Journal of the Mathematical Society of Japan

Universal curvature identities and Euler–Lagrange formulas for Kähler manifolds

Peter B. Gilkey, JeongHyeong Park, Kouei Sekigawa

2016 Volume 68 Issue 2 Pages 459-487
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820459

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We relate certain universal curvature identities for Kähler manifolds to the Euler–Lagrange equations of the scalar invariants which are defined by pairing characteristic forms with powers of the Kähler form.

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Classes of weights and second order Riesz transforms associated to Schrödinger operators

Fu Ken Ly

2016 Volume 68 Issue 2 Pages 489-533
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820489

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We consider the Schrödinger operator −Δ + V on ℝⁿ with n ≥ 3 and V a member of the reverse Hölder class ℬ_s for some s > n/2. We obtain the boundedness of the second order Riesz transform ∇² (−Δ + V)⁻¹ on the weighted spaces L^p(w) where w belongs to a class of weights related to V. To prove this, we develop a good-λ inequality adapted to this setting along with some new heat kernel estimates.

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Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory

Yuzuru Inahama

2016 Volume 68 Issue 2 Pages 535-577
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820535

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In this paper we study short time asymptotics of a density function of the solution of a stochastic differential equation driven by fractional Brownian motion with Hurst parameter H (1/2 < H < 1) when the coefficient vector fields satisfy an ellipticity condition at the starting point. We prove both on-diagonal and off-diagonal asymptotics under mild additional assumptions. Our main tool is Malliavin calculus, in particular, Watanabe's theory of generalized Wiener functionals.

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Long time existence of classical solutions for the 3D incompressible rotating Euler equations

Ryo Takada

2016 Volume 68 Issue 2 Pages 579-608
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820579

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We consider the initial value problem of the 3D incompressible rotating Euler equations. We prove the long time existence of classical solutions for initial data in H^s(ℝ³) with s > 5/2. Also, we give an upper bound of the minimum speed of rotation for the long time existence when initial data belong to H^7/2(ℝ³).

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The kernel of Ribet's isogeny for genus three Shimura curves

Josep González, Santiago Molina

2016 Volume 68 Issue 2 Pages 609-635
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820609

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There are exactly nine reduced discriminants D of indefinite quaternion algebras over ℚ for which the Shimura curve X_D attached to D has genus 3. We present equations for these nine curves and, moreover, for each D we determine a subgroup c(D) of cuspidal divisors of degree zero of Jac(X₀(D))^new such that the abelian variety Jac(X₀(D))^new/c(D) is the jacobian of the curve X_D.

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Triple chords and strong (1, 2) homotopy

Noboru Ito, Yusuke Takimura

2016 Volume 68 Issue 2 Pages 637-651
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820637

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A triple chord is a sub-diagram of a chord diagram that consists of a circle and finitely many chords connecting the preimages for every double point on a spherical curve. This paper describes some relationships between the number of triple chords and an equivalence relation called strong (1, 2) homotopy, which consists of the first and one kind of the second Reidemeister moves involving inverse self-tangency if the curve is given any orientation. We show that a knot projection is trivialized by strong (1, 2) homotopy, if it is a simple closed curve or a prime knot projection without 1- and 2-gons whose chord diagram does not contain any triple chords. We also discuss the relation between Shimizu's reductivity and triple chords.

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Lindelöf theorem for harmonic mappings

David Kalaj

2016 Volume 68 Issue 2 Pages 653-667
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820653

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We extend the classical Lindelöf theorem for harmonic mappings. Assume that f is an univalent harmonic mapping of the unit disk U onto a Jordan domain with C¹ boundary. Then the function arg(∂_ϕ(f(z))/z), where z = re^iϕ, has continuous extension to the boundary of the unit disk, under certain condition on f|_T.

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Milnor-type theorems for left-invariant Riemannian metrics on Lie groups

Takahiro Hashinaga, Hiroshi Tamaru, Kazuhiro Terada

2016 Volume 68 Issue 2 Pages 669-684
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820669

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For all left-invariant Riemannian metrics on three-dimensional unimodular Lie groups, there exist particular left-invariant orthonormal frames, so-called Milnor frames. In this paper, for any left-invariant Riemannian metrics on any Lie groups, we give a procedure to obtain an analogous of Milnor frames, in the sense that the bracket relations among them can be written with relatively smaller number of parameters. Our procedure is based on the moduli space of left-invariant Riemannian metrics. Some explicit examples of such frames and applications will also be given.

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Moduli spaces of α-stable pairs and wall-crossing on ℙ²

Jinwon Choi, Kiryong Chung

2016 Volume 68 Issue 2 Pages 685-709
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820685

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We study the wall-crossing of the moduli spaces M^α(d,1) of α-stable pairs with linear Hilbert polynomial dm + 1 on the projective plane ℙ² as we alter the parameter α. When d is 4 or 5, at each wall, the moduli spaces are related by a smooth blow-up morphism followed by a smooth blow-down morphism, where one can describe the blow-up centers geometrically. As a byproduct, we obtain the Poincaré polynomials of the moduli spaces M(d,1) of stable sheaves. We also discuss the wall-crossing when the number of stable components in Jordan–Hölder filtrations is three.

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Some remarks on cubature formulas with linear operators

Masatake Hirao, Takayuki Okuda, Masanori Sawa

2016 Volume 68 Issue 2 Pages 711-735
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820711

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In this paper we consider a novel type of cubature formulas called operator-type cubature formulas. The notion originally goes back to a famous work by G. D. Birkhoff in 1906 on Hermite interpolation problem. A well-known theorem by Sobolev in 1962 on invariant cubature formulas is generalized to operator-type cubature, which provides a systematic treatment of Lebedev's works in the 1970s and some related results by Shamsiev in 2006. We give a lower bound for the number of points needed, and discuss analytic conditions for equality, together with tight illustrations for Laplacian-type cubature.

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An obstruction for codimension two contact embeddings in the odd dimensional Euclidean spaces

Naohiko Kasuya

2016 Volume 68 Issue 2 Pages 737-743
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820737

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We prove that the first Chern class of a codimension two closed contact submanifold of the odd dimensional Euclidean space is trivial. For any closed co-oriented contact 3-manifold with trivial Chern class, we prove that there is a contact structure on the 5-dimensional Euclidean space which admits a contact embedding of it.

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Spaces of algebraic maps from real projective spaces to toric varieties

Andrzej Kozlowski, Masahiro Ohno, Kohhei Yamaguchi

2016 Volume 68 Issue 2 Pages 745-771
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820745

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The problem of approximating the infinite dimensional space of all continuous maps from an algebraic variety X to an algebraic variety Y by finite dimensional spaces of algebraic maps arises in several areas of geometry and mathematical physics. An often considered formulation of the problem (sometimes called the Atiyah–Jones problem after [1]) is to determine a (preferably optimal) integer n_D such that the inclusion from this finite dimensional algebraic space into the corresponding infinite dimensional one induces isomorphisms of homology (or homotopy) groups through dimension n_D, where D denotes a tuple of integers called the “degree” of the algebraic maps and n_D → ∞ as D → ∞. In this paper we investigate this problem in the case when X is a real projective space and Y is a smooth compact toric variety.

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Kiselman's principle, the Dirichlet problem for the Monge–Ampère equation, and rooftop obstacle problems

Tamás Darvas, Yanir A. Rubinstein

2016 Volume 68 Issue 2 Pages 773-796
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820773

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First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data. This yields a new relation between solutions of the Dirichlet problem for the homogeneous real and complex Monge–Ampère equations and Kiselman's minimum principle. More generally, it establishes partial regularity for a Bremermann envelope whether or not it solves the Monge–Ampère equation. Second, we prove the second order regularity of the solution of the free-boundary problem for the Laplace equation with a rooftop obstacle, based on a new a priori estimate on the size of balls that lie above the non-contact set. As an application, we prove that convex- and plurisubharmonic-envelopes of rooftop obstacles have bounded second derivatives.

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Coefficient multipliers of H¹ into ℓ^q associated with Laguerre expansions

Yehao Shi, Zhongkai Li

2016 Volume 68 Issue 2 Pages 797-805
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820797

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The purpose of the paper is to study coefficient multipliers of the Hardy space H¹([0,∞)) associated with Laguerre expansions. As a consequence, a Paley type inequality is obtained.

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Zeta functions of 𝔽₁-buildings

Anton Deitmar, Ming-Hsuan Kang

2016 Volume 68 Issue 2 Pages 807-822
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820807

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The analogue of the Bruhat–Tits building of a p-adic group in F1-geometry is a single apartment. In this setting, the trace formula gives rise to a several variable zeta function analogously to the p-adic case. The analogy carries on to the fact that the restriction to certain lines yield zeta functions which are defined in geometrical terms. Also, the classical formula of Ihara has an analogue in this setting.

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On the blow-analytic equivalence of tribranched plane curves

Cristina Valle

2016 Volume 68 Issue 2 Pages 823-838
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820823

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We prove the finiteness of the number of blow-analytic equivalence classes of embedded plane curve germs for any fixed number of branches and for any fixed value of μ′ —a combinatorial invariant coming from the dual graphs of good resolutions of embedded plane curve singularities. In order to do so, we develop the concept of standard form of a dual graph. We show that, fixed μ′ in ℕ, there are only a finite number of standard forms, and to each one of them correspond a finite number of blow-analytic equivalence classes. In the tribranched case, we are able to give an explicit upper bound to the number of graph standard forms. For μ′ ≤ 2, we also provide a complete list of standard forms.

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Special Lagrangian submanifolds invariant under the isotropy action of symmetric spaces of rank two

Kaname Hashimoto, Katsuya Mashimo

2016 Volume 68 Issue 2 Pages 839-862
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820839

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We study special Lagrangian submanifolds of the cotangent bundle T^*Sⁿ of the sphere in the tangent space of Riemannian symmetric space of rank two. We show that the special Lagrangian submanifolds correspond to the solution of a differential equation on ℝ² under the assumption that the submanifold is of cohomogeneity one. Our result is the generalization of the former work of Sakai and the first author [5]. We study the qualitative properties of the solution for the special Lagrangian submanifolds and give some examples.

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Classification of conformal minimal immersions of constant curvature from S² to Q₃

Mingyan Li, Xiaoxiang Jiao, Ling He

2016 Volume 68 Issue 2 Pages 863-883
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820863

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In this paper, we study geometry of conformal minimal two-spheres immersed in complex hyperquadric Q₃. We firstly use Bahy-El-Dien and Wood's results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from S² to G(2,5;ℝ). Then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from S² to G(2,5;ℝ), or equivalently, a complex hyperquadric Q₃.

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Hydrodynamic limit for a certain class of two-species zero-range processes

Kenkichi Tsunoda

2016 Volume 68 Issue 2 Pages 885-898
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820885

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Großkinsky and Spohn [5] studied several-species zero-range processes and gave a necessary and sufficient condition for translation invariant measures to be invariant under such processes. Based on this result, they investigated the hydrodynamic limit. In this paper, we consider a certain class of two-species zero-range processes which are outside of the family treated by Großkinsky and Spohn. We prove a homogenization property for a tagged particle and apply it to derive the hydrodynamic limit under the diffusive scaling.

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On the class number divisibility of pairs of quadratic fields obtained from points on elliptic curves

Yoshichika Iizuka, Yutaka Konomi, Shin Nakano

2016 Volume 68 Issue 2 Pages 899-915
Published: 2016
Released on J-STAGE: May 03, 2016

DOIhttps://doi.org/10.2969/jmsj/06820899

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Let l be the prime 3,5 or 7 and let m be a nonzero integer. We give a method for constructing an infinite family of pairs of quadratic fields ℚ(√D) and ℚ(√(mD)) with both class numbers divisible by l. Such quadratic fields are parametrized by rational points on a specified elliptic curve.

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