Abstract. The present paper is concerned with extensions of complex analysis on the complex plane C to conformally flat 4-manifolds. We shall give in an explicit form a fundamental system of spinors that will serve as the basis vectors for the Laurent expansion. Restricted to a sphere around the center of the expansion these spinors form a complete orthonormal system of eigenspinors of the tangential Dirac operator on the sphere, and give a basis of the representations of Spin (4). We shall also give the definition of meromorphic spinors and residues, and prove under some hypothesis that, on a compact conformally flat 4-manifold, the sum of the residues of a meromorphic spinor is zero.

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In this paper we consider the n dimensional Navier-Stokes equations and we prove a new regularity criterion for weak solutions. More precisely, if n=3, 4 we show that the “smallness” of at least n-1 components of the velocity in L^∞ (O, T; Lⁿ_w (Rⁿ)) is sufficient to ensure regularity of the weak solutions.

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For a given symmetric space of compact type, it is known that a certain Riemannian submersion of a Hilbert space onto the symmetric space is nat-urally defined. In this paper, we describe the principal curvatures and the principal distributions of the inverse image (which becomes a proper Fredholm submanifold) of a curvature adapted submanifold in the symmetric space under this Riemannian submer-sion. The curvature adapted submanifold is minimal if and only if the inverse image is formally minimal in certain sense.

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The pre-Tango structure is an invertible subsheaf of the sheaf of locally exact differentials on a curve. Some of them induce surfaces having non-closed global differential 1-forms. In the present article, we show that, in characteristic two, there exists such a pre-Tango structure on every curve, even ordinary one, of genus greater than 11.

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In this paper, we first present the complete list of the singularity types of the Picard number one Gorenstein log del Pezzo surface and the number of the isomorphism classes with the given singularity type. Then we give out a method to find out all singularity types of Gorenstein log del Pezzo surface. As an application, we present the complete list of the Dynkin type of the Picard number two Gorenstein log del Pezzo surfaces. Finally we present the complete list of the singularity type of the relatively minimal Gorenstein log del Pezzo surface and the number of the isomorphism classes with the given singularity type.

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Abstract. In the theory of deformation of Okamoto-Painleve pair (S, Y), a local cohomology group H¹_D (_??__s (-log D)) plays an important role. In this paper, we estimate the local cohomology group of pair (S, Y) for several types, and obtain the following results. For a pair (S, Y) corresponding to the space of initial conditions of the Painlevé equations, we show that the local cohomology group H¹_D (_??__s (-log D)) is at least 1 dimensional. This fact is the key for understanding Painlevé equation related to (S, Y). Moreover we show that, for the pairs (S, Y) of type Ã₈, the local cohomology group H¹_D (_??_S (-log D)) vanishes. Therefore in this case, there is no differential equation on S-Y in the sense of the theory.

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