The Bando-Calabi-Futaki character of a compact Kähler manifold is an obstruction to the existence of Kähler metrics with constant scalar curvature, which is a generalization of the Futaki character of a Fano manifold. In this paper, we study the Bando-Calabi-Futaki character of a compact toric manifold. In particular, we shall prove that the Bando-Calabi-Futaki character of a compact toric manifold vanishes on the Lie algebra of the unipotent radical of the automorphism group.

A hyperelliptic variety is by definition a complex projective variety, not isomorphic to an abelian variety, which admits an abelian variety as a finite étale covering. The main contribution of this paper is a classification of hyperelliptic threefolds.

We study affine immersions, as introduced by Nomizu and Pinkall, of $M^n$ into $\R^{n+p}$. We call $M^n$ linearly full if the image of $M$ is not contained in a lower dimensional affine space. Typical examples of affine immersions are the Euclidean and semi-Riemannian immersions. A classification, under an additional assumption that the rank of the second fundamental form is at least two, of the hypersurfaces with parallel second fundamental form was obtained by Nomizu and Pinkall. If we assume that the second fundamental form is parallel and $M$ is linearly full, then $p \le n(n+1)/2$. In this paper we completely classify the affine immersions with parallel second fundamental form in $\R^{n+n(n+1)/2}$, obtaining amongst others the generalized Veronese immersions.

The geometric local monodromy of a plane curve singularity is a diffeomorphism of a compact oriented surface with non empty boundary. The monodromy diffeomorphism is a product of right Dehn twists, where the number of factors is equal to the rank of the first homology of the surface. The core curves of the Dehn twists are quadratic vanishing cycles of the singularity. Moreover, the monodromy diffeomorphism decomposes along reduction curves into pieces, which are invariant, such that the restriction of the monodromy on each piece is isotopic to a diffeomorphism of finite order. In this paper we determine the mutual positions of the core curves of the Dehn twists, which appear in the decomposition of the monodromy, together with the positions of the reduction curves of the monodromy.

Let $x:\M^m\to \hs^n$ be a submanifold in the $n$-dimensional sphere $\hs^n$ without umbilics. Two basic invariants of $x$ under the Möbius transformation group in $\hs^n$ are a 1-form $\F$ called the Möbius form and a symmetric $(0,2)$ tensor $\bf A$ called the Blaschke tensor. $x$ is said to be Möbius isotropic in $\hs^n$ if $\F\equiv 0$ and ${\bf A}=\l dx\cdot dx$ for some smooth function $\l$. An interesting property for a Möbius isotropic submanifold is that its conformal Gauss map is harmonic. The main result in this paper is the classification of Möbius isotropic submanifolds in $\hs^n$. We show that (i) if $\l >0$, then $x$ is Möbius equivalent to a minimal submanifold with constant scalar curvature in $\hs^n$; (ii) if $\l=0$, then $x$ is Möbius equivalent to the pre-image of a stereographic projection of a minimal submanifold with constant scalar curvature in the $n$-dimensional Euclidean space $\R^n$; (iii) if $\l <0$, then $x$ is Möbius equivalent to the image of the standard conformal map $\t: \text{\bi H}^n\to \hs^n_+$ of a minimal submanifold with constant scalar curvature in the $n$-dimensional hyperbolic space $\text{\bi H}^n$. This result shows that one can use Möbius differential geometry to unify the three different classes of minimal submanifolds with constant scalar curvature in $\hs^n$, $\R^n$ and $\text{\bi H}^n$.

We study in this paper the positive solution of the Cauchy problem for a fast diffusion equation with source. We derive a secondary critical exponent of the behavior of the initial value at infinity for the existence of global (in time) and nonglobal solutions of the Cauchy problem. Furthermore, the large time behaviors of those global solutions are also studied.

We verify a special case of V. V. Shokurov's conjecture about characterization of toric varieties. More precisely, we consider three-dimensional log varieties with only purely log terminal singularities and numerically trivial log canonical divisor. In this situation we prove an inequality connecting the rank of the group of Weil divisors modulo algebraic equivalence and the sum of coefficients of the boundary. We describe such varieties for which the equality holds and show that all of them are toric.

Representations of arbitrary real or complex invertible matrices as products of matrices of special type have been used for many purposes. The matrix form of the Gram-Schmidt orthonormalization procedure and the Gauss elimination process are instances of such matrix factorizations. For arbitrary, finite-dimensional, semisimple Lie groups, the corresponding matrix factorizations are known as Iwasawa decomposition and Bruhat decomposition. The work of Matsuki and Rossmann has generalized the Iwasawa decomposition for the finite-dimensional, semisimple Lie groups. In infinite dimensions, for affine loop groups/Kac-Moody groups, the Bruhat decomposition has an, also classical, competitor, the Birkhoff decomposition. Both decompositions (in infinite dimensions), the Iwasawa decomposition and the Birkhoff decomposition, have had important applications to analysis, e.g., to the Riemann-Hilbert problem, and to geometry, like to the construction of harmonic maps from Riemann surfaces to compact symmetric spaces and compact Lie groups. The Matsuki/Rossmann decomposition has been generalized only very recently to untwisted affine loop groups by Kellersch and facilitates the discussion of harmonic maps from Riemann surfaces to semisimple symmetric spaces.

In the present paper we extend the decompositions of Kellersch and Birkhoff for untwisted affine loop groups to general Lie groups. These generalized decompositions have already been used in the discussion of harmonic maps from Riemann surfaces to arbitrary loop groups.

In this paper we prove the existence of families of complete mean curvature one surfaces in the hyperbolic three-space. We show that for each Costa-Hoffman-Meeks embedded minimal surface of positive genus in Euclidean three-space, we can produce, by cousin correspondence, a family of complete mean curvature one surfaces in the hyperbolic three-space. These surfaces have positive genus, three ends and the same group of symmetry of the original minimal surfaces. Furthermore, two of the ends approach the same point in the ideal boundary of hyperbolic three-space and the third end is asymptotic to a horosphere. The method we use to produce these results were developed in a recent paper by W. Rossman, M. Umehara and K. Yamada.