Kazdan–Warner problem on compact Riemann surfaces with boundary

Motivated by a recent work of Chen and Zhu (Commun Math Stat 1:369–385. 2013), we establish a Trudinger–Moser inequality on a compact Riemannian surface without boundary. The proof is based on blow-up analysis together with Carleson–Chang’s result (Bull Sci Math 110:113–127. 1986). This inequality is different from the classical one, which is due to Fontana (Comment Math Helv 68:415–454. 1993), since the Gaussian curvature is involved. As an application, we improve Chen–Zhu’s result as follows: a modified Liouville energy of conformal Riemannian metric has a uniform lower bound, provided that the Euler characteristic is nonzero and the volume of the conformal surface has a uniform positive lower bound.

Let (M,g) be a compact smooth Riemannian surface with boundary. In this paper, we use blowing-up analysis to prove that some Moser?Trudinger trace inequalities hold on certain function spaces, and that the extremal functions exist in those function spaces without any additional hypothesis on (M,g).

This big chapter is devoted to the theory of compact Riemann surfaces. Tori are examples of compact Riemann surfaces. This means that we generalize the theory of elliptic functions here. A compact Riemann surface can be associated with any algebraic function, and in this way we obtain all compact Riemann surfaces. The compact Riemann surfaces achieve the same result for the integration of algebraic functions as does the theory of elliptic functions for the elliptic integrals. The triumph of the theory of Riemann surfaces was that it made the “integrals of the first kind” understandable and solved the so-called Jacobi inversion problem. We have to go a long way to achieve this aim. At the end, we shall arrive at the best-known theorems of the theory of Riemann surfaces, such as the the Riemann–Roch theorem, Abel’s theorem, and the Jacobi inversion theorem. On the way, we must also understand the topology of compact Riemann surfaces. We shall treat the topological classification completely here.

The volume of the space of holomorphic maps from [formula omitted] to [formula omitted

We introduce a family of variational functionals for spinor fields on a compact Riemann surface M that can be used to find close-to-conformal immersions of M into \(\mathbb {R}^3\) in a prescribed regular homotopy class. Numerical experiments indicate that, by taking suitable limits, minimization of these functionals can also yield piecewise smooth isometric immersions of a prescribed Riemannian metric on M.

On each compact Riemann surface Σ of genusp≥1, we have the Bergman metric obtained by pulling back the flat metric on its Jacobian via the Albanese map. Taking theL 2-product of holomorphic quadratic differentials w.r.t. this metric induces a Riemannian metric on the Teichmuller spaceT p that is invariant under the action of the modular group. We investigate geometric properties of this metric as an alternative to the usually employed Weil-Petersson metric.

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition. In the previous paper, we show that the Chern-Simons Higgs equation with parameter λ > 0 has at least two solutions (uλ1, uλ2) for λ sufficiently large, which satisfy that uλ1 → −u0 almost everywhere as λ → ∞, and that uλ2 → −∞ almost everywhere as λ → ∞, where u0 is a (negative) Green function on M. In this paper, we study the asymptotic behavior of the solutions as λ → ∞, and prove that uλ2 − \(\overline {u_\lambda ^2 } \) converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary ∂M is negative, or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.

A Hodge theoretic projective structure on compact Riemann surfaces

Since the early 1950's, when Kodaira “discovered” positive line bundles, the notion of positivity has undergone a continuous evolution. This paper is intended as an introduction to the study of positivity notions. More specifically, I consider the simplest case - line bundles over compact Riemann surfaces - and compare five positivity notions for such bundles. The results obtained are certainly not new; they are, in fact, known in much greater generality. However, by restricting to the dimension one case, I am able to make use of Riemann surface techniques to significantly simplify the proofs. In fact, this article should be easily understood by anyone familiar with the contents of Gunning's Lectures on Riemann surfaces.

We prove that given any meromorphic function / on a compact Riemann surface M' there exists another meromorphic function g on M' such that {df, g} is the Weierstrass pair defining a complete conformai minimal immersion of finite total curvature into Euclidean 3-space defined on M' punctured at a finite set of points.As corollaries we obtain i) any compact Riemann surface can be immersed in Euclidean 3-space as in the above with at most 4p + 1 punctures, where p is the genus of the Riemann surface; ii) any hyperelliptic Riemann surface of genus p can be so immersed with at most 3p + 4 punctures.

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (Ho and Liu in Commun Anal Geom 16(3):617–679, 2008).

The Uniformization Theorem due to Koebe and Poincare implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.

Every non-trivial closed curve $C$ on a compact Riemann surface $R$ is freely homotopic to the $r$-fold iterate ${C_0}^r$ of some primitive closed geodesic $C_0$ on $R$. We call $r$ the multiplicity of $C$, and denote it by $N_{R} (C)$. Let $f$ be a non-constant holomorphic map of a compact Riemann surface $R_1$ of genus $g_1$ onto another compact Riemann surface $R_2$ of genus $g_2$ with $g_1 \\geq g_2 > 1$, and $C$ a simple closed geodesic of hypebolic length $l_{R_1} (C)$ on $R_1$. In this paper, we give an upper bound for $N_{R_2} (f(C))$ depending only on $g_1$, $g_2$ and $l_{R_1} (C)$.

In the previous papers of the present author, a general theory of multiplicative Weierstrass points on compact Riemann surfaces for arbitrary characters was developed. In the present paper, some additional relations between multiplicative Weierstrass points on a compact Riemann surface for an arbitrary character and special subsets in the Jacobi variety, the canonical embedding of a compact Riemann surface into a projective space, are established. We not distinction between classical Weierstrass points and multiplicative Weierstrass points on a compact Riemann surface.

We give a general definition of the exponents of a meromorphic connection ∇ on a holomorphic vector bundle E of rank n over a compact Riemann surface X. We prove that they can be computed as invariants of a vector bundle E L canonically attached to E, which we construct and call the Levelt bundle of E, and whose degree (equal to the sum of the exponents) we estimate by upper and lower bounds (Fuchs' relations). We use this definition to construct, for every linear differential equation on a compact Riemann surface (with regular or irregular singularities), the companion bundle of the equation, a vector bundle endowed with a meromorphic connection that is equivalent to the given equation and has precisely the same singularities and the same set of exponents.