In this article, we prove a local energy estimate for the linear wave equation on metrics with slow decay to a Kerr metric with small angular momentum. As an application, we study the quasilinear wave equation \(\Box _{g(u, t, x)} u = 0\) where the metric g(u, t, x) is close (and asymptotically equal) to a Kerr metric with small angular momentum g(0, t, x). Under suitable assumptions on the metric coefficients, and assuming that the initial data for u is small enough, we prove global existence and decay of the solution u.

中文翻译：

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渐近接近Kerr的度量上波动方程的局部能量估计

在本文中，我们证明了线性波形方程的局部能量估计，该方程基于缓慢衰减的度量到小角动量的Kerr度量。作为一种应用，我们研究准线性波动方程\（\ Box _ {g（u，t，x）} u = 0 \），其中度量g（u，  t，  x）接近于（且渐近地等于）a小角动量g（0，  t，  x）的Kerr度量。根据该度量系数适当的假设，并假设初始数据ü足够小，我们证明解的存在性和衰减ü。