Consider a relatively compact region Ω of a Riemann surface R. The term Dirichlet integral over Ω, DΩ(·), is used for the variation whose Euler-Lagrange equation is Δu = 0 on Ω and the term energy integral over Ω, , is used for the variation with Euler-Lagrange equation

The purpose of this paper is to study the extension problem of holomorphic maps of a complex manifold into a taut complex space, which is defined by analogy with a taut complex manifold given by H. Wu ([11]).

Let X(t), t≥0, be a real Gaussian process with mean 0, stationary increments, and a2(t) = E|X(t) - X(0)|2. Here dH(λ), for some bounded monotone H. We summarize the main results.

The main purpose of this paper is to derive conclusions about the structure of a nilpotent group of algebra automorphisms and, in the case of a Lie algebra, about the influence of this nilpotence on the structure of the algebra. A motivation for this study is a well known theorem due to Kolchin: A unipotent linear group can be triangularized and is thus nilpotent. The converse is manifestly false, but we have (as an immediate consequence of Theorem 2.7):

Let k denote the quotient field of a complete discrete rank one valuation ring R of unequal characteristic and let p denote the characteristic of R̅; assume that R contains a primitive pth root of unity, so that the absolute ramification index e of R is a multiple of p — 1, and each Gallois extension K ⊃ k of degree p may be obtained by the adjunction of a pth root.

1. By the method of Dolzhenko’s paper, we studied relations between non-tangetial (angular) boundary behaviors and horocyclic boundary behaviors of arbitrary functions defined in the open unit disc of the complex plane in [8]. Vessey [5], [6] investigated the behavior of arbitrary functions on paths which are “more tangential” than horocycles. The purpose of the present paper is to prove the fact that is sharper than the results in Vessey [5], [6], and generalize the results in [8] to obtain the connection between behaviors on two “more tangential” angles.

Let G be a reductive algebraic Lie group acting linearly on a (finite-dimensional) real vector-space U with a maximal compact isotropy subgroup K and suppose that the quotient Ω = G/K is a self-dual homogeneous cone in U. Let (G′, K′) be another such pair corresponding to a self-dual homogeneous cone Ω′ in U′.

Let R be a commutative ring and K be the total quotient ring of R. Let Σ be a separable K-algebra which is a finitely generated projective, faithful K-module and Λ be an R-order in DΛ/R. We denote by DΛ/R the Dedekind different of Λ and by NΛ/R the Noetherian different of Λ.

In a recent paper, Sato [6] has shown that for every Gaussian measure n on a real separable or reflexive Banach space (X, ‖ • ‖) there exists a separable closed sub-space X〵 of X such that and is the σ-extension of the canonical Gaussian cylinder measure of a real separable Hilbert space such that the norm is contiunous on and is dense in The main purpose of this note is to prove that ‖ • ‖ x〵 is measurable (and not merely continuous) on .

The main purpose of this note is to characterize a compact Káhler-Einstein manifold in terms of curvature form. The curvature form Q is an EndT valued differential form of type (1,1) which represents the curvature class of the manifold. We shall prove that the curvature form of a Káhler metric is the harmonic representative of the curvature class if and only if the Káhler metric is an Einstein metric in the generalized sense (g.s.), that is, if the Ricci form of the metric is parallel. It is well known that a Káhler metric is an Einstein metric in the g. s. if and only if it is locally product (globally, if the manifold is simply connected and complete) of Kàhler-Einstein metrics. We obtain an integral formula, involving the integral of the trace of some operators defined by the curvature tensor, which measures the deviation of a Káhler-Einstein metric from a Hermitian symmetric metric. In the final section we shall prove the uniqueness up to equivalence of Kãhler-Einstein metrics in a simply connected compact complex homogeneous space. This result was proved by Berger in the case of a complex projective space and our proof is completely different from Berger’s.