Asymptotics of ACH-Einstein Metrics

We study the boundary asymptotics of ACH metrics which are formally Einstein. In terms of the partially integrable almost CR structure induced on the boundary at infinity, existence and uniqueness of such formal asymptotic expansions are studied. It is shown that there always exist formal solutions to the Einstein equation if we allow logarithmic terms, and that a local CR-invariant tensor arises as the obstruction to the existence of a log-free solution. Some properties of this new CR invariant, the CR obstruction tensor, are discussed.     

Conformal Compactification of Asymptotically Locally Hyperbolic Metrics II: Weakly ALH Metrics

In this paper we pursue the work initiated in [6 Bahuaud , E. ( 2009 ). Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics . Pacific J. Math. 239 : 231 – 249 .[Crossref], [Web of Science ®] , [Google Scholar], 7 Bahuaud , E. , Gicquaud , R. ( 2011 ). Conformal compactification of asymptotically locally hyperbolic metrics . J. Geom. Anal. 21 : 1085 – 1118 .[Crossref], [Web of Science ®] , [Google Scholar]]: study the extent to which conformally compact asymptotically hyperbolic metrics can be characterized intrinsically. We show how the decay rate of the sectional curvature to ?1 controls the Hölder regularity of the compactified metric. To this end, we construct harmonic coordinates that satisfy some Neumann-type condition at infinity. Combined with a new integration argument, this permits us to recover to a large extent our previous result without any decay assumption on the covariant derivatives of the Riemann tensor.     

Invariant Metrics on G-Spaces

Let X be a G-space such that the orbit space X/G is metrizable. Suppose a family of slices is given at each point of X. We study a construction which associates, under some conditions on the family of slices, with any metric on X/G an invariant metric on X. We show also that a family of slices with the required properties exists for any action of a countable group on a locally compact and locally connected metric space.     

Metrics and undirected cuts

Suppose thatG is an undirected graph whose edges have nonnegative integer-valued lengthsl(e), and that {s ₁,t ₁},?, {s _m ,t _m } are pairs of its vertices. Can one assign nonnegative weights to the cuts ofG such that, for each edgee, the total weight of cuts containinge does not exceedl(e) and, for eachi, the total weight of cuts ‘separating’s _i andt _i is equal to the distance (with respect tol) betweens _i andt _i ? Using linear programming duality, it follows from Papernov's multicommodity flow theorem that the answer is affirmative if the graph induced by the pairs {s ₁,t ₁},?, {s _m ,t _m } is one of the following: (i) the complete graph with four vertices, (ii) the circuit with five vertices, (iii) a union of two stars. We prove that if, in addition, each circuit inG has an even length (with respect tol) then there exists a suitable weighting of the cuts with the weights integer-valued; moreover, an algorithm of complexity O(n ³) (n is the number of vertices ofG) is developed for solving such a problem. Also a class of metrics decomposable into a nonnegative linear combination of cut-metrics is described, and it is shown that the separation problem for cut cones isNP-hard.     

Rigid Properties of Quasi-almost-Einstein Metrics

In this paper,quasi-almost-Einstein metrics on complete manifolds are studied.Two examples are given and several formulas are established.With the help of these formulas,the author proves rigid results on compact or noncompact manifolds,in which some basic tools,such as the weighted volume comparison theorem and the weak maximum principle at infinity,are used.A lower bound estimate for the scalar curvature is also obtained.     

Metrics with harmonic spinors

We show that every closed spin manifold of dimensionn 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheresS ⁿ ,n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sumM ₁#M ₂ with certain metrics is close to the union of the spectra ofM ₁ and ofM ₂.Partially supported by SFB 256 and by the GADGET program of the EU     

Metrics on complex manifolds

In this note we discuss various canonical metrics on complex manifolds. A generalization of the Bergman metric is proposed and the relations of metrics on moduli spaces are commented. At last, we review some existence theorems of solutions to the Strominger system.     

Reasoning-and-Proving in School Mathematics Textbooks

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.     

Deformation of Asymptotically Isospectral Metrics

In this note, we introduce the notion of asymptotically isospectral deformation of order . Then, we study the asymptotically isospectral problems with model spaces. Some techniques in the conformal geometry and Kähler geometry are used to investigate the asymptotically isospectral problems with suitable medol spaces.     

Some Properties of Sasakian Metrics in Cotangent Bundles

The main aim of this paper is to investigate curvature properties of the Sasakian metrics in the cotangent bundle.     

On Some Almost-Periodicity Problems in Various Metrics

The Bohr-type and the Bochner-type definitions for almost periodic functions are examined in various metrics (Stepanov, Weyl and Besicovitch). The correct definitions of Besicovitch-like multifunctions are given. Weak almost-periodic solutions are proved for differential equations and inclusions. This problem is also discussed as a fixed-point problem in function spaces.     

Characterization of Weakly Berwald Fourth-Root Metrics

Ukrainian Mathematical Journal - In recent studies, it has been shown that the theory of fourth-root metrics plays a very important role in physics, theory of space-time structures, gravitation,...     

共形双曲度量的孤立奇点

Nitsche证明了共形双曲度量的孤立奇点要么是锥奇点,要么是尖奇点,二者必其一(Nitsche J.über die isolierten singularit?ten der L?sungen von△u=e~u [J]. Math Z, 1957, 68(3):316-324.).本文利用展开映射证明了在孤立奇点附近存在复坐标z,使得度量要么为(4α~2|z|~(2α-2))/((1-|z|~(2α))~2)|dz|~2,其中α 0,要么为|z|~(-2)(ln|z|)~(-2)|dz|~2.     

On Order-Preserving Properties of Probability Metrics

Stochastic orders and probability metrics are two areas that have received considerable attention in recent literature. A question of interest on its own and for certain applications is the preservation of metrics with respect to orders. This property does not seem to have been examined explicitly so far. Our purpose in this paper is to provide some useful remarks and results related to several orders and metrics which are standard in probability.     

Tropical Mixtures of Star Tree Metrics

We study tree metrics that can be realized as a mixture of two star tree metrics. We prove that the only trees admitting such a decomposition are the ones coming from a tree with at most one internal edge, and whose weights satisfy certain linear inequalities. We also characterize the fibers of the corresponding mixture map. In addition, we discuss the general framework of tropical secant varieties and we interpret our results within this setting. Finally, we show that the set of tree metric ranks of metrics on fixed taxa is unbounded.     

Homogeneous Metrics with Nonnegative Curvature

Given compact Lie groups H⊂G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics.