论文标题
ALE指标的lojasiewicz不平等
A Łojasiewicz inequality for ALE metrics
论文作者
论文摘要
我们引入了一个新的功能，灵感来自Perelman的$λ$功能功能，该功能适用于渐近的本地欧几里得（ALE）设置，并表示$λ_ {\ permatatorName {ale}} $。它的表达式包括一个边界项，该术语被证明是ADM质量。我们证明$λ_{\ permatatorName {ale}} $是在Ricci-flat Ale ALE指标的方便社区上定义和分析的，我们表明它沿RICCI流是单调的。例如，这使我们确定具有非负标态曲率的可集成和稳定的Ricci-flat ALE指标的小扰动具有非负质量。然后，我们引入了一条通用的证据方案，以在非compact歧管上的lojasiewicz-simon不等式，并证明它适用于RICCI-FLAT围绕RICCI-FLAT指标的$λ_ {\ propatatorName {ale}} $。此外，我们为具有可集成的RICCI-FLAT变形的指标获得了最佳加权Lojasiewicz指数。
We introduce a new functional inspired by Perelman's $λ$-functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted $λ_{\operatorname{ALE}}$. Its expression includes a boundary term which turns out to be the ADM-mass. We prove that $λ_{\operatorname{ALE}}$ is defined and analytic on convenient neighborhoods of Ricci-flat ALE metrics and we show that it is monotonic along the Ricci flow. This for example lets us establish that small perturbations of integrable and stable Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a general scheme of proof for a Lojasiewicz-Simon inequality on non-compact manifolds and prove that it applies to $λ_{\operatorname{ALE}}$ around Ricci-flat metrics. We moreover obtain an optimal weighted Lojasiewicz exponent for metrics with integrable Ricci-flat deformations.
