Suppose that $[\mu {]}_{T(\Delta )}$ is a point of the universal Teichmüller space $T(\Delta )$. In 1998, Božin, Lakic, Marković, and Mateljević showed that there exists $\mu$ such that $\mu$ is uniquely extremal in $[\mu {]}_{T(\Delta )}$ and has a nonconstant modulus. It is a natural problem whether there is always an extremal Beltrami coefficient of constant modulus in $[\mu {]}_{T(\Delta )}$ if $[\mu {]}_{T(\Delta )}$ admits infinitely many extremal Beltrami coefficients; the purpose of this paper is to show that the answer is negative. An infinitesimal version is also obtained. Extremal sets of extremal Beltrami coefficients are considered, and an open problem is proposed. The key tool of our argument is Reich’s construction theorem.

We give an explicit construction of the Ree groups of type $G_2$ as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject (see [7] and [8]). We also give a short proof that the norm of a Ree group is anisotropic.

In this paper we consider the birational classification of pairs $\left(S,\mathcal{L}\right)$, with $S$ a rational surface and $\mathcal{L}$ a linear system on $S$. We give a classification theorem for such pairs, and we determine, for each irreducible plane curve $B$, its Cremona minimal models, that is, those plane curves which are equivalent to $B$ via a Cremona transformation and have minimal degree under this condition.

Let $\left(A,\mathfrak{m}\right)$ be a Noetherian local ring with $d=dimA\ge 2$. Then, if $A$ is a Buchsbaum ring, the first Hilbert coefficients ${\mathrm{e}}_Q^1\left(A\right)$ of $A$ for parameter ideals $Q$ are constant and equal to $-\sum _{i=1}^{d-1}\left(\genfrac{}{}{0ex}{}{d-2}{i-1}\right)h^i\left(A\right)$, where $h^i\left(A\right)$ denotes the length of the ith local cohomology module ${\mathrm{H}}_{\mathfrak{m}}^i\left(A\right)$ of $A$ with respect to the maximal ideal $\mathfrak{m}$. This paper studies the question of whether the converse of the assertion holds true, and proves that $A$ is a Buchsbaum ring if $A$ is unmixed and the values ${\mathrm{e}}_Q^1\left(A\right)$ are constant, which are independent of the choice of parameter ideals $Q$ in $A$. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.

In this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphism $f:X\to S$ with connected fibers such that a general fiber has an ample canonical bundle, and for a positive integer $m$, we construct a canonical singular Hermitian metric $h_{E,m}$ on $f_{\ast }{\mathcal{O}}_X\left(m{K}_{X/S}\right)$ with semipositive curvature in the sense of Nakano.

By a theorem of Wahl, for canonically embedded curves which are hyperplane sections of K3 surfaces, the first Gaussian map is not surjective. In this paper we prove that if $C$ is a general hyperplane section of high genus (> 280) of a general polarized K3 surface, then the second Gaussian map of $C$ is surjective. The resulting bound for the genus $g$ of a general curve with surjective second Gaussian map is decreased to $g>152$.

In this paper we consider complete, noncompact, negatively curved surfaces that are twice continuously differentiably embedded in Euclidean three-space, showing that if such surfaces have square integrable second fundamental form, then their topology must, by the index method, be an annulus. We then show how this relates to some minimal surface theorems and has a corollary on minimal surfaces with finite total curvature. In addition, we discuss, by the index method, the relation between the topology and asymptotic curves. Finally, we apply the results yielded to the problem of isometrical immersions into Euclidean three-space of black hole models.

In this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the book Supersonic Flow and Shock Waves by Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed.