Hokkaido Mathematical Journal

Vol. 55,2026

    No. 3

    HAN, Jing; YAN, Qiming; ZHOU, Chaohui;
    A Note on Meromorphic Maps from Complete Kähler Manifolds into Projective Spaces Intersecting Hyperlpanes.
    Hokkaido Mathematical Journal, 55 (2026) pp.325-352

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    Abstract

    In this paper, we establish the truncated non-integrated defect relation for meromorphic maps from complete Kähler manifolds into the projective space intersecting hyperlpanes with the distributive constant of these hyperplanes. As an application, the uniqueness theorem is also given.

    Keywords Non-integrated defect relation; uniqueness theorem; meromorphic map; Kähler manifold; hyperlpane;
    msc2020(primary) 32H30; 32A22;
    msc2020(secondary) 30D35;

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    Abstract

    We study the $L^\infty$-decay estimate and the asymptotic behavior of solutions to dissipative nonlinear Schrödinger equations. In some known results, Kita-Nakamura established an $L^{\infty}$-decay result under a weak dissipative condition for nonlinear power $p \gt 2.686$. In this paper, we extend the lower bound of the admissible range to $p \gt 5/2$, still under weak dissipation and with small initial data in $H^2$ satisfying $x^2 u_0 \in L^2$. Our result also contrasts with the work of Jin-Jin-Li, Chinese mathematicians, who treated the same problem under a strong dissipative condition and removed the size restriction on the initial data. Notably, we have solved the problem of extending the nonlinear power $p$ range for the modified scattering and decay estimates without the strong dissipative condition.

    Keywords dissipative nonlinear Schrödinger equations; subcritical nonlinearity; $L^{\infty}$-decay estimate;
    msc2020(primary) 35Q55;
    msc2020(secondary) 35B40;
    JUAREZ-CAMPOS, Beatriz; NAUMKIN, Pavel I.;
    Modified Scattering for the Derivative Fractional Cubic Nonlinear Schrödinger Equation with Dispersion Order $\alpha\in(4,5)$.
    Hokkaido Mathematical Journal, 55 (2026) pp.377-424

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    Abstract

    We consider the Cauchy problem for the derivative fractional cubic nonlinear Schrödinger equation \[ \begin{cases} \partial_{t}u-\dfrac{1}{\alpha}\left\vert \partial_{x}\right\vert ^{\alpha}u=i\lambda\partial_{x}(\left\vert u\right\vert ^{2}u), & t\gt 0,~ x\in\mathbb{R}, \\ u\left( 0,x\right) =u_{0}\left( x\right) , & x\in\mathbb{R}, \end{cases} \] where $\alpha\in\left( 4,5\right) ,$ and $\lambda\in\mathbb{R}.$ We show that the large time asymptotics of solutions has a logarithmic correction in the phase comparing with the corresponding linear case. We develop the method based on the factorization techniques. Also we apply the known results on the $\mathbf{L}^{2}$ - boundedness of pseudodifferential operators.

    Keywords Derivative fractional cubic nonlinear Schrödinger equation; Modified scattering; Asymptotics for large time;
    msc2020(primary) 35B40; 35Q92;
    LÓPEZ, Rafael;
    Maximal translation surfaces in Lorentz-Minkowski space.
    Hokkaido Mathematical Journal, 55 (2026) pp.425-456

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    Abstract

    A translation surface in Lorentz-Minkowski space $\mathbb{L}^3$ is a surface defined as the sum of two spatial curves in $\mathbb{L}^3$. In this paper, we present a classification of maximal translation surfaces. We prove that if one of the generating curves is planar, then the other must also be planar. Moreover, we provide a complete description of these surfaces. When both planar curves are of Frenet type, we generalize the known examples of Scherk surfaces. In the case where one of the generating curves is a pseudo-null, we obtain examples of maximal surfaces. These surfaces do not have counterparts in Euclidean space.

    Keywords maximal surface; translation surface; Scherk surface; Frenet curve; pseudo-null curve;
    msc2020(primary) 53A10; 53C42; 53C50;
    IKAWA, Osamu; TASAKI, Hiroyuki;
    Intersections of real and quaternionic flag manifolds in complex flag manifolds, and their sum of Betti numbers.
    Hokkaido Mathematical Journal, 55 (2026) pp.457-474

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    Abstract

    We show that the sum of Betti numbers of the intersections of two real flag manifolds in a complex flag manifold consisting of sequences of complex subspaces in a complex vector space are invariants of such real flag manifolds. We also show that similar claims hold for the intersections of two quaternionic flag manifolds and the intersections of real and quaternionic flag manifolds.

    Keywords complex flag manifold; real flag manifold;
    msc2020(primary) 15A66; 53C35;
    AOKI, Masamitsu;
    Quasimorphisms of free products of racks and quandles.
    Hokkaido Mathematical Journal, 55 (2026) pp.475-489

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    Abstract

    We show that the second bounded cohomology of the free product of racks and quandles is infinite-dimensional as a real vector space. This is similar to the case of groups. As a corollary, we show that the second bounded cohomology of the free rack and the free quandle is infinite-dimensional. We also give another proof of this corollary using homogeneous group quasimorphisms.

    Keywords quasimorphism; quandle; rack; free product; bounded cohomology;
    msc2020(primary) 57K12; 20J06; 20F65;