Bach-flat asymptotically locally Euclidean metrics

We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kähler metrics with zero scalar curvature, and metrics with harmonic curvature. Similar results were obtained for Einstein metrics in [And89], [BKN89], [Tia90], but our analysis differs from the Einstein case in that (1) we consider more generally a fourth order system in the metric, and (2) we do not assume any pointwise Ricci curvature bound.     

Concerning the Strauss conjecture on asymptotically Euclidean manifolds

In this paper we verify the Strauss conjecture for semilinear wave equations on asymptotically Euclidean manifolds when n=3,4. We also give an almost sharp lifespan for the subcritical case 2?p<pc when n=3. The main ingredients include a Keel-Smith-Sogge type estimate with 0<μ<1/2 and weighted Strichartz estimates of order two.     

Concerning the wave equation on asymptotically Euclidean manifolds

We obtain KSS, Strichartz and certain weighted Strichartz estimates for the wave equation on (ℝ ^d , g), d ≥ 3, when the metric g is non-trapping and approaches the Euclidean metric like 〈x〉^−ρ with ρ > 0. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for ρ > 1 and d = 3. Also, we establish the Strauss conjecture when the metric is radial with ρ > 1 for d = 3.     

Global existence of null-form wave equations on small asymptotically Euclidean manifolds

We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on (R³,g)$({\mathbb{R}}^3,\mathfrak{g})$, where the metric g$\mathfrak{g}$ is a small perturbation of the flat metric and approaches the Euclidean metric like (1+|x|²)^−ρ/2${(1+{|x|}^2)}^{-\rho /2}$ with ρ>1$\rho >1$. Global and almost global existence for systems without the null condition are also discussed for certain small time-dependent perturbations of the flat metric in Appendix?A.     

Isometric imbedding of a complete two-dimensional locally Euclidean manifold in a Euclidean space

In this article we consider the problem of isometric imbedding of a complete two-dimensional locally Euclidean manifold in a Euclidean space. For each of the possible topological types the corresponding minimal dimension of the extending Euclidean space is indicated. See [3].Translated from Matematicheskie Zametki, Vol. 13, No. 3, pp. 427–429, March, 1973.     

A locally asymptotically powerful test for nonlinear autoregressive models

We propose a locally asymptotically powerful test to simultaneously examine hypotheses relative to the parametric form of the conditional mean and the conditional variance functions in a class of nonlinear semi-parametric time series models without a specified error law. On the basis of a modified version of the Le Cam method of Hwang and Basawa (2001), we establish the local asymptotic normality relative to the model. The main result shows that the test statistic built by substituting consistent estimated residuals and parameters for the theoretical ones is asymptotically normal. Its asymptotic power is computed and the result is illustrated by some simulations. To cite this article: F. Chebana, N. Laïb, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Characterization of locally connected continua in Euclidean spaces

We prove that, for any locally connected bounded continuum in the Euclidean space E ⁿ ,n2, there exists a sequence of imbeddings of the segment [0, 1] into E ⁿ uniformly convergent to a continuous mapping of [0, 1] onto this continuum.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1080–1088, August, 1995.     

On estimation and adaptive estimation for locally asymptotically normal families

Summary Locally asymptotically minimax (LAM) estimates are constructed for locally asymptotically normal (LAN) families under very mild additional assumptions. Adaptive estimation is also considered and a sufficient condition is given for an estimate to be locally asymptotically minimax adaptive. Incidently, it is shown that a well known lower bound due to Hájek (1972) for the local asymptotic minimax risk is not sharp.Research partially supported by NSF grants no. MCS 78-02846 and MCS 77-03493-01     

On gaps between quadratic non-residues in the Euclidean and Hamming metrics

The authors have recently introduced and studied a modification of the classical number theoretic question about the largest gap between consecutive quadratic non-residues and primitive roots modulo a prime p$p$, where the distances are measured in the Hamming metric on binary representations of integers. Here we continue to study the distribution of such gaps. In particular we prove the upper bound

_?p≤(0.117198…+o(1))logp/log2${?}_p\le (0.117198\dots +o\left(1\right))logp/log2$

for the smallest Hamming weight _?p${?}_p$ among prime quadratic non-residues modulo a sufficiently large prime p$p$. The Burgess bound on the least quadratic non-residue only gives _?p≤(0.15163…+o(1))logp/log2${?}_p\le (0.15163\dots +o\left(1\right))logp/log2$.     

Complex spaces with locally product metrics: special spaces

Summary Certain special spaces are considered within the framework of the general theory of complex Riemannian spaces with locally product metrics. Spaces of constant curvature are discussed and a counterpart of Schur's Theorem is established. It is also shown that there exist what may be regarded as complex Schwarzschild metrics. No physical interpretation is suggested. Entrata in Redazione il 30 ottobre 1968. On leave of absence from the Department of Mathematics, The University, Bristol.     

On locally conformally flat critical metrics for quadratic functionals

We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudo-convex pseudo-Hermitian structure \(\theta \) on the CR sphere \(\mathbb {S}^{2n+1}\subset \mathbb {C}^{n+1}\), achieves its maximum when \(\theta \) is the standard contact form.     

Complex spaces with locally product metrics: general theory

Summary Locally product complex spaces are introduced with particular reference to the Calculus of Variations on complex manifolds. The goedesics of such spaces possess unusual features associated with the fact that many of the connection coefficients are tensorial in character. A restricted partial covariant derivative is introduced together with a curvature tensor. The latter is found to be invariant under a large class of gauge-like transformations. This invariance property leads naturally to the introduction of almost totally decomposable complex spaces. Necessary and sufficient conditions for a locally product complex Riemannian space to be almost totally decomposable are discussed. The counterpart of the Einstein tensor is also exhibited. On leave of absence from the Department of Mathematics, The University. Bristol. Entrata in Redazione il 30 ottobre 1968.     

TT-eigentensors for the Lichnerowicz Laplacian on some asymptotically hyperbolic manifolds with warped products metrics

Let (M =]0, ∞[×N, g) be an asymptotically hyperbolic manifold of dimension n + 1 ≥ 3, equipped with a warped product metric. We show that there exist no TT L ²-eigentensors with eigenvalue in the essential spectrum of the Lichnerowicz Laplacian Δ _L . If (M, g) is the real hyperbolic space, there is no symmetric L ²-eigentensors of Δ _L .     

A renormalized index theorem for some complete asymptotically regular metrics: The Gauss-Bonnet theorem

We study the Gauss-Bonnet theorem as a renormalized index theorem for edge metrics. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence and the asymptotically cylindrical metrics of the Atiyah-Patodi-Singer index theorem. We use renormalization to make sense of the curvature integral and the dimensions of the L²-cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod xm, we show that the finite time supertrace of the heat kernel on conformally compact manifolds renormalizes independently of the choice of special boundary defining function.