Norihito Koiso

On Rotationally Symmetric Hamilton's Equation for Kähler-Einstein Metrics

Publisher Summary This chapter focuses on rotationally symmetric Hamiltons equation for Kahler–Einstein metric. Any Riemannian metric g0 with positive Ricci curvature on a compact three-dimensional manifold is deformed to an Einstein metric along the equation , where rt denotes the Ricci tensor of gt and the mean value of the scalar curvature. The chapter explains how the solution of an equation converges to a Kahler–Einstein metric if it exists, even on a compact Kahler manifold with positive first Chern class.

Elasticae in a Riemannian submanifold

For a curve 7(s) in a riemannian manifold M we define two quantities: the length L(γ) and the total square curvature E(<γ). A curve γ is called an elastica if it is a critical point of the functional E restricted to the space of curves of a fixed Inegth L0. The notion of elastica is quite old. But modern approaches to it in differential geometry are rather new. J. Langer and D.A. Singer classified all closed elasticae in the euclidean space ([!]), and showed that Palais-Smales condition (C) holds for the space of curves in a riemannian manifold ([2]). In this paper we consider elasticae restricted in a submanifold. For example, let M be a compact surface of the euclidean space and C the set of all closed curves of given length in the surface. Is there a closed curve in C which minimizes the elastic energy E (defined as curves of the euclidean space) ? We will affirmatively answer to the question in a more general situation.

The vortex filament equation and a semilinear Schrödinger equation in a hermitian symmetric space

In this paper, we will prove the existence and the uniqueness of a classical solution for the initial value problem, and generalize it to the case of curves in 3-dimensional space forms. We also consider related semilinear Schrodinger equations for curves in Kahler manifolds. It is remarkable that we need symmetric spaces as manifolds for infinite time existence of solutions. More precisely, we will get the following results.

Uniqueness and non-existence of metrics with prescribed Ricci curvature

Abstract We investigate whether the Ricci tensor uniquely determines the Riemannian structure, and we give conditions that a doubly covariant tensor has to satisfy in order to be the Ricci tensor for some Riemannian structure.

Long time existence for vortex filament equation in a Riemannian manifold

Vortex filament equation in the Euclidean space has a long time solution for any closed initial data, because it can be converted into a standard nonlinear Schrodinger equation. While the Riemannian version of vortex filament equation is not integrable at all, we prove that it has a long time solution for any closed initial data.

On singular perturbation of a semilinear hyperbolic equation

AbstractWe consider a hyperbolic version of Eells-Sampsons equation:

ON MOTION OF AN ELASTIC WIRE IN A RIEMANNIAN MANIFOLD AND SINGULAR PERTURBATION

D_\tau u_\tau + \mu u_\tau + \tilde \Delta u = 0

Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds. II

. This equation is semilinear with respect to space derivative and time derivative. Letuμ(τx) be the solution with initial data u(0) andτμ(0), and putvμ(t,x)=uμ(μt,x). We show that when the resistance μ → ∞,Vμ(t,x) converges to a solution of the original parabolic Eells-Sampsons equation: