In this paper, we first prove that every Mori dream surface X satisfies the bounded cohomology property (BCP). Namely, there exists a constant c(X) > 0 such that h(1)(& Oscr; (X)(C)) <= c(X)h(0)(& Oscr; X(C)) for every curve C on X. We then prove that there is a positive constant m(Y ) such that l(C) := (KY & sdot; C)(C-2)(-1) <= m(Y ) for every ample curve C on a geometrically ruled surface Y over a curve of genus g, and Y satisfies the BCP if g <= 1.
We study the Cauchy problem for the modified Camassa-Holm positive flow with step-like initial data. Using the Riemann-Hilbert approach, we first establish the basic Riemann-Hilbert problem, which involves three distinct forms of jump matrices defined on corresponding intervals of the real axis. The reconstruction formula for the solution is then derived by analyzing the behavior of the eigenfunctions at λ = 0 and λ = ∞. Based on the nonlinear steepest descent method, we deform the basic Riemann-Hilbert problem and isolate the parts that make the dominant contribution to the solution through rigorous error estimation. Finally, the long-time asymptotic behavior of the solution is obtained.
A symplectic Lie algebra (& gfr;, [& sdot;,& sdot;],omega) is called Ricci-flat if kappa(g)(x,y) = omega([u(g),x],y), where kappa(g )denotes its Killing form and u(g) its unimodularity vector defined by omega(u(g),x) = tr(ad x). A symplectic Lie algebra (& gfr;, [& sdot;,& sdot;],omega) is called flat if its natural symplectic product, i.e. 3 omega(x star y,z) := omega([x,y],z) + omega([x,z],y) is left-symmetric. In this paper, we investigate the Ricci-flatness property of two important classes of symplectic Lie algebras: almost-abelian symplectic Lie algebras and Novikov symplectic Lie algebras. We first prove that an almost-abelian or Novikov symplectic Lie algebra is flat if and only if it is 2-step nilpotent. Then we provide necessary and sufficient conditions for almost-abelian and Novikov symplectic Lie algebras to be Ricci-flat. Finally, we characterize Ricci-flat symplectic Lie algebras that are both almost-abelian and Novikov by proving that they are precisely the flat ones.
Lee and the second named author studied involutions on smooth minimal surfaces S of general type with p(g)(S) = 0 and K-S(2) = 7. They gave the possibilities of the birational models W of the quotients and the branch divisors B0 induced by involutions sigma on the surfaces S. In this paper, we improve and refine the results of Lee and the second named author. We exclude the case of the Kodaira dimension kappa(W) = 1 when the number k of isolated fixed points of an involution sigma on S is nine. The possibilities of branch divisors B-0 are reduced for the case k = 9, and are newly given for the case k = 11. Moreover, we show that if the branch divisor B-0 has three irreducible components, then S is an Inoue surface.
