Related papers: A short proof of Hara and Nakai's theorem

In this paper continuing our work started in our earlier papers we prove the corona theorem for the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of finite type.

By modifying Cole's example, we construct explicit Riemann surfaces with large bounds on corona solutions in an elementary way.

In this paper, we prove an upper bound on the $\widehat{A}$ genus of a smooth, closed, spin Riemannian manifold using its scalar curvature lower bound, Neumann isoperimetric constant, and volume. The proof of this result relies on spectral…

We prove the corona theorem for domains whose boundary lies in certain smooth quasicircles. These curves, which are not necessarily Dini-smooth, are defined by quasiconformal mappings whose complex dilatation verifies certain conditions.…

Assuming the Riemann hypothesis, we obtain upper and lower bounds for moments of the Riemann zeta-function averaged over the extreme values between its zeros on the critical line. Our bounds are very nearly the same order of magnitude. The…

Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in.…

We prove under RH the existence of a very large positive and negative values of the argument of the Riemann zeta function on a very short intervals.

In this paper, we give some estimates of the sum of the square norm of the sections of the pluricanonical bundles over a Riemann surface with genus greater than 2 and Gauss curvature (-1). Using these estimate, we give a uniform estimate of…

We prove two versions of a boundary Harnack principle in which the constants do not depend on the domain.

Let X be a curve over the finite field of q elements and let N(X), g(X) be its number of rational points and genus respectively. The Ihara constant A(q) is defined by the limit superior of N(X)/g(X) as the genus of X goes to infinity. In…

It is pointed out that current conservation alone does not suffice to prove Hara's theorem as it was claimed recently. By explicit calculation we show that the additional implicit assumption made in such "proofs" is that of a sufficiently…

We give new upper bounds for the number of nonconstant holomorphic maps depending only on the genus. Our estimates improve previously known bounds. The proof is based on the study of pullbacks of holomorphic differentials, together with…

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal…

Let $D\subset\Co$ be a bounded domain, whose boundary $B$ consists of $k$ simple closed continuous curves and $H^{\infty}(D)$ be the algebra of bounded analytic functions on $D$. We prove the matrix-valued corona theorem for matrices with…

Assuming the Riemann Hypothesis, we establish lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$. Our proof is based upon a recent method of Rudnick and Soundararajan…

We give an elementary and self-contained proof of the uniformization theorem for non-compact simply-connected Riemann surfaces.

The Riemann-Roch theorem is of utmost importance in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of…

Involutions without fixed points on hyperbolic closed Riemann surface are discussed. For an orientable surface $X$ of even genus with an arbitrary Riemannian metric $d$ admitting an involution $\tau$, it is known that $\min_{p\in…

The Riemann Theorem states, that for any nontrivial connected and simply connected domain on the Riemann sphere there exists some its conformal bijection to the exterior of the unit disk. In this paper we find an explicit form of this map…

Assuming the Riemann Hypothesis, we obtain an upper bound for the 2k-th moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$ for every positive integer k. Our bounds are nearly as sharp as…