Annals of mathematics

Đề tài " The space of embedded minimal surfaces of fixed genus in a 3manifold III; Planar domains "
Annals of Mathematics This paper is the third in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed 3manifold. In [CM3]–[CM5] we describe the case where the surfaces are topologically disks on any ﬁxed small scale. Although the focus of this paper, general planar domains, is more in line with [CM6], we will prove a result here (namely, Corollary III.
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Annals of Mathematics In our ﬁrst article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number ﬁelds, while the spaces underlying those composition laws were closely related to certain exceptional Lie groups.
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Annals of Mathematics By S. Artstein, V. Milman, and S. J. Szarek For two convex bodies K and T in Rn , the covering number of K by T , denoted N (K, T ), is deﬁned as the minimal number of translates of T needed to cover K. Let us denote by K ◦ the polar body of K and by D the euclidean unit ball in Rn . We prove that the two functions of t, N (K, tD) and N (D, tK ◦ ), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K ⊂...
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Annals of Mathematics By Curtis T. McMullen* .Annals of Mathematics, 165 (2007), 397–456 Dynamics of SL2(R) over moduli space in genus two By Curtis T. McMullen* Abstract This paper classiﬁes orbit closures and invariant measures for the natural action of SL2 (R) on ΩM2 , the bundle of holomorphic 1forms over the moduli space of Riemann surfaces of genus two. Contents 1. Introduction 2. Dynamics and Lie groups 3. Riemann surfaces and holomorphic 1forms 4. Abelian varieties with real multiplication 5. Recognizing eigenforms 6. Algebraic sums of 1forms 7.
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We prove that if f (x) = n−1 ak xk is a polynomial with no cyclotomic k=0 factors whose coeﬃcients satisfy ak ≡ 1 mod 2 for 0 ≤ k 1 + log 3 , 2n resolving a conjecture of Schinzel and Zassenhaus [21] for this class of polynomials. More generally, we solve the problems of Lehmer and Schinzel and Zassenhaus for the class of polynomials
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Annals of Mathematics We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler’s equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a “physical condition”, related to the fact that the pressure of a ﬂuid has to be positive. ...
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Annals of Mathematics In this paper we will solve one of the central problems in dynamical systems: Theorem 1 (Density of hyperbolicity for real polynomials). Any real polynomial can be approximated by hyperbolic real polynomials of the same degree. Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of inﬁnity.
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We verify an old conjecture of G. P´lya and G. Szeg˝ saying that the o o regular ngon minimizes the logarithmic capacity among all ngons with a ﬁxed area. 1. Introduction The logarithmic capacity cap E of a compact set E in R2 , which we identify with the complex plane C, is deﬁned by (1.1) − log cap E = lim (g(z, ∞) − log z), z→∞ where g(z, ∞) denotes the Green function of a connected component Ω(E) ∞ of C \ E having singularity at z = ∞; see [4, Ch. 7], [7, §11.1]. By an ngon with...
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We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c
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We show that unital simple C ∗ algebras with tracial topological rank zero which are locally approximated by subhomogeneous C ∗ algebras can be classiﬁed by their ordered Ktheory. We apply this classiﬁcation result to show that certain simple crossed products are isomorphic if they have the same ordered Ktheory. In particular, irrational higher dimensional noncommutative tori of the form C(Tk ) ×θ Z are in fact inductive limits of circle algebras. Introduction In recent years there has been rapid progress in classiﬁcation of nuclear simple C ...
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This paper is the ﬁrst in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed Riemannian 3manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R3 (with the ﬂat metric). This study is undertaken here and completed in [CM6]. These local results are then applied in [CM7] where we describe the general structure of ﬁxed genus surfaces in 3manifolds. There are two local models for...
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The uniform spanning forest (USF) in Zd is the weak limit of random, uniformly chosen, spanning trees in [−n, n]d . Pemantle [11] proved that the USF consists a.s. of a single tree if and only if d ≤ 4. We prove that any two components of the USF in Zd are adjacent a.s. if 5 ≤ d ≤ 8, but not if d ≥ 9. More generally, let N (x, y) be the minimum number of edges outside the USF in a path joining x and y in Zd . Then max N (x, y) : x, y ∈ Zd = (d − 1)/4 a.s.
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This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed 3manifold. The key is to understand the structure of an embedded minimal disk in a ball in R3 . This was undertaken in [CM3], [CM4] and the global version of it will be completed here; see the discussion around Figure 12 for the local case and [CM15] for some more details. Our main results are Theorem 0.1 (the lamination theorem) and Theorem 0.2 (the onesided curvature estimate). ...
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Đề tài " Extension properties of meromorphic mappings with values in nonK¨ahler complex manifolds "
Statement of the main result. Denote by Δ(r) the disk of radius r in C, Δ := Δ(1), and for 0
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Reduction of the singularities of codimension one singular foliations in dimension three By Felipe Cano Contents 0. Introduction 1. Blowingup singular foliations 1.1. Adapted singular foliations 1.2. Permissible centers 1.3. Vertical invariants 1.4. First properties of presimple singularities 2. Global strategy 2.1. Reduction to presimple singularities. Statement 2.2. Good points. Bad points. Equireduction 2.3. Finiteness of bad points 2.4. The inﬂuency locus 2.5. The local control theorem 2.6. Destroying cycles 2.7. Global criteria of blowingup 3. Local control 3.1.
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We consider a specialization of an untwisted quantum aﬃne algebra of type ADE at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its ﬁnite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of “computable” polynomials, similar to KazhdanLusztig polynomials. At the same time we “compute” qcharacters for all simple modules. The result is based on “computations” of Betti numbers of graded/cyclic quiver varieties.
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We introduce and study “isomonodromy” transformations of the matrix linear diﬀerence equation Y (z + 1) = A(z)Y (z) with polynomial A(z). Our main result is construction of an isomonodromy action of Zm(n+1)−1 on the space of coeﬃcients A(z) (here m is the size of matrices and n is the degree of A(z)). The (birational) action of certain rank n subgroups can be described by diﬀerence analogs of the classical Schlesinger equations, and we prove that for generic initial conditions these diﬀerence Schlesinger equations have a unique solution. ...
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In this paper we introduce a method for partial description of the Poisson boundary for a certain class of groups acting on a segment. As an application we ﬁnd among the groups of subexponential growth those that admit nonconstant bounded harmonic functions with respect to some symmetric (inﬁnitely supported) measure µ of ﬁnite entropy H(µ). This implies that the entropy h(µ) of the corresponding random walk is (ﬁnite and) positive. As another application we exhibit certain discontinuity for the recurrence property of random walks. ...
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In this paper we prove C k structural stability conjecture for unimodal maps. In other words, we shall prove that Axiom A maps are dense in the space of C k unimodal maps in the C k topology. Here k can be 1, 2, . . . , ∞, ω. 1. Introduction 1.1. The structural stability conjecture. The structural stability conjecture was and remains one of the most interesting and important open problems in the theory of dynamical systems. This conjecture states that a dynamical system is structurally stable if and only if it satisﬁes Axiom A and the...
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Surfaces of sections are a classical tool in the study of 3dimensional dynamical systems. Their use goes back to the work of Poincar´ and Birkhoﬀ. e In the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of ﬁnitely many distinguished periodic solutions. Such a system is established for nondegenerate Reeb ﬂows on the tight 3sphere by means of pseudoholomorphic curves.
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