Showing papers in "Tokyo Journal of Mathematics in 1998"

The Maslov Index: a Functional Analytical Definition and the Spectral Flow Formula

Abstract: We give a functional analytical definition of the Maslov index for continuous curves in the Fredholm-Lagrangian Grassmannian. Our definition does not require assumptions either at the endpoints or at the crossings of the curve with the Maslov cycle. We demonstrate an application of our definition by developing the symplectic geometry of self-adjoint extensions of unbounded symmetric operators. We discuss continuous variations of the form $A_D+C_t$, where $A_D$ is a fixed self-adjoint unbounded Fredholm operator and $\{C_t\}$ a family of bounded self-adjoint operators. We extend the definition of the spectral flow to such families of unbounded operators in a purely functional analytical way. We then prove that the spectral flow is equal to the Maslov index of the corresponding family of abstract Cauchy data spaces.

Timelike Surfaces of Constant Mean Curvature in Minkowski 3-Space

Abstract: Since the discovery of constant mean curvature tori in Euclidean 3-space, methods in the theory of integrable systems (or soliton theory) have been applied frequently in differential geometry. In particular, surfaces of constant mean or Gaussian curvature in Euclidean 3-space have been studied extensively [2], [3]. On the other hand, the geometry of surfaces in Minkowski 3-space has been a subject ofwide interest [9], [10]. For example, a Kenmotsu-type representation formula for spacelike surfaces with prescribed mean curvature has been obtained by K. Akutagawa and S. Nishikawa, and M. A. Magid [9] has obtained such a representation formula for timelike surfaces. L. McNertney studied spacelike maximal surfaces, timelike extremal surfaces and timelike surfaces of constant positive Gaussian curvature by classical methods [10]. In our previous paper [5], we have studied spacelike surfaces with constant mean or Gaussian curvature in Minkowski 3-space via the theory of finite-type harmonic maps. To adapt the methods in the theory of integrable systems for our purposes, namely to study the geometry of surfaces, we reformulated the fundamental equations of spacelike surfaces. This allowed us to considerably simplify the computations in the study of such spacelike surfaces. The main tool of this reformulation was our use of split-quaternion numbers. In addition we obtained representation formulae for immersions in terms of loop group theory [5]. The purpose of this paper is to develop a corresponding setting for timelike surfaces of constant mean curvature in Minkowski 3-space. In contrast to the case of spacelike surfaces or timelike extremal surfaces, no systematic theory exists for timelike constant (nonzero) mean curvature surfaces. For this reason, we shall devote Section 1 to preliminary materials. The reformulation of the fundamental equations will be carried out in Section 2. As a result we obtain a

Complex Multiplication Formulae for Hyperelliptic Curves of Genus Three

Abstract: p.387, `.5, “[1]” should be “[2]”. p.389, `.−12, “ d dt`+1 ” should be “ d ` dt` ”. p.389, `.−9, “(i, n1 + · · · + nj−1 + `)-entry” should be “(n1 + · · · + nj−1 + `, i)-entry with 1 ≤ ` 5 nj”. p.389, `.−4, “0 5 ` 5 n” should be “1 5 ` 5 nj − 1”. p.389, `.−2, “Γ(C,O(D))” should be “Γ(C,Ω1(−D))”. p.390, `.8, “non-singular” should be “singular” and “(0, · · · , 0)” should be “(0, 0, 0)”. p.390, `.−5, “−λ2u1u2” should be “−2 2 u1u2”. p.390, `.−5, “−3 3 u1u3” should be “−3 3 u1u2”. p.390, `.−4, “− 2λ2 3 u1u3” should be “+ λ2 6 u1u3”. p.391, `.−9, “ ∂ ∂u4 2 ” should be “ ∂ 4 ∂u4 3 ”.

The Relationship between Entropy and Strong Orbit Equivalence for the Minimal Homeomorphisms (II)

Abstract: Every minimal homeomorphism of a Cantor set is strongly orbit equivalent to a homeomorphism of infinite entropy.

On a Variety of Algebraic Minimal Surfaces in Euclidean 4-Space

Abstract: In this paper, we show that the moduli space of the Weierstrass data for algebraic minimal surfaces in Euclidean 4-space with fixed topological type, orders of branched points and ends, and total curvature, has the structure of a real analytic variety. We provide the lower bounds of its dimension. We also show that the moduli space of the Weierstrass data for stable algebraic minimal surfaces in Euclidean 4-space has the structure of a complex analytic variety.

Fredholm Determinant for Piecewise Linear Transformations on a Plane

Abstract: Certain piecewise linear expanding maps on a finite union of polygons in $\mathbf{R}^2$ are considered. The Perron-Frobenius operator associated with a map is considered on a locally convex linear space which is an extension of the space of bounded variation functions, and the spectrum of it is determined by Fredholm matrices. New signed symbolic dynamics are defined by using screens, and the Fredholm matrices are constructed by renewal equations on this signed symbolic dynamics.

A Construction of Everywhere Good $\mathbf{Q}$-Curves with $p$-Isogeny

Abstract: An elliptic curve $E$ defined over $\bar{\mathbf{Q}}$ is called a $\mathbf{Q}$-curve, if $E$ and $E^\sigma$ are isogenous over $\bar{\mathbf{Q}}$ for any $\sigma$ in $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$. For a real quadratic field $K$ and a prime number $p$, we consider a $\mathbf{Q}$-curve $E$ with the following properties: 1) $E$ is defined over $K$, 2) $E$ has everywhere good reduction over $K$, 3) there exists a $p$-isogeny between $E$ and its conjugate $E^\sigma$. In this paper, a method to construct such a $\mathbf{Q}$-curve $E$ for some $p$ will be given.

A Degree Condition for the Existence of $[a,b]$-Factors in $K_{1,n}$-Free Graphs

Abstract: A graph is called $K_{1,n}$-free if it contains no $K_{1,n}$ as an induced subgraph. Let $a$, $b$ $(0\leq a

Tangle Decompositions of Doubled Knots

Abstract: We show that a decomposing 2-sphere which gives an essential tangle decomposition of a doubled knot gives also an essential tangle decomposition of its companion knot.