论文标题
cartan-hadamard歧管上Sobolev不等式的径向最佳函数的不存在
Nonexistence of radial optimal functions for the Sobolev inequality on Cartan-Hadamard manifolds
论文作者
论文摘要
众所周知，Euclidean Sobolev的不平等在任何cartan-Hadamard歧管上$ n \ ge 3 $，即任何完整的，简单地连接的Riemannian歧管，具有非阳性截面曲率。作为cartan-hadamard猜想的副产品，数学文献中的一个长期问题仅在Ghomi和Ghomi和Spruck的一份突破性论文中解决，我们现在可以断言，最佳常数也是欧几里得的，即它与Euclidean Space $ \ Mathbbbb {r的euclidean Space $ \ Mathbb {r}^n $相吻合。有人可能会问是否在通用的cartan-hadamard歧管$ \ mathbb {m}^n $上是否存在所有最佳功能。我们在这里证明的是，没有利用cartan-hadamard猜想的有效性的临时参数，这至少对于与距固定极距离的距离径向对称的函数至少是错误的。更确切地说，我们表明，如果Sobolev不等式中的最佳量是通过某种径向功能实现的，则$ \ Mathbb {M}^n $必须等于$ \ Mathbb {r}^n $。
It is well known that the Euclidean Sobolev inequality holds on any Cartan-Hadamard manifold of dimension $ n\ge 3 $, i.e. any complete, simply connected Riemannian manifold with nonpositive sectional curvature. As a byproduct of the Cartan-Hadamard conjecture, a longstanding problem in the mathematical literature settled only very recently in a breakthrough paper by Ghomi and Spruck, we can now assert that the optimal constant is also Euclidean, namely it coincides with the one achieved in the Euclidean space $ \mathbb{R}^n $ by the Aubin-Talenti functions. One may ask whether there exist at all optimal functions on a generic Cartan-Hadamard manifold $ \mathbb{M}^n $. What we prove here, with ad hoc arguments that do not take advantage of the validity of the Cartan-Hadamard conjecture, is that this is false at least for functions that are radially symmetric with respect to the geodesic distance from a fixed pole. More precisely, we show that if the optimum in the Sobolev inequality is achieved by some radial function, then $\mathbb{M}^n $ must be isometric to $ \mathbb{R}^n $.
