The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature
The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature
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DOI：
10.4310/cag.2000.v8.n4.a5
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发表时间：
2000
影响因子：
0.7
通讯作者：
Zheng-chao Han;Yanyan Li
中科院分区：
文献类型：
--
作者：
Zheng-chao Han;Yanyan Li
设(M，g)是n维紧、光滑、无边界的黎曼流形。对于n=2，Poincare的统一定理指出，M上存在与g逐点共形且具有常高斯曲率的度量。对于n>；3，著名的Yamabe猜想指出M上存在与g逐点共形且具有常数量曲率的度量。Yamabe猜想已经被Yamabe[Y]，Trudinger[T]，Aubin[A]和Schoen[SI]的工作所证明。请看Lee and Parker[LP]的调查。关于这一问题的著作和相关著作，另见巴赫里和布雷齐斯[BB]、巴赫里[B]和舍恩[S2-3]。带边界的紧致黎曼流形的Yamabe问题的类似问题已由Cherrier，EScott bar等人研究过。特别地，埃斯科瓦尔在[E2]中证明了一大类具有边界的紧致黎曼流形与边界上具有常数量曲率且平均曲率为零的流形共形等价。有关结果，另见[E3][E5]。从现在起，除非另有说明，否则本文中的(M，g)表示某种光滑紧的n维黎曼流形。我们用M表示M的内部，用Dm表示M的边界。用n-2dn-2La表示An-c(N)Ra，其中c(N)是--，Bq表示-i-Hq，y y*4(n-1)y Du 2y其中u是Dm上关于5的外单位法线，HG表示Dm相对于内法线的平均曲率(R中的球具有正的平均曲率)。
Let (M,g) be an n dimensional compact, smooth, Riemannian manifold without boundary. For n = 2, the Uniformization Theorem of Poincare says that there exist metrics on M which are pointwise conformal to g and have constant Gauss curvature. For n > 3, the well known Yamabe conjecture states that there exist metrics on M which are pointwise conformal to g and have constant scalar curvature. The Yamabe conjecture has been proved through the work of Yamabe [Y], Trudinger [T], Aubin [A], and Schoen [SI]. See Lee and Parker [LP] for a survey. See also Bahri and Brezis [BB], Bahri [B], and Schoen [S2-3] for works on the problem and related ones. Analogues of the Yamabe problem for compact Riemannian manifolds with boundary have been studied by Cherrier, Escobar, and others. In particular, Escobar proved in [E2] that a large class of compact Riemannian manifolds with boundary are conformally equivalent to one with constant scalar curvature and zero mean curvature on the boundary. See also [E3][E5] for related results. From now on in the paper, (M, g) denotes some smooth compact n dimensional Riemannian manifold with boundary, unless we specify otherwise. We use M to denote the interior of M, and dM the boundary of M. We use n — 2 d n — 2 La to denote An—c(n)Ra, where c(n) is — —, BQ to denote ——I—-—hQ, y y * 4(n -1) y du 2 y where u is the outward unit normal on dM with respect to 5, and hg to denote the mean curvature of dM with respect to the inner normal (balls in R have positive mean curvatures).