Take e.g. two disjoint nondegenerate compact continua A and B in the complex plane C with connected complements and pick a simple arc γ in the complex sphere Ĉ disjoint from A ∪ B, which we call a pasting arc for A and B. Construct a covering Riemann surface Ĉ_γ over Ĉ by pasting two copies of $\hat{\mathbf C}\setminus\gamma$ crosswise along γ. We embed A in one sheet and B in another sheet of two sheets of Ĉ_γ which are copies of $\hat{\mathbf C}\setminus\gamma$ so that $\hat{\mathbf C}_{\gamma}\setminus A \cup B$ is understood as being obtained by pasting ($\hat{\mathbf C}\setminus A)\setminus \gamma$ with ($\hat{\mathbf C}\setminus B)\setminus \gamma$ crosswise along γ. In the comparison of the variational 2 capacity cup(A,$\hat{\mathbf C}_{\gamma}\setminus B$) of the compact set A considered in the open set $\hat{\mathbf C}_{\gamma}\setminus B$ with the corresponding cap(A,$\hat{\mathbf C}\setminus B$), we say that the pasting arc γ for A and B is subcritical, critical, or supercritical according as cap(A,$\hat{\mathbf C}_{\gamma}\setminus B$) is less than, equal to, or greater than cap(A,$\hat{\mathbf C}\setminus B$), respectively. We have shown in our former paper [4] the existence of pasting arc γ of any one of the above three types but that of supercritical and critical type was only shown under the additional requirment on A and B that A and B are symmetric about a common straight line simultaneously. The purpose of the present paper is to show that in the above mentioned result the additional symmetry assumption is redundant: we will show the existence of supercritical and hence of critical arc γ starting from an arbitrarily given point in $\hat{\mathbf C}\setminus A \cup B$ for any general admissible pair of A and B without any further requirment whatsoever.

We calculate some invariants determined by the spectrum of the Jacobi operator J of totally complex submanifolds of the quaternionic projective space QPⁿ. We use these invariants, and the ones determined by the spectrum of the Laplace-Beltrami operator Δ, to characterize parallel submanifolds of QPⁿ.

In this paper, we compute L_r (S_r) for an isometric immersion x : Mⁿ → $\overline M^{n+1}_c$, from an n-dimensional Riemannian manifold Mⁿ into an (n+1)-dimensional Riemannian manifold $\overline M^{n+1}_c$, of constant sectional curvature c. Here, by L_r we mean the linearization of the second order differential operator associated to the (r+1)-th elementary symmetric function S_r+1 on the eigenvalues of the second fundamental form A of x. The resulting formulae are then applied to study how the behavior of higher-order mean curvature functions of Mⁿ influence its geometry.

We are interested in the stochastic property of some "Anosov-like" system. In this paper we will treat a transitive and partially hyperbolic diffeomorphism f of a 2-dimensional torus with uniformly contracting direction, and show that if f is of C² and admits an SRB measure, then f is an Anosov diffeomorphism. In our proof we use the Pujals-Sambarino theorem for C² diffeomorphisms with dominated splitting. In the case of C^1+α the above statement is not true in general, i.e. we can construct a C^1+α counter example of Maneville-Pomeau type.

Conformal measures are measures satisfying a certain transformation rule for elements of a Kleinian group G and are normally supported by the limit set of G. They are usually constructed by a method due to S. J. Patterson as weak limits of measures supported by a fixed orbit of G in the hyperbolic space, often identified with the unit ball Bⁿ. We call such limit measures Patterson measures. This has been the predominant way to obtain conformal measures and one may get the impression that all conformal measures are Patterson measures. We show in this note that this is not the case and two concrete examples are given in the last section.

This paper investigates the Castelnuovo-Mumford regularity of generic hyperplane section of projective curve. The classical method of Castelnuovo plays an important role in order to study the extremal examples for the bounds for the Castelnuovo-Mumford regularity.

We study the minimum moduli of weighted composition operators on the disk algebra and the space of bounded analytic functions.

In this paper, we investigate when a smooth complete toric variety of positive characteristic has a maximal wild hypersurface bundle over it. In particular, we determine the possibilities for toric varieties with Picard number at most three and for toric Fano varieties of dimension at most four. Moreover, we construct maximal wild hypersurface bundles over almost all of them.

In this paper, we introduce the definition of weakly weighted-sharing which is between "CM" and "IM". Using the notion of weakly weighted-sharing, we study the uniqueness problems on meromorphic function and its kth order derivative f^(k) satisfying certain sharing set properties. As consequences, we are able to answer questions posed by Kit-wing Yu, which were also studied by I. Lahiri and A. Sarkar, L. P. Liu and Y. X. Gu. Our results sharpen the above results.

A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kähler manifolds as well as Hermitian surfaces with vanishing associated conformal curvature tensor are characterized as locally conformal to a Kähler manifold of constant holomorphic sectional curvatures.

Some consequences of equality in the Castelnuovo-Severi inequality are discussed. In particular, it is shown that if a Riemann surface of genus ten, W₁₀, admits four coverings of tori, each in three sheets, then W₁₀ admits an elementary abelian group of order 27. By previous work this last result is then characterized by the vanishing of certain thetanulls. An elementary discussion of the direct product of monodromy groups is an essential part of the proofs.