Abstract In this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W-functional, both on a complete non-compact Riemannian manifold ( M , g ) {(M,g)} . One consequence of our estimation is that, on an ALE (or asymptotic flat) manifold ( M , g ) {(M,g)} , if the scalar curvature s of ( M , g ) {(M,g)} is non-negative and has quadratical decay at infinity, then M is scalar flat, i.e., s = 0 {s=0} in M. We also introduce a new constant d ⁢ ( M , g ) {d(M,g)} . For the existence of the ground states, we use Lions’ concentration-compactness method.

中文翻译：

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关于 𝑊 泛函的 Lambda 常数和基态的新结果

摘要 在本文中，我们从变分公式研究了 lambda 常数的性质和 Perelman 著名的 W-泛函的基态的存在性。我们有两种结果。一个是关于 G. Perelman 的 lambda 常数的估计，另一个是关于他的 W-泛函的基态的存在，两者都在一个完整的非紧黎曼流形 ( M , g ) {(M, g ) } . 我们估计的一个结果是，在 ALE（或渐近平面）流形 ( M , g ) {(M,g)} 上，如果 ( M , g ) {(M,g)} 的标量曲率 s 是非-负并且在无穷远处有二次衰减，那么 M 是标量平坦的，即 M 中的 s = 0 {s=0}。我们还引入了一个新的常数 d ⁢ ( M , g ) {d(M,g)} 。对于基态的存在，我们使用Lions的浓度-紧凑性方法。