Mostrando recursos 1 - 20 de 63

  1. Erratum to “$L^p$ measure of growth and higher order Hardy–Sobolev–Morrey inequalities on $\mathbb{R}^{N}$”

    RABIER, Patrick J.

  2. The graded structure induced by operators on a Hilbert space

    GUO, Kunyu; WANG, Xudi
    In this paper we define a graded structure induced by operators on a Hilbert space. Then we introduce several concepts which are related to the graded structure and examine some of their basic properties. A theory concerning minimal property and unitary equivalence is then developed. It allows us to obtain a complete description of $\mathcal{V}^\ast(M_{z^k})$ on any $H^2(\omega)$. It also helps us to find that a multiplication operator induced by a quasi-homogeneous polynomial must have a minimal reducing subspace. After a brief review of multiplication operator $M_{z+w}$ on $H^2(\omega,\delta)$, we prove that the Toeplitz operator $T_{z+\overline{w}}$ on $H^2(\mathbb{D}^2)$, the Hardy...

  3. Reflections at infinity of time changed RBMs on a domain with Liouville branches

    CHEN, Zhen-Qing; FUKUSHIMA, Masatoshi
    Let $Z$ be the transient reflecting Brownian motion on the closure of an unbounded domain $D\subset \mathbb{R}^d$ with $N$ number of Liouville branches. We consider a diffuion $X$ on $\overline{D}$ having finite lifetime obtained from $Z$ by a time change. We show that $X$ admits only a finite number of possible symmetric conservative diffusion extensions $Y$ beyond its lifetime characterized by possible partitions of the collection of $N$ ends and we identify the family of the extended Dirichlet spaces of all $Y$ (which are independent of time change used) as subspaces of the space $\mathrm{BL}(D)$ spanned by the extended Sobolev...

  4. Arithmetic exceptionality of generalized Lattès maps

    We consider the arithmetic exceptionality problem for the generalized Lattès maps on $\mathbf{P}^2$. We prove an existence result for maps arising from the product $E \times E$ of elliptic curves $E$ with CM.

  5. Exponential mixing for generic volume-preserving Anosov flows in dimension three

    TSUJII, Masato
    Let $M$ be a closed 3-dimensional Riemann manifold and let $3\le r\le \infty$. We prove that there exists an open dense subset in the space of $C^r$ volume-preserving Anosov flows on $M$ such that all the flows in it are exponentially mixing.

  6. Spaces of nonnegatively curved surfaces

    We determine the homeomorphism type of the space of smooth complete nonnegatively curved metrics on $S^2$, $RP^2$, and $\mathbb{C}$ equipped with the topology of $C^\gamma$ uniform convergence on compact sets, when $\gamma$ is infinite or is not an integer. If $\gamma=\infty$, the space of metrics is homeomorphic to the separable Hilbert space. If $\gamma$ is finite and not an integer, the space of metrics is homeomorphic to the countable power of the linear span of the Hilbert cube. We also prove similar results for some other spaces of metrics including the space of complete smooth Riemannian metrics on an arbitrary...

  7. A functional equation with Borel summable solutions and irregular singular solutions

    ŌUCHI, Sunao
    A Functional equation $\sum_{i=1}^{m}a_{i}(z)u(\varphi_{i}(z))=f(z)$ is considered. First we show the existence of solutions of formal power series. Second we study the homogeneous equation $(f(z)\equiv 0)$ and construct formal solutions containing exponential factors. Finally it is shown that there exists a genuine solution in a sector whose asymptotic expansion is a formal solution, by using the theory of Borel summability of formal power series. The equation has similar properties to those of irregular singular type in the theory of ordinary differential equations.

  8. Rank two jump loci for solvmanifolds and Lie algebras

    PAPADIMA, Ştefan; PAUNESCU, Laurenţiu
    We consider representation varieties in $SL_2$ for lattices in solvable Lie groups, and representation varieties in $\mathfrak{sl}_2$ for finite-dimensional Lie algebras. Inside them, we examine depth 1 characteristic varieties for solvmanifolds, respectively resonance varieties for cochain Differential Graded Algebras of Lie algebras. We prove a general result that leads, in both cases, to the complete description of the analytic germs at the origin, for the corresponding embedded rank 2 jump loci.

  9. Needle decompositions and isoperimetric inequalities in Finsler geometry

    OHTA, Shin-ichi
    Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Lévy–Gromov, Bakry–Ledoux, Bayle and Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by optimal transport theory. Cavalletti and Mondino subsequently generalized the localization method, in a different way more directly along optimal transport theory, to essentially non-branching metric measure spaces satisfying the curvature-dimension condition. This class in particular includes reversible (absolutely homogeneous) Finsler manifolds. In this paper, we construct needle decompositions of non-reversible (only positively homogeneous) Finsler manifolds, and show an isoperimetric...

  10. Curvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metrics

    BURSTALL, Francis E.; HERTRICH-JEROMIN, Udo; SUYAMA, Yoshihiko
    There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat 3-metric with the Guichard condition in the interior of the space, an evolution of orthogonal (local-)Riemannian 2-metrics with constant Gauss curvature $-1$ is determined; for a 2-metric belonging to a certain class of orthogonal analytic 2-metrics with constant Gauss curvature $-1$, a one-parameter family of conformally flat 3-metrics with the Guichard condition is determined as evolutions...

  11. Birational maps preserving the contact structure on $\mathbb{P}^3_\mathbb{C}$

    CERVEAU, Dominique; DÉSERTI, Julie
    We study the group of polynomial automorphisms of $\mathbb{C}^3$ (resp. birational self-maps of $\mathbb{P}^3_\mathbb{C}$) that preserve the contact structure.

  12. Homogenisation on homogeneous spaces

    LI, Xue-Mei
    Motivated by collapsing of Riemannian manifolds and inhomogeneous scaling of left invariant Riemannian metrics on a real Lie group $G$ with a sub-group $H$, we introduce a family of interpolation equations on $G$ with a parameter $\epsilon>0$, interpolating hypo-elliptic diffusions on $H$ and translates of exponential maps on $G$ and examine the dynamics as $\epsilon\to 0$. When $H$ is compact, we use the reductive homogeneous structure of Nomizu to extract a converging family of stochastic processes (converging on the time scale $1/\epsilon$), proving the convergence of the stochastic dynamics on the orbit spaces $G/H$ and their parallel translations, providing also...

  13. On the Galois structure of arithmetic cohomology II: ray class groups

    BURNS, David; KUMON, Asuka
    We investigate the explicit Galois structure of ray class groups. We then derive consequences of our results concerning both the validity of Leopoldt’s Conjecture and the existence of families of explicit congruence relations between the values of Dirichlet $L$-series at $s=1$.

  14. Spacelike Dupin hypersurfaces in Lorentzian space forms

    LI, Tongzhu; NIE, Changxiong
    Similar to the definition in Riemannian space forms, we define the spacelike Dupin hypersurface in Lorentzian space forms. As conformal invariant objects, spacelike Dupin hypersurfaces are studied in this paper using the framework of the conformal geometry of spacelike hypersurfaces. Further we classify the spacelike Dupin hypersurfaces with constant Möbius curvatures, which are also called conformal isoparametric hypersurface.

  15. Dynamics and the Godbillon–Vey class of $C^1$ foliations

    HURDER, Steven; LANGEVIN, Rémi
    Let $\mathcal{F}$ be a codimension-one, $C^2$-foliation on a manifold $M$ without boundary. In this work we show that if the Godbillon–Vey class $GV(\mathcal{F}) \in H^3(M)$ is non-zero, then $\mathcal{F}$ has a hyperbolic resilient leaf. Our approach is based on methods of $C^1$-dynamical systems, and does not use the classification theory of $C^2$-foliations. We first prove that for a codimension-one $C^1$-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points $E(\mathcal{F})$ has positive Lebesgue measure. We then prove that if $E(\mathcal{F})$ has positive measure for a $C^1$-foliation, then $\mathcal{F}$ must have a hyperbolic resilient leaf, and hence its geometric entropy...

  16. Weighted Bott–Chern and Dolbeault cohomology for LCK-manifolds with potential

    ORNEA, Liviu; VERBITSKY, Misha; VULETESCU, Victor
    A locally conformally Kähler (LCK) manifold is a complex manifold, with a Kähler structure on its universal covering $\tilde M$, with the deck transform group acting on $\tilde M$ by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kähler form taking values in a local system $L$, called the conformal weight bundle. The $L$-valued cohomology of $M$ is called Morse–Novikov cohomology; it was conjectured that (just as it happens for Kähler manifolds) the Morse–Novikov complex satisfies the $dd^c$-lemma, which (if true) would have far-reaching consequences for the geometry of LCK manifolds. In...

  17. A characterization of regular points by Ohsawa–Takegoshi extension theorem

    GUAN, Qi'an; LI, Zhenqian
    In this article, we present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related Ohsawa–Takegoshi extension theorem holds. We also obtain a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.

  18. Topology of mixed hypersurfaces of cyclic type

    INABA, Kazumasa; KAWASHIMA, Masayuki; OKA, Mutsuo
    Let $f_{II}({\boldsymbol{z}}, \bar{{\boldsymbol{z}}}) = z_{1}^{a_{1}+b_{1}}\bar{z}_{1}^{b_{1}}z_{2} + \cdots + z_{n-1}^{a_{n-1}+b_{n-1}}\bar{z}_{n-1}^{b_{n-1}}z_{n} + z_{n}^{a_{n}+b_{n}}\bar{z}_{n}^{b_{n}}z_{1}$ be a mixed weighted homogeneous polynomial of cyclic type and $g_{II}({\boldsymbol{z}}) = z_{1}^{a_{1}}z_{2} + \cdots + z_{n-1}^{a_{n-1}}z_{n} + z_{n}^{a_{n}}z_{1}$ be the associated weighted homogeneous polynomial where $a_{j} \geq 1$ and $b_{j} \geq 0$ for $j = 1, \dots, n$. We show that two links $S^{2n-1}_{\varepsilon} \cap f_{II}^{-1}(0)$ and $S^{2n-1}_{\varepsilon} \cap g_{II}^{-1}(0)$ are diffeomorphic and their Milnor fibrations are isomorphic.

  19. On the sequential polynomial type of modules

    GOTO, Shiro; NHAN, Le Thanh
    Let $M$ be a finitely generated module over a Noetherian local ring $R$. The sequential polynomial type $\mathrm{sp}(M)$ of $M$ was recently introduced by Nhan, Dung and Chau, which measures how far the module $M$ is from the class of sequentially Cohen–Macaulay modules. The present paper purposes to give a parametric characterization for $M$ to have $\mathrm{sp}(M)\le s$, where $s\ge -1$ is an integer. We also study the sequential polynomial type of certain specific rings and modules. As an application, we give an inequality between $\mathrm{sp}(S)$ and $\mathrm{sp}(S^G) $, where $S$ is a Noetherian local ring and $G$ is a...

  20. The selection problem for discounted Hamilton–Jacobi equations: some non-convex cases

    GOMES, Diogo A.; MITAKE, Hiroyoshi; TRAN, Hung V.
    Here, we study the selection problem for the vanishing discount approximation of non-convex, first-order Hamilton–Jacobi equations. While the selection problem is well understood for convex Hamiltonians, the selection problem for non-convex Hamiltonians has thus far not been studied. We begin our study by examining a generalized discounted Hamilton–Jacobi equation. Next, using an exponential transformation, we apply our methods to strictly quasi-convex and to some non-convex Hamilton–Jacobi equations. Finally, we examine a non-convex Hamiltonian with flat parts to which our results do not directly apply. In this case, we establish the convergence by a direct approach.

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