Motivated by the superlinear behavior of the Barzilai-Borwein (BB) method for two-dimensional quadratics, we propose two gradient methods which adaptively choose a small step-size or a large step-size at each iteration. The small step-size is primarily used to induce a favorable descent direction for the next iteration, while the large step-size is primarily used to produce a sufficient reduction. Although the new algorithms are still linearly convergent in the quadratic case, numerical experiments on some typical test problems indicate that they compare favorably with the BB method and some other efficient gradient methods.

In this paper, we give a sufficient condition for both the relative K-stability and the properness of modified K-energy associated to Calabi's extremal metrics on toric manifolds. In addition, several examples of toric manifolds which satisfy the sufficient condition are presented.

In this paper we prove the existence and regularity of solutions to the first boundary value problem for Abreu's equation, which is a fourth order nonlinear partial differential equation closely related to the Monge-Ampère equation. The first boundary value problem can be formulated as a variational problem for the energy functional. The existence and uniqueness of maximizers can be obtained by the concavity of the functional. The main ingredients of the paper are the a priori estimates and an approximation result, which enable us to prove that the maximizer is smooth in dimension 2.1991 Mathematics Subject Classification. 35J60.

Abstract. In this note, we prove that on polarized toric manifolds the relative K-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and we also show that the modified K-energy is proper in the space of G 0 -invariant Kähler potentials in the case of toric surfaces which admit the extremal metrics. IntroductionAround the existence of Calabi's extremal metrics, there is a well-known conjecture (cf. For the "only if" part of this conjecture, the first breakthrough was made by Tian [Ti]. By introducing the concept of K-stability, he gave an answer to the "only if" part for Kähler-Einstein manifolds. Remarkable progress was made by Donaldson, who showed the Chow-Mumford stability for a polarized Kähler manifold with constant scalar curvature when the holomorphic automorphisms group Aut(M ) of M is finite [D1]. Donaldson's result was later generalized by Mabuchi to any polarized Kähler manifold M which admits an extremal metric without any assumption for Aut(M ) [M1], [M2]. However, it is still unknown whether there is a generalization of Tian's result for the K-stability on the Kähler-Einstein manifolds analogous to Donaldson-Mabuchi's result for the Chow-Mumford stability.As we know, the definition of K-stability on a polarization Kähler manifold is associated to degenerations (or test configurations in the sense of Donaldson [D2]) on the underlying manifold. In order to study the relation between the K-stability and K-energy on a polarized toric manifold, Donaldson in [D2] introduced a class of special degenerations induced by rational, piecewise linear functions, called toric degenerations. He also proved that for the surfaces' case the K-energy is bounded from below in the space of G 0 -invariant Kähler potentials under the assumption of K-stability for any toric degeneration, where G 0 is a maximal compact torus group. In this note we focus on polarized toric manifolds as in [D2], and we shall give an