A complex manifold is called an Oka manifold if every holomorphic map from a convex set in a Euclidean space to the manifold is a limit of entire maps. We show that every Oka manifold admits families of noncompact complex curves of any given topological type, endowed with a prescribed family of conformal structures, with approximation on compact Runge subsets. A similar result holds for families of minimal surfaces and null holomorphic curves in Euclidean spaces.

In a recent joint work with David Kalaj (2021), we introduced a new Finsler pseudometric on any domain in the real Euclidean space $\mathbb R^n$ for $n\ge 3$, defined in terms of conformal harmonic discs, by analogy with the Kobayashi pseudometric on complex manifolds. This "minimal pseudometric" describes the maximal rate of growth of hyperbolic conformal minimal surfaces in a given domain. On the unit ball, the minimal metric coincides with the classical Beltrami-Cayley-Klein metric. I will discuss sufficient geometric conditions for a domain to be (complete) hyperbolic, meaning that its minimal pseudometric is a (complete) metric.

We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc ${\mathbb D}$ in $\mathbb C$ to the unit ball ${\mathbb B}^n$ in $\mathbb R^n$, $n\ge 2$, at any point where the map is conformal. In dimension $n=2$ this generalizes the classical Schwarz-Pick lemma to harmonic maps $\mathbb D\to\mathbb D$ which are conformal only at the reference point. In dimensions $n\ge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs $\mathbb D\to {\mathbb B}^n$. Let ${\mathcal M}$ denote the restriction of the Bergman metric on the complex $n$-ball to the real $n$-ball ${\mathbb B}^n$. We show that conformal harmonic immersions $M \to ({\mathbb B}^n,{\mathcal M})$ from any hyperbolic open Riemann surface $M$ with its natural Poincaré metric are distance-decreasing, and the isometries are precisely the conformal embeddings of $\mathbb D$ onto affine discs in ${\mathbb B}^n$. (Joint work with David Kalaj.)

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We show that if R is a compact Riemann surface and M is a domain in R whose complement is a union of countably many pairwise disjoint smoothly bounded closed discs, then M is the complex structure of a complete bounded minimal surface in $\mathbb{R}^3$. More precisely, we prove that there is a complete conformal minimal immersion $X:M→\mathbb{ℝ}^3$ extending to a continuous map from the closure of $M$ such that $X(\partial M)$ is a union of pairwise disjoint Jordan curves. This extends a result for finite bordered Riemann surfaces proved in 2015. (Joint work with Antonio Alarcon.)

I will describe the construction of a properly embedded holomorphic disc in the unit ball of $\mathbb{C}^2$ having the following surprising combination of properties: - on the one hand, the disc has finite area, and hence is the zero set of a bounded holomorphic function on the ball; - on the other hand, its real analytic boundary curve is everywhere dense in the sphere.

We prove that every meromorphic function on an open Riemann surface $M$ is the complex Gauss map of a conformal minimal immersion $f:M\to \mathbb R^3$; furthermore, $f$ may be chosen as the real part of a holomorphic null curve $F:M\to\mathbb C^3$. Analogous results are proved for conformal minimal immersions $M\to\mathbb R^n$ for any $n>3$. We also show that every conformal minimal immersion $M\to\mathbb R^n$ is isotopic to a flat one, and we identify the path connected components of the space of all conformal minimal immersions $M\to\mathbb R^n$ for any $n\ge 3$. (Joint work with Antonio Alarcón and Francisco J. López,

Let $M$ be an open Riemann surface. It was proved by Alarcón and Forstneric that every conformal minimal immersion $M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve $M\to\mathbb C^3$. We prove the following substantially stronger result in this direction: for any $n\ge 3$, the inclusion of the space of real parts of non flat null holomorphic immersions $M\to\mathbb C^n$ into the space of non flat conformal minimal immersions $M\to \mathbb R^n$ satisfies the parametric h-principle with approximation; in particular, it is a weak homotopy equivalence. Analogous results hold for several other related maps. For an open Riemann surface $M$ of finite topological type, we obtain optimal results by showing that the above inclusion and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences. (Joint work with Finnur Lárusson.)

Two of the most classical theorems in the theory of holomorphic functions are the Runge approximation theorem and the Weierstrass interpolation theorem. In higher dimensions these correspond to the Oka-Weil approximation theorem and the Cartan extension theorem. A complex manifold X is said to be an Oka manifold if these classical results, and some of their natural extensions, hold for holomorphic maps from any Stein manifold (in particular, from complex Euclidean spaces) to X. After a brief historical review, beginning with the classical Oka-Grauert theory and continuing with the seminal work of Gromov, I will describe some recent developments and future challenges in this field of complex geometry. In particular, I shall describe a recently discovered connection between Oka theory and the classical theory of minimal surfaces in Euclidean spaces.

We establish the Hodge conjecture for the top dimensional cohomology group with integer coefficients of any $q$-complete complex manifold $X$ with $q<{\rm dim} X$. This holds in particular for the complement $X=\mathbb{CP}^n\setminus A$ of any complex projective manifold defined by $q< n$ independent equations.

We will construct approximate solutions to Riemann-Hilbert boundary value problems for null holomorphic curves in the complex Euclidean 3-space $\mathbb{C}^3$. Using this technique, we will prove that every bordered Riemann surface admits a complete proper null holomorphic embedding into a ball of $\mathbb{C}^3$, hence a complete conformal minimal immersion into $\mathbb{R}^3$ with bounded image. We will also construct properly embedded null curves in $\mathbb{C}^3$ with a bounded coordinate function; these give rise to properly embedded null curves in $SL_2(\mathbb{C})$ and to properly immersed Bryant surfaces in the hyperbolic 3-space $\mathbb{H}^3$ that are conformally equivalent to any bordered Riemann surface. In particular, we give the first examples of proper Bryant surfaces with finite topology and of hyperbolic conformal type.

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 $\mathbb{C}^3$.

Two of the classical theorems in the theory of holomorphic functions are the Runge approximation theorem and the Weierstrass interpolation theorem. A complex manifold X is said to be an Oka manifold if these results, and some of their natural extensions, are valid for holomorphic maps from any Stein manifold (in particular, from complex Euclidean spaces) to X. After a brief review of the development of this subject, beginning with the classical Oka-Grauert theory and continuing with the seminal work of Gromov, I will describe some of the recent developments and future challenges in this field of complex geometry.