 27 r（n-2），i=1，（2）Zai a <27 r（n2），对所有j = 1，2，. ., n.由于IzIa/21 ldzl在0处的锥角是α，因此根据奇异度量，该定理可以被陈述为定理2。在穿孔复平面Au = Cl，VI.，Vn}，使得在每个Vi附近，u（z）= 8ilogIz Vi + a连续函数，其中/8i E（-1，0）和u = -2 log Iz + a连续1991年数学学科分类。小学52 A55;中学53 C25。? 1992美国数学学会0002-9939/92 $1.00 + $.25每页
We study the Liouville equation Au = -e2u in the complex plane with prescribed singularities and obtain a necessary and sufficient condition for the existence of the solution. The proof is based on the continuity method and a uniqueness theorem. Suppose M is a punctured Riemann sphere with n points V1, ..., Vn removed and ai, ... , an E (0, 27r) are n given numbers. We are interested in seeking a convex polytope with boundary P in S3 having n vertices so that P {V1, ..., Vn} is the Riemann surface M and the cone angle at Vi is ai. Our result is the following Theorem 1. Let M be an n-punctured Riemann sphere (n > 3) and ai be n numbers in (0, 27) . Then there is a (necessarily unique) convex polytope in S3 with n vertices whose boundary P satisfies (a) P { V1, ..., Vn } is conformally equivalent to M and (b) the cone angle at Vi is ai, if and only if n (1) Zai>27r(n-2), i=1 and (2) Zai a < 27r(n 2), for all j = 1,2, .. .,n. iAj Since the cone angle of IzIa/21ldzl at 0 is a, in terms of the singular metric, the theorem can be stated as Theorem 2. The Liouville equation Au = -exp(2u) in the punctured complex plane M = CI,VI ..., Vn} so that near each Vi, u(z) = 8ilogIz Vi + a continuous function, where /8i E (-1, 0) and u = -2 log IZ + a continuous Received by the editors April 22, 1991. 1991 Mathematics Subject Classification. Primary 52A55; Secondary 53C25. ? 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page