论文标题
Sobolev级渐近双曲线歧管和Yamabe问题
Sobolev-class asymptotically hyperbolic manifolds and the Yamabe problem
论文作者
论文摘要
我们考虑渐近的双曲线歧管，其指标具有Sobolev级规律性，并引入了几种技术工具，用于研究此类歧管上的PDE。我们的结果采用了两个新型的功能空间系列，适用于可能具有大量内部可不同性的指标，但Sobolev的规律性仅意味着无穷大的Hölder连续结构结构。我们为这些家庭中的指标建立了用于椭圆运算符的弗雷霍姆定理。 为了证明我们方法的实用性，我们解决了这一类别的Yamabe问题。作为一种特殊情况，我们表明，只要公制承认$ w^{1，p} $保形性压实，渐近yamabe问题就可以解决，其$ p $大于歧管的尺寸。
We consider asymptotically hyperbolic manifolds whose metrics have Sobolev-class regularity, and introduce several technical tools for studying PDEs on such manifolds. Our results employ two novel families of function spaces suitable for metrics potentially having a large amount of interior differentiability, but whose Sobolev regularity implies only a Hölder continuous conformal structure at infinity. We establish Fredholm theorems for elliptic operators arising from metrics in these families. To demonstrate the utility of our methods, we solve the Yamabe problem in this category. As a special case, we show that the asymptotically hyperbolic Yamabe problem is solvable so long as the metric admits a $W^{1,p}$ conformal compactification, with $p$ greater than the dimension of the manifold.
