论文标题
在相结合变形下紧凑的歧管上的规定标量曲率
Prescribed Scalar Curvature on Compact Manifolds Under Conformal Deformation
论文作者
论文摘要
对于封闭式的流形和带边界的紧凑型歧管，我们为规定的标量曲率问题提供了足够和“几乎”的必要条件共形拉普拉斯（必要时有适当的边界条件下）的第一个特征值是正的。当歧管不是$ \ mathbb {s}^{n} $的某些商时，我们表明，一方面，在歧管的某个开放子集中，任何一个正常的光滑函数都以任意种如态的歧视，并且对歧管的其余部分没有限制，并且对某些规定的标量曲率函数在同型下方的范围内；另一方面，任何平滑函数$ s $几乎都是yamabe class $ [g] $中Yamabe指标的规定标量曲率函数，因为在$ s $的适当扰动中，在任意小的开放子集中使用$ s $的适当扰动是yamabe Metric的规定标量曲率函数。当歧管是$ \ mathbb {s}^{n} $或$ \ mathbb {s}^n /γ$带有kleinian group $γ$的$ \ mathbb {s}^n /γ$时，我们表明的是，任何满足技术分析条件B的正面功能，称为条件B，都可以用作规定的curvature在这些歧管上的规定曲率功能。
We give sufficient and "almost" necessary conditions for the prescribed scalar curvature problems within the conformal class of a Riemannian metric $ g $ for both closed manifolds and compact manifolds with boundary, including the interesting cases $ \mathbb{S}^{n} $ or some quotient of $ \mathbb{S}^{n} $, in dimensions $ n \geqslant 3 $, provided that the first eigenvalues of conformal Laplacian (with appropriate boundary conditions if necessary) are positive. When the manifold is not some quotient of $ \mathbb{S}^{n} $, we show that, on one hand, any smooth function that is a positive constant within some open subset of the manifold with arbitrary positive measure, and has no restriction on the rest of the manifold, is a prescribed scalar curvature function of some metric under conformal change; on the other hand, any smooth function $ S $ is almost a prescribed scalar curvature function of Yamabe metric within the conformal class $ [g] $ in the sense that an appropriate perturbation of $ S $ that defers with $ S $ within an arbitrarily small open subset is a prescribed scalar curvature function of Yamabe metric. When the manifold is either $ \mathbb{S}^{n} $ or $ \mathbb{S}^n / Γ$ with Kleinian group $ Γ$ we show that any positive function that satisfies a technical analytical condition, called CONDITION B, can be realized as a prescribed scalar curvature functions on these manifolds.
