We study holomorphic immersions of open Riemann surfaces into ${\mathbb C}^n$ whose derivative lies in a conical algebraic subvariety $A$ of ${\mathbb C}^n$ that is smooth away from the origin. Classical examples of such $A$-immersions include null curves in ${\mathbb C}^3$ which are closely related to minimal surfaces in ${\mathbb R}^3$, and null curves in $SL_2(\mathbb{C})$ that are related to Bryant surfaces. We establish a basic structure theorem for the set of all $A$-immersions of a bordered Riemann surface, and we prove several approximation and desingularization theorems. Assuming that $A$ is irreducible and is not contained in any hyperplane, we show that every $A$-immersion can be approximated by $A$-embeddings; this holds in particular for null curves in ${\mathbb C}^3$. If in addition $A \setminus \{0\}$ is an Oka manifold, then $A$-immersions are shown to satisfy the Oka principle, including the Runge and the Mergelyan approximation theorems. Another version of the Oka principle holds when $A$ admits a smooth Oka hyperplane section. This lets us prove in particular that every open Riemann surface is biholomorphic to a properly embedded null curve in ${\mathbb C}^3$.
COBISS.SI-ID: 16655705
We prove that every circled domain in the Riemann sphere admits a proper holomorphic embedding into the affine plane ${\mathbb C}^2$.
COBISS.SI-ID: 16645209
In this paper we give an extension of the Cartier-Gabriel-Kostant structure theorem to Hopf algebroids.
COBISS.SI-ID: 16432473
By using an explicit Bellman function, we prove a bilinear embedding theorem for the Laplacian associated with a weighted Riemannian manifold $(M,\mu_\varphi)$ having the Bakry-Emery curvature bounded from below. The embedding, acting on the cartesian product of $L^p(M,\mu_\varphi)$ and $L^q(T^ast M, \mu_\varphi)$, $1/p + 1/q = 1$, involves estimates which are independent of the dimension of the manifold and linear in $p$. As a consequence we obtain linear dimension-free estimates of the $L^p$ norms of the corresponding shifted Riesz transform. All our proofs are analytic.
COBISS.SI-ID: 16719705
We study the nonlinear Fourier transform $\mathcal{F}$ associated with the integrable nonlinear partial differential equations of AKNS-ZS type. We show that $\mathcal{F}$ is a real analytic operator between the appropriate Hilbert spaces, and that it has a real analytic local inverse near the origin. We construct a convergent iterative scheme by means of which one can calculate the inverse ${\mathcal{F}}^{-1}$ to any desired degree of accuracy.
COBISS.SI-ID: 16833369