Mostrando recursos 1 - 20 de 364

  1. On the estimate of the first positive eigenvalue of a sublaplacian in a pseudohermitian manifold

    Fan, Yen-Wen; Kuo, Ting-Jung
    In this paper, we first obtain a CR version of Yau’s gradient estimate for eigenfunctions of a sublaplacian. Second, by using CR analogue of Li-Yau’s eigenvalue estimate, we are able to obtain a lower bound of the first positive eigenvalue in a pseudohermitian manifold of nonvanishing pseudohermitian torsion and nonpositive lower bound on pseudohermitian Ricci curvature.

  2. Tame Fréchet structures for affine Kac-Moody groups

    Freyn, Walter
    We construct holomorphic loop groups and their associated affine Kac-Moody groups and prove that they are tame Fréchet manifolds; furthermore we study the adjoint action of these groups. These results form the functional analytic core for a theory of affine Kac-Moody symmetric spaces, that will be developed in forthcoming papers. Our construction also solves the problem of complexification of completed Kac-Moody groups: we obtain a description of complex completed Kac-Moody groups and, using this description, deduce constructions of their non-compact real forms.

  3. Existence of approximate Hermitian-Einstein structures on semi-stable bundles

    Jacob, Adam
    The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle $E$ over a compact Kähler manifold $X$. It is shown that if $E$ is semi-stable, then Donaldson’s functional is bounded from below. This implies that $E$ admits an approximate Hermitian-Einstein structure, generalizing a classic result of Kobayashi for projective manifolds to the Kähler case. As an application some basic properties of semi-stable vector bundles over compact Kähler manifolds are established, such as the fact that semi-stability is preserved under certain exterior and symmetric products.

  4. Periodic constant mean curvature surfaces in $\mathbb{H}^2 \times \mathbb{R}$

    Mazet, Laurent; Rodríguez, M. Magdalena; Rosenberg, Harold

  5. Irreducible quasifinite modules over a class of Lie algebras of block type

    Chen, Hongjia; Guo, Xiangqian; Zhao, Kaiming
    For any nonzero complex number $q$, there is a Lie algebra of Block type, denoted by $\mathcal{B}(q)$. In this paper, a complete classification of irreducible quasifinite modules is given. More precisely, an irreducible quasifinite module is a highest weight or lowest weight module, or a module of intermediate series. As a consequence, a classification for uniformly bounded modules over another class of Lie algebras, the semi-direct product of the Virasoro algebra and a module of intermediate series, is also obtained. Our method is conceptional, instead of computational.

  6. Dirac Lie Groups

    Li-Bland, David; Meinrenken, Eckhard
    A classical theorem of Drinfel'd states that the category of simply connected Poisson Lie groups $H$ is isomorphic to the category of Manin triples $(\mathfrak{d, g, h})$, where $\mathfrak{h}$ is the Lie algebra of $H$. In this paper, we consider Dirac Lie groups, that is, Lie groups $H$ endowed with a multiplicative Courant algebroid $A$ and a Dirac structure $E \subseteq \mathbb{A}$ for which the multiplication is a Dirac morphism. It turns out that the simply connected Dirac Lie groups are classified by so-called Dirac Manin triples. We give an explicit construction of the Dirac Lie group structure defined by a Dirac...

  7. $\mathcal{F}$-stability for self-shrinking solutions to mean curvature flow

    Andrews, Ben; Li, Haizhong; Wei, Yong
    In this paper, we formulate the notion of the $\mathcal{F}$-stability of self-shrinking solutions to mean curvature flow in arbitrary codimension. Then we give some classifications of the $\mathcal{F}$-stable self-shrinkers in arbitrary codimension. We show that the only $\mathcal{F}$-stable self-shrinking solution which is a closed minimal submanifold in a sphere must be the shrinking sphere. We also prove that the spheres and planes are the only $\mathcal{F}$-stable self-shrinkers with parallel principal normal. In the codimension one case, our results reduce to those of Colding and Minicozzi.

  8. A no breathers theorem for some noncompact Ricci flows

    Zhang, Qi S.
    Under suitable conditions near infinity and assuming boundedness of curvature tensor, we prove a no breathers theorem in the spirit of Ivey-Perelman for some noncompact Ricci flows. These include Ricci flows on asymptotically flat (AF) manifolds with positive scalar curvature, which was studied in "Mass under the Ricci flow," [X. Dai and L. Ma, Comm. Math. Phys. 274:1 (2007), pp. 65–80] and "Rotationally symmetric Ricci flow on asymptotically flat manifolds," [T. A. Oliynyk, and E. Woolgar, Comm. Anal. Geom., 15:3 (2007), pp. 535–568] in connection with general relativity. Since the method for the compact case faces a difficulty, the proof involves solving a new non-local...

  9. CM elliptic curves and primes captured by quadratic polynomials

    Ji, Qingzhong; Qin, Hourong
    Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with complex multiplication. For a prime $p$, some formulas for $a_p = p + 1 \sharp E(\mathbb{F}_p)$ are given in terms of the binomial coefficients. We show that the equality $a_p = r$ holds for some fixed integer $r$ if and only if a certain quadratic polynomial represents the prime $p$. In particular, for $E \colon y^2 = x^3 + x, a_p = 2$ holding for an odd prime $p$ if and only if $p$ is of the form $n^2 + 1$ and for $E \colon y^2 = x^3 - 11x + 14,...

  10. Boundaries of cycle spaces and degenerating Hodge structures

    Hayama, Tatsuki
    We study a property of cycle spaces in connection with degenerating Hodge structures of odd-weight, and we construct maps from some partial compactifications of period domains to the Satake compatifications of Siegel spaces. These maps are a generalization of the maps from the toroidal compactifications of Siegel spaces to the Satake compactifications. We also show continuity of these maps for the case for the Hodge structure of Calabi-Yau threefolds with $h^{2,1} = 1$.

  11. Asymptotic spectral flow for Dirac operators of disjoint Dehn twists

    Tsai, Chung-Jun
    Let $Y$ be a compact, oriented 3-manifold with a contact form $a$. For any Dirac operator $\mathcal{D}$, we study the asymptotic behavior of the spectral flow between $\mathcal{D}$ and $\mathcal{D} + \mathrm{cl}(-\frac{ir}{2}a)$ as $r \to \infty$. If $a$ is the Thurston-Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the next order term of the spectral flow function is $\mathcal{O}(r)$.

  12. Hypoellipticity of the $\overline{\partial}$-Neumann problem at a point of infinite type

    Baracco, Luca; Khanh, Tran Vu; Zampieri, Giuseppe
    We prove local hypoellipticity of the complex Laplacian $\square$ in a domain which has superlogarithmic estimates outside a curve transversal to the CR directions and for which the holomorphic tangential derivatives of a defining function are superlogarithmic multipliers in the sense of "A general method of weights in the $\overline{\partial}$-Neumann problem," [T. V. Khanh, Ph.D. Thesis, Padua (2009)].

  13. Small four-manifolds without non-singular solutions of normalized Ricci flows

    Ishida, Masashi
    It is known that connected sums $X\# K 3 \# (\Sigma_g \times \Sigma_h) \# \ell_1 (S^1 \times S^3) \# \ell_2 \overline{\mathbb{C}P^2}$ satisfy the Gromov-Hitchin-Thorpe type inequality, but can not admit non-singular solutions of the normalized Ricci flow for any initial metric, where $\Sigma_g \times \Sigma_h$ is the product of two Riemann surfaces of odd genus, $\ell_1, \ell_2 \gt 0$ are sufficiently large positive integers, $g, h \gt 3$ are also sufficiently large positive odd integers, and $X$ is a certain irreducible symplectic 4-manifold. These examples are closely related with a conjecture of Fang, Zhang and Zhang. In the current article, we point out...

  14. A result on Ricci curvature and the second Betti number

    Wan, Jianming
    We prove that the second Betti number of a compact Riemannian manifold vanishes under certain Ricci curved restriction. As consequences we obtain an interesting curved restriction for compact Kähler-Einstein manifolds and a homology sphere theorem in ${\rm dim}=4,5$.

  15. Projective completions of affine varieties via degree-like functions

    Mondal, Pinaki
    We study projective completions of affine algebraic varieties induced by filtrations on their coordinate rings. In particular, we study the effect of the “multiplicative” property of filtrations on the corresponding completions and introduce a class of projective completions (of arbitrary affine varieties) which generalizes the construction of toric varieties from convex rational polytopes. As an application we recover (and generalize to varieties over algebraically closed fields of arbitrary characteristics) a “finiteness” property of divisorial valuations over complex affine varieties proved in “Divisorial valuations via arcs” [T. de Fernex, L. Ein, and S. Ishii, Publ. Res. Inst. Math. Sci., 44:2 (2008), pp. 425–448]. We...

  16. Structure of Hochschild cohomology of path algebras and differential formulation of Euler's polyhedron formula

    Guo, Li; Li, Fang
    This article studies the Lie algebra $\mathrm{Der}(\mathrm{k}\Gamma)$ of derivations on the path algebra $\mathrm{k}\Gamma$ of a quiver $\Gamma$ and the Lie algebra on the first Hochschild cohomology group $HH1(\mathrm{k}\Gamma)$. We relate these Lie algebras to the algebraic and combinatorial properties of the path algebra. Characterizations of derivations on a path algebra are obtained, leading to a canonical basis of $\mathrm{Der}(\mathrm{k}\Gamma)$ and its Lie algebra properties. Special derivations are associated to the vertices, arrows and faces of a quiver, and the concepts of a connection matrix and boundary matrix are introduced to study the relations among these derivations, giving rise to an interpretation of...

  17. Symmetry defect of algebraic varieties

    Janeczko, S.; Jelonek, Z.; Ruas, M. A. S.
    Let $X, Y \subset k^m(k = \mathbb{R},\mathbb{C})$ be smooth manifolds. We investigate the central symmetry of the configuration of $X$ and $Y$. For $p \in k^m$ we introduce a number $\mu(p)$ of pairs of points $x \in X$ and $y \in Y$ such that $p$ is the center of the interval $\overline{xy}$. We show that if $X, Y$ (including the case $X = Y$ ) are algebraic manifolds in a general position, then there is a closed (semi-algebraic) set $B \subset k^m$, called symmetry defect set of the $X$ and $Y$ configuration, such that the function $\mu$ is locally constant and not...

  18. Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces

    De Jong, Robin
    Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces. As a corollary we state precise asymptotic formulas for the beta-invariant introduced around 2000 by R. Hain and D. Reed. These formulas are a refinement of a result Hain and Reed prove in their paper. We...

  19. A holographic principle for the existence of parallel Spinor fields and an inequality of Shi-Tam type

    Hijazi, Oussama; Montiel, Sebastián
    Suppose that $\Sigma = \partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $(M, \langle , \rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $\Sigma$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric $\langle , \rangle {}_H = H^2 \langle , \rangle$ is at least $n/2$ and equality holds if and only if there exists a non-trivial parallel spinor field on $M$. As a consequence, if $\Sigma$ admits an isometric and isospin immersion $F$ with mean curvature $H_0$ as a hypersurface...

  20. Stable logarithmic maps to Deligne-Faltings pairs II

    Abramovich, Dan; Chen, Qile
    We make an observation which enables one to deduce the existence of an algebraic stack of logarithmic maps for all generalized Deligne-Faltings logarithmic structures (in particular simple normal crossings divisors) from the simplest case with characteristic generated by $\mathbb{N}$ (essentially the smooth divisor case).

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