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# A properly embedded holomorphic disc in the ball with finite area and dense boundary curve

## Franc Forstneric Univerza v Ljubljani

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.

Seminario 1ª Planta, IEMATH

# Every meromorphic function is the Gauss map of a conformal minimal surface

## Franc Forstneric Univerza v Ljubljani

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,

Seminario 1ª planta, IEMath

# The parametric h-principle for minimal surfaces in $\mathbb{R}^n$ and null curves in $\mathbb{C}^n$

## Franc Forstneric Univerza v Ljubljani

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.)

Seminario 1ª planta, IEMath

# Oka Theory and Minimal Surfaces

## Franc Forstneric Univerza v Ljubljani

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.

Seminario 1ª Planta, IEMath-Gr

# On the Hodge conjecture for $q$-complete manifolds.

## Franc Forstneric Univerza v Ljubljani

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.

Seminario 1ª Planta, IEMath-Gr

# The Calabi-Yau problem, null curves, and Bryant surfaces.

## Franc Forstneric Univerza v Ljubljani

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.

Seminario de Matemáticas, 1ª planta

# Null curves and directed immersions of Riemann surfaces

## Franc Forstneric Univerza v Ljubljani

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 $\mathbb{C}^3$.

# What is an Oka manifold?

## Franc Forstneric Univerza v Ljubljani

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.

Seminario de Matemáticas. 1ª Planta. Sección de Matemáticas

# Franc Forstneric

## Univerza v Ljubljani

Number of talks
8
Number of visits
8
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Country of origin
Eslovenia