Metrical Theorems Research Articles

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Kollár's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.

A Volume Comparison Theorem for Asymptotically Hyperbolic Manifolds

We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.

A positive mass theorem for Lipschitz metrics with small singular sets

We prove that the positive mass theorem applies to Lipschitz metrics as long as the singular set is low-dimensional, with no other conditions on the singular set. More precisely, let $g$ be an asymptotically flat Lipschitz metric on a smooth manifold $M^n$, such that $n<8$ or $M$ is spin. As long as $g$ has bounded $C^2$ norm and nonnegative scalar curvature on the complement of some singular set $S$ of Minkowski dimension less than $n/2$, the mass of $g$ must be nonnegative. We conjecture that the dimension of $S$ need only be less than $n-1$ for the result to hold. These results complement and contrast with earlier results of H. Bray, P. Miao, and Y. Shi and L.-F. Tam, where $S$ is a hypersurface.

Obstruction-flat asymptotically locally Euclidean metrics

We show that any asymptotically locally Euclidean (ALE) metric which is obstruction-flat or extended obstruction-flat must be ALE of a certain optimal order. Moreover, our proof applies to very general elliptic systems and in any dimension n ≥ 3. The proof is based on the technique of Cheeger–Tian for Ricci-flat metrics. We also apply this method to obtain a singularity removal theorem for (extended) obstruction-flat metrics with isolated C 0-orbifold singular points.

The semi-classical ergodic Theorem for discontinuous metrics

The semi-classical ergodic Theorem for discontinuous metrics

On homothetic balanced metrics

In this article, we study the set of balanced metrics given in Donaldson’s terminology (J. Diff. Geometry 59:479–522, 2001) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when M admits a non-positive Kahler–Einstein metric, in the case of non-homogenous toric Kahler-Einstein manifolds of dimension ≤ 4 and in the case of the constant scalar curvature metrics found in Arezzo and Pacard (Acta. Math. 196(2):179–228, 2006; Ann. Math. 170(2):685–738, 2009).

Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics

Two metrics g and ḡ are geodesically equivalent if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics, such as the classification of local normal forms and the Lie problem (the description of projective vector fields), to the case when the (1, 1)-tensor G := g gkj has one real eigenvalue, or two complex conjugate eigenvalues, and give first applications. As a part of the proof of the main result, we generalise the Topalov-Sinjukov (hierarchy) Theorem for pseudo-Riemannian metrics. © 2011 American Mathematical Society.

A compactness theorem for scalar-flat metrics on manifolds with boundary

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this set is compact for dimensions greater than or equal to 7 under the generic condition that the trace-free 2nd fundamental form of the boundary is nonzero everywhere.

On the structure of conformally compact Einstein metrics

Let M be an (n + 1)-dimensional manifold with non-empty boundary, satisfying π 1(M, ∂ M) = 0. The main result of this paper is that the space of conformally compact Einstein metrics on M is a smooth, infinite dimensional Banach manifold, provided it is non-empty. We also prove full boundary regularity for such metrics in dimension 4 and a local existence and uniqueness theorem for such metrics with prescribed metric and stress–energy tensor at conformal infinity, again in dimension 4. This result also holds for Lorentzian–Einstein metrics with a positive cosmological constant.

Metrical theorems for inhomogeneous Diophantine approximation in positive characteristic

We consider inhomogeneous Diophantine approximation for formal Laurent series over a finite base field. We establish an analogue of a strong law of large numbers due to W. M. Schmidt with a better error term than in the real case. A special case of our result improves upon a recent result by H. Nakada and R. Natsui and completes a result of M. M. Dodson, S. Kristensen, and J. Levesley. Moreover, we prove various results for inhomogeneous Diophantine approximation with restricted denominators.

A DECOMPOSITION OF A QUASI-OSCILLATOR

A decomposition of a simplest mechanical system – a q-oscillator, is considered. A q-oscillator (quasi-oscillator) is defined as a system consisting of a particle moving on a linear segment with a constant speed and reflecting from the segment's end-points. A theorem, stating that a discrete-time version of q-oscillator can be decomposed into a countable set of discrete-time rotators (a rotator consists of a particle rotating on a circle with a constant angular rate) as well as a metrical theorem, concerning the rate of such decomposition, are proved. Some physics-related examples are discussed. A phase-shifting perturbed rotator and a number-theoretical matrix system modelling the quantum oscillator, are presented.

Gallot–Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist

We generalize for complete pseudo-Riemannian metrics a classical result of Gallot (1979) [3] and Tanno (1978) [13]: we show that if a closed complete manifold admits a nonconstant function λ satisfying ∇ k ∇ j ∇ i λ + 2 ∇ k λ ⋅ g i j + ∇ i λ ⋅ g j k + ∇ j λ ⋅ g i k = 0 , then the metric is the Riemannian metric of constant curvature +1. We use this result to give a simple proof of a recent result of Alekseevsky, Cortes, Galaev and Leistner (2009) [1]. Certain generalizations for higher Gallot equations are given.

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 L 2 -cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod x m , 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.

Positional Cloning by Linkage Disequilibrium

Recently, metric linkage disequilibrium (LD) maps that assign an LD unit (LDU) location for each marker have been developed (Maniatis et al. 2002). Here we present a multiple pairwise method for positional cloning by LD within a composite likelihood framework and investigate the operating characteristics of maps in physical units (kb) and LDU for two bodies of data (Daly et al. 2001; Jeffreys et al. 2001) on which current ideas of blocks are based. False-negative indications of a disease locus (type II error) were examined by selecting one single-nucleotide polymorphism (SNP) at a time as causal and taking its allelic count (0, 1, or 2, for the three genotypes) as a pseudophenotype, Y. By use of regression and correlation, association between every pseudophenotype and the allelic count of each SNP locus (X) was based on an adaptation of the Malecot model, which includes a parameter for location of the putative gene. By expressing locations in kb or LDU, greater power for localization was observed when the LDU map was fitted. The efficiency of the kb map, relative to the LDU map, to describe LD varied from a maximum of 0.87 to a minimum of 0.36, with a mean of 0.62. False-positive indications of a disease locus (type I error) were examined by simulating an unlinked causal SNP and the allele count was used as a pseudophenotype. The type I error was in good agreement with Wald's likelihood theorem for both metrics and all models that were tested. Unlike tests that select only the most significant marker, haplotype, or haploset, these methods are robust to large numbers of markers in a candidate region. Contrary to predictions from tagging SNPs that retain haplotype diversity, the sample with smaller size but greater SNP density gave less error. The locations of causal SNPs were estimated with the same precision in blocks and steps, suggesting that block definition may be less useful than anticipated for mapping a causal SNP. These results provide a guide to efficient positional cloning by SNPs and a benchmark against which the power of positional cloning by haplotype-based alternatives may be measured.

The comparison theorem for Bergman and Kobayashi metrics on Cartan-Hartogs domain of the second type*

Abstract In this paper, the holomorphic sectional curvature under invariant metric on a Cartan-Hartogs domain of the second type YII(N, p,K) is presented and an invariant Kalher metric which is complete and not less than the Bergman metric is constructed, such that its holomorphic sectional curvature is bounded above by a negative constant. Hencea comparison theorem for the Bergman and Kobayashi metrics on YII(N, p, K) is obtained.

Local Classification of Conformally-Einstein Kähler Metrics in Higher Dimensions

The requirement that a (non-Einstein) Kahler metric in any given complex dimension $m>2$ be almost-everywhere conformally Einstein turns out to be much more restrictive, even locally, than in the case of complex surfaces. The local biholomorphic-isometry types of such metrics depend, for each $m>2$, on three real parameters along with an arbitrary Kahler-Einstein metric $h$ in complex dimension $m-1$. We provide an explicit description of all these local-isometry types, for any given $h$. That result is derived from a more general local classification theorem for metrics admitting functions we call {\it special Kahler-Ricci potentials}.

Metrical Theorems on Restricted Diophantine Approximations to Points on a Curve

We investigate rational approximations (r/p,q/p) or (r/p,q/r) to points on the curve (α,ατ) for almost all α>0, where p,q,r are all primes. We immediately obtain corollaries on making p,[αp], [ατp] simultaneously prime.

The Comparison Theorem for Bergman and Kobayashi Metrics on Cartan-Hartogs Domain of the Third Type

In this paper, we give the holomorphic sectional curvature under invariant Kähler metric on a Cartan-Hartogs domain of the third type Y III (N,q,K) and construct an invariant Kähler metric, which is complete and not less than the Bergman metric, such that its holomorphic sectional curvature is bounded above by a negative constant. Hence we obtain a comparison theorem for the Bergman and Kobayashi metrics on Y III (N,q,K).

Extraction of optimally unbiased bits from a biased source

We explore the problem of transforming n independent and identically biased {-1,1}-valued random variables X/sub 1/,...,X/sub n/ into a single {-1,1} random variable f(X/sub 1/,...,X/sub n/), so that this result is as unbiased as possible. In general, no function f produces a completely unbiased result. We perform the first study of the relationship between the bias b of these X/sub i/ and the rate at which f(X/sub 1/,...,X/sub n/) can converge to an unbiased {-1,1} random variable (as n/spl rarr//spl infin/). A {-1,1} random variable has bias b if E(X/sub i/)=b. Fixing a bias b, we explore the rate at which the output bias |E(f(X/sub 1/,...,X/sub n/))| can tend to zero for a function f:{-1,1}*/spl rarr/{-1,1}. This is accomplished by classifying the behavior of the natural normalized quantity /spl Xi/(b)/spl Delta/inf/sub f/[lim/sub n/spl rarr//spl infin//n/spl radic/(|E(f(X/sub 1/,...,X/sub n/))|] this infimum taken over all such f. We show that for rational b, /spl Xi/(b)=(1/s), where (1+b/2)=(r/s) (r and s relatively prime). Developing the theory of uniform distribution of sequences to suit our problem, we then explore the case where b is irrational. We prove a new metrical theorem concerning multidimensional Diophantine approximation type from which we show that for (Lebesgue) almost all biases b, /spl Xi/(b)=0. Finally, we show that algebraic biases exhibit curious boundary behavior, falling into two classes. Class 1. Those algebraics b for which /spl Xi/(b)>0 and, furthermore, c/sub 1//spl les//spl Xi/(b)/spl les/c/sub 2/ where c/sub 1/ and c/sub 2/ are positive constants depending only on b's algebraic characteristics. Class 2. Those algebraics b for which there exist n>0 and f: {-1,1}/sup n//spl rarr/{-1,1} so that E(f(X/sub 1/,...,X/sub n/))=0. Notice that this classification excludes the possibility that n/spl radic/(|E(f(X/sub 1/,...,X/sub n/))| limits to zero (for algebraics). For rational and algebraic biases, we also study the computational problem by restricting f to be a polynomial time computable function. Finally, we discuss natural extensions where output distributions other than the uniform distribution on {-1,1} are sought.

Metrical Theorems on Prime Values of the Integer Parts of Real Sequences

Let an be an increasing sequence of positive reals with an → ∞ as n → ∞. Necessary and sufficient conditions are obtained for each of the sequences [ α a n ] , [ α a n ] , [ a n α ] to take on infinitely many prime values for almost all α > rβ. For example, the sequence αan is infinitely often prime for almost all α > 0 if and only if there is a subsequence of the an, say bn, with bn + 1 > bn + 1 and with the series ∑ 1 / b n divergent. Asymptotic formulae are obtained when the sequences considered are lacunary. An earlier result of the author reduces the problem to estimating the measure of overlaps of certain sets, and sieve methods are used to obtain the correct order upper bounds. 1991 Mathematics Subject Classification: primary 11N05; secondary 11K99, 11N36.