We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations (“linearised gravity”) around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior, and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components). These latter statements are in fact deduced in the course of the proof by exploiting associated quantities shown to satisfy the Regge–Wheeler equation, for which appropriate decay can be obtained easily by adapting previous work on the linear scalar wave equation. The bounds on the rate of decay to linearised Kerr are inverse polynomial, suggesting that dispersion is sufficient to control the non-linearities of the Einstein equations in a potential future proof of non-linear stability. This paper is self-contained and includes a physical-space derivation of the equations of linearised gravity around Schwarzschild from the full non-linear Einstein vacuum equations expressed in a double null gauge.

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Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. This paper considers both minimax and adaptive estimation of the principal subspace in the high dimensional setting. Under mild technical conditions, we first establish optimal rates of convergence for estimating the principal subspace which sharp with respect to all the parameters, thus providing a complete character ization of the difficulty of the estimation problem in term of the convergence rate. The lower bound is obtained by calculating the local metric entropy an application of Fano's lemma. The rate optimal estimator is constructed using aggregation, which, however, might not be computationally feasible. We then introduce an adaptive procedure for estimating the principal sub space which is fully data driven and can be computed efficiently. It is shown that the estimator attains the optimal rates of convergence simultaneously over a large collection of the parameter spaces. A key idea in our construc tion is a reduction scheme which reduces the sparse PCA problem to a high dimensional multivariate regression problem. This method is potentially useful for other related

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