A family of S¹-equivariant hypersurfaces of constant mean curvature can be obtained by using the Lagrangians with suitable potentials in the unit 3-sphere equipped with a certain parameterized metric. The conservation law is effectively applied to the construction of S¹-equivariant hypersurfaces of constant mean curvature.

Kada proved in a previous paper (Topology Appl., 2009) that the collection of compatible metrics on a locally compact separable metrizable space has the same cofinal type, in the sense of Tukey relation, as the set of functions from ω to ω with respect to eventually dominating order. By generalizing this result, we characterize the order structure of the collection of compatible metrics on a separable metrizable space in terms of generalized Galois-Tukey connection.

Let X be a proper metric space. The sublinear Higson compactification h_LX is a variant of the Higson compactification. Its boundary h_LX\X is denoted ν_LX, and is called the sublinear Higson corona of X. The sublinear Higson corona is a functor from the category of coarse spaces to that of compact Hausdorff spaces. Let P be a compact metric space and X be an unbounded proper metric space. We show that the sublinear Higson corona of a product space P × X equipped with a cone metric is homeomorphic to a product P × ν_LX. Especially, the sublinear Higson corona of the n-dimensional Euclidean space is homeomorphic to the product of an (n − 1)-dimensional sphere and the sublinear Higson corona of natural numbers.

We determine the group presentations of universal central extensions derived from loop groups, where loop groups are Chevalley groups over Laurent polynomial rings. We also show that the universal central extensions have Tits systems. For our purpose, we introduce additive Steinberg symbols.

We classify pseudo-parallel proper CR submanifolds of a non-flat complex space form with semi-flat normal connection under the condition that the dimension of the holomorphic tangent space is greater than 2.

We give an efficient bimodule projective resolution of a maximal Z-order Λ of the ordinary quaternion algebra over Q, and therefore we determine the ring structure of the Hochschild cohomology of Λ by calculating the Yoneda products using this resolution.

Replacing the integrand of the Yang-Mills functional by F (||R^∇||²/2), we define an F-Yang-Mills functional, and hence F-Yang-Mills fields, where F is a non-negative function. The gaps of F-Yang-Mills fields on some submanifolds of the Euclidean spaces and the spheres are investegated in this paper.

In this paper, we give some results about a classification of reductive prehomogeneous vector spaces with two irreducible components. In particular, we give the complete classification of 3-simple prehomogeneous vector spaces with two irreducible components. We consider everything over the complex number field C.