Showing papers in "Tohoku Mathematical Journal in 2020"

Algebraic cycles and Verra fourfolds

Abstract: This note is about the Chow ring of Verra fourfolds. For a general Verra fourfold, we show that the Chow group of homologically trivial 1-cycles is generated by conics. We also show that Verra fourfolds admit a multiplicative Chow–Kunneth decomposition, and draw some consequences for the intersection product in the Chow ring of Verra fourfolds.

Dynamical degree and arithmetic degree of endomorphisms on product varieties

Abstract: For a dominant rational self-map on a smooth projective variety defined over a number field, Shu Kawaguchi and Joseph H. Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at an algebraic point whose forward orbit is well-defined and Zariski dense. We give some examples of self-maps on product varieties and rational points on them for which the Kawaguchi-Silverman conjecture holds.

A sufficient condition for a hypersurface to be isoparametric

Abstract: Let $M^n$ be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose $\mathfrak{a}$ is a symmetric $(0,2)$ tensor field whose dual $(1,1)$ tensor $\mathcal{A}$ has $n$ distinct eigenvalues, and $\mathrm{tr}(\mathcal{A}^k)$ are constants for $k=1,\ldots, n-1$. We show that all the eigenvalues of $\mathcal{A}$ are constants, generalizing a theorem of de Almeida and Brito [dB90] to higher dimensions. As a consequence, a closed hypersurface $M^n$ in $S^{n+1}$ is isoparametric if one takes $\mathfrak{a}$ above to be the second fundamental form, giving affirmative evidence to Chern's conjecture.

Uniqueness theorems for the Boussinesq system

Abstract: We address the uniqueness problem for mild solutions of the Boussinesq system in $\mathbb{R}^3$ We provide several uniqueness classes on the velocity and the temperature, generalizing in this way the classical $C([0,T]; L^3(\mathbb{R}^3))$-uniqueness result for mild solutions of the Navier–Stokes equations

Metaplectic categories, gauging and property $F$

Abstract: $N$-Metaplectic categories, unitary modular categories with the same fusion rules as $SO(N)_2$, are prototypical examples of weakly integral modular categories generalizing the model for the Ising anyons, i.e. metaplectic anyons. A conjecture of the second author would imply that images of the braid group representations associated with metaplectic categories are finite groups, i.e. have property $F$. While it was recently shown that $SO(N)_2$ itself has property $F$, proving property $F$ for the more general class of metaplectic modular categories is an open problem. We verify this conjecture for $N$-metaplectic modular categories when $N$ is odd, exploiting their recent enumeration together with a characterization in terms of Galois conjugation and twisting. In another direction, we prove that when $N$ is divisible by 8 the $N$-metaplectic categories have 3 non-trivial bosons, and the boson condensation procedure applied to 2 of these bosons yields $\frac{N}{4}$-metaplectic categories. Otherwise stated: any $8k$-metaplectic category is a $\mathbb{Z}_2$-gauging of a $2k$-metaplectic category, so that the $N$ even metaplectic categories lie towers of $\mathbb{Z}_2$-gaugings commencing with $2k$- or $4k$-metaplectic categories with $k$ odd.

Molecular characterization of anisotropic variable Hardy-Lorentz spaces

Abstract: Let $A$ be an expansive dilation on $\mathbb{R}^n$, $q\in(0, \infty]$ and $p(\cdot):\mathbb{R}^n\rightarrow(0, \infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition Let $H^{p(\cdot), q}_A({\mathbb{R}}^n)$ be the anisotropic variable Hardy-Lorentz space defined via the radial grand maximal function In this paper, the authors first establish its molecular characterization via the atomic characterization of $H^{p(\cdot), q}_A(\mathbb{R}^n)$ Then, as applications, the authors obtain the boundedness of anisotropic Calderon-Zygmund operators from $H^{p(\cdot), q}_{A}(\mathbb{R}^n)$ to $L^{p(\cdot), q}(\mathbb{R}^n)$ or from $H^{p(\cdot), q}_{A}(\mathbb{R}^n)$ to itself All these results are still new even in the classical isotropic setting

A large family of projectively equivalent $C^0$-Finsler manifolds

Abstract: A $C^0$-Finsler structure on a differentiable manifold is a continuous real valued function defined on its tangent bundle such that its restriction to each tangent space is a norm. In this work we present a large family of projectively equivalent $C^0$-Finsler manifolds $(\hat M,\hat F)$, where $\hat M$ is diffeomorphic to the Euclidean plane. The structures $\hat F$ don't have partial derivatives and they aren't invariant by any transformation group of $\hat M$. For every $p,q \in (\hat M,\hat F)$, we determine the unique minimizing path connecting $p$ and $q$. They are line segments parallel to the vectors $(\sqrt{3}/2,1/2)$, $(0,1)$ or $(-\sqrt{3}/2,1/2)$, or else a concatenation of two of these line segments. Moreover $(\hat M,\hat F)$ aren't Busemann $G$-spaces and they don't admit any bounded open $\hat F$-strongly convex subsets. Other geodesic properties of $(\hat M,\hat F)$ are also studied.

A sixth order flow of plane curves with boundary conditions

Abstract: For small-energy initial regular planar curves with generalised Neumann boundary conditions, we consider the steepest-descent gradient flow for the $L^2$-norm of the derivative of curvature with respect to arc length. We show that such curves between parallel lines converge exponentially in the $C^\infty$ topology in infinite time to straight lines.

Functional equations of zeta functions associated with homogeneous cones

Abstract: In this paper, we focus on solvable prehomogeneous vector spaces associated with homogeneous cones, and consider the associated zeta functions in several variables. We discuss $\mathbb{Q}$-structures of these prehomogeneous vector spaces, and give explicit formulas of functional equations of the zeta functions by using the data of homogeneous cones. The associated $b$-functions are also described explicitly.

Higher-order nonlinear schrödinger equation in 2D case

Abstract: We consider the Cauchy problem for the higher-order nonlinear Schrodinger equation in two dimensional case \[ \left\{\!\!\! \begin{array}{c} i\partial _{t}u+\frac{b}{2}\Delta u-\frac{1}{4}\Delta ^{2}u=\lambda \left\vert u\right\vert u,\text{ }t>0,\quad x\in \mathbb{R}^{2}\,\mathbf{,} \\ u\left( 0,x\right) =u_{0}\left( x\right) ,\quad x\in \mathbb{R}^{2} \,\mathbf{,} \end{array} \right. \] where $\lambda \in \mathbb{R}\mathbf{,}$ $b>0.$ We develop the factorization techniques for studying the large time asymptotics of solutions to the above Cauchy problem. We prove that the asymptotics has a modified character.

Calderón-Zygmund singular integrals in central Morrey-Orlicz spaces

Abstract: We prove the strong-type and weak-type estimates for the Calderon–Zygmund singular integrals on central Morrey–Orlicz and weak central Morrey–Orlicz spaces defined in our earlier paper [26] Next

New functional equations of finite multiple polylogarithms

Abstract: We give a finite analogue of the well-known formula $\mathrm{Li}_{\underbrace{1, \ldots, 1}_n}(t)=\frac{1}{n!}\mathrm{Li}_1(t)^n$ of multiple polylogarithms for any positive integer $n$ by using the shuffle relation of finite multiple polylogarithms of Ono–Yamamoto type. Unlike the usual case, the terms regarded as error terms appear in this formula. As a corollary, we obtain “$t \leftrightarrow 1-t$” type new functional equations of finite multiple polylogarithms of Ono–Yamamoto type and Sakugawa-Seki type.

Invariant Einstein metrics on $\mathrm{SU}(n)$ and complex Stiefel manifolds

Abstract: We study existence of invariant Einstein metrics on complex Stiefel manifolds $G/K = \operatorname{SU}(\ell+m+n)/\operatorname{SU}(n) $ and the special unitary groups $G = \operatorname{SU}(\ell+m+n)$ We decompose the Lie algebra $\frak g$ of $G$ and the tangent space $\frak p$ of $G/K$, by using the generalized flag manifolds $G/H = \operatorname{SU}(\ell+m+n)/\operatorname{S}(\operatorname{U}(\ell)\times\operatorname{U}(m)\times\operatorname{U}(n))$ We parametrize scalar products on the 2-dimensional center of the Lie algebra of $H$, and we consider $G$-invariant and left invariant metrics determined by $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(\ell)\times\operatorname{U}(m)\times\operatorname{U}(n))$-invariant scalar products on $\frak g$ and $\frak p$ respectively Then we compute their Ricci tensor for such metrics We prove existence of $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(1)\times\operatorname{U}(2)\times\operatorname{U}(2))$-invariant Einstein metrics on $V_3\mathbb{C}^{5}=\operatorname{SU}(5)/\operatorname{SU}(2)$, $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(2)\times\operatorname{U}(2)\times\operatorname{U}(2))$-invariant Einstein metrics on $V_4\mathbb{C}^{6}=\operatorname{SU}(6)/\operatorname{SU}(2)$, and $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(m)\times\operatorname{U}(m)\times\operatorname{U}(n))$-invariant Einstein metrics on $V_{2m}\mathbb{C}^{2m+n}=\operatorname{SU}(2m+n)/\operatorname{SU}(n)$ We also prove existence of $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(1)\times\operatorname{U}(2)\times\operatorname{U}(2))$-invariant Einstein metrics on the compact Lie group $\operatorname{SU}(5)$, which are not naturally reductive The Lie group $\operatorname{SU}(5)$ is the special unitary group of smallest rank known for the moment, admitting non naturally reductive Einstein metrics Finally, we show that the compact Lie group $\operatorname{SU}(4+n)$ admits two non naturally reductive $\operatorname{Ad}(\operatorname{S}(\operatorname{U}(2)\times\operatorname{U}(2)\times\operatorname{U}(n)))$-invariant Einstein metrics for $ 2 \leq n \leq 25$, and four non naturally reductive Einstein metrics for $n\ge 26$ This extends previous results of K Mori about non naturally reductive Einstein metrics on $\operatorname{SU}(4+n)$ ($n \geq 2$)

Closed almost Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler

Abstract: We show that a closed almost Kahler 4-manifold of pointwise constant holomorphic sectional curvature $k\geq 0$ with respect to the canonical Hermitian connection is automatically Kahler. The same result holds for $k<0$ if we require in addition that the Ricci curvature is $J$-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.

On infiniteness of integral overconvergent de Rham-Witt cohomology modulo torsion

Abstract: In this article, we give examples of smooth varieties of positive characteristic whose first integral overconvergent de Rham–Witt cohomology modulo torsion is not finitely generated over the Witt ring of the base field.

Zeta-functions of root systems and Poincaré polynomials of Weyl groups

Abstract: We consider a certain linear combination of zeta-functions of root systems for a root system. Showing two different expressions of this linear combination, we find that a certain signed sum of zeta-functions of root systems is equal to a sum involving Bernoulli functions of root systems. This identity gives a non-trivial functional relation among zeta-functions of root systems, if the signed sum does not identically vanish. This is a generalization of the authors' previous result (Proc. London Math. Soc. 100 (2010), 303–347). We present several explicit examples of such functional relations. We give a criterion of the non-vanishing of the signed sum, in terms of Poincare polynomials of associated Weyl groups. Moreover we prove a certain converse theorem.

A note on an anabelian open basis for a smooth variety

Abstract: Schmidt and Stix proved that every smooth variety over a field finitely generated over the field of rational numbers has an open basis for the Zariski topology consisting of “anabelian” varieties. This was predicted by Grothendieck in his letter to Faltings. In the present paper, we generalize this result to smooth varieties over generalized sub-$p$-adic fields. Moreover, we also discuss an absolute version of this result.

Rough integration via fractional calculus

Abstract: On the basis of fractional calculus, the author's previous study [9] introduced an approach to the integral of controlled paths against Holder rough paths. The integral in [9] is defined by the Lebesgue integrals for fractional derivatives without using any arguments based on discrete approximation. In this paper, we revisit the approach of [9] and show that, for a suitable class of Holder rough paths including geometric Holder rough paths, the integral in [9] is consistent with that obtained by the usual integration theory of rough path analysis, given by the limit of the compensated Riemann–Stieltjes sums.

Constant mean curvature trinoids with one irregular end

Abstract: We construct a new five parameter family of constant mean curvature trinoids with two asymptotically Delaunay ends and one irregular end

Boundedness of maximal operators on Herz spaces with radial variable exponent

Abstract: Our aim in this paper is to deal with the boundedness of the central Hardy-Littlewood maximal operator on Herz spaces with radial variable exponent. As a special case, we show the boundedness of the maximal operator in weighted Lebesgue spaces with radial variable exponent. We also discuss the integrability of the maximal functions in such spaces.

Erratum: Stability of stationary solution for the Lugiato-Lefever equation

Abstract: Rainer Mandel [2] pointed out a mistake about Remark 3.1 (i) on page 654 in [4], which was published in Tohoku Mathematics Journal, Vol. 63, 2011. Here and hereafter, the numbers of formulas, theorems, sections and pages correspond to those in the paper [4] except for (C1), (C2), (C3) and (C4) below.

Convergence theorems of a modified iteration process for generalized nonexpansive mappings in hyperbolic spaces

Abstract: In this paper, we introduce a modified Picard-Mann hybrid iterative process for a finite family of mappings in the framework of hyperbolic spaces. Furthermore, we establish $\Delta$-convergence and strong convergence results for a sequence generated by a modified Picard-Mann hybrid iterative process involving mappings satisfying the condition $(E)$ in the setting of hyperbolic spaces which more general than one mapping in the setting of CAT(0) spaces in Ritika and Khan [19]. Our results are the extension and improvement of the results in Ritika and Khan [19]. Moreover, in the numerical example we also illustrate an example for supporting our main result.

A formula for the heat kernel coefficients on Riemannian manifolds

Abstract: Based on the idea of adiabatic expansion theory, we will present a new formula for the asymptotic expansion coefficients of every derivative of the heat kernel on a compact Riemannian manifold It will be very useful for having systematic understanding of the coefficients, and, furthermore, by using only a basic knowledge of calculus added to the formula, one can describe them explicitly up to an arbitrarily high order