# Nonclassical minimizing surfaces with smooth boundary

## TL;DR

This paper constructs a near-Euclidean Riemannian metric and a smooth closed curve in four-dimensional space such that the unique area-minimizing surface spanned by the curve has infinite topology, demonstrating nonclassical behavior.

## Contribution

It introduces a method to create a Riemannian metric close to Euclidean space with a boundary curve whose minimal surface has infinite topology, a novel example in geometric analysis.

## Key findings

- Existence of a smooth boundary curve with an infinite topology minimal surface.
- Construction of a near-Euclidean metric that admits such a surface.
- The minimal surface is calibrated and almost Kähler.

## Abstract

We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $\Gamma\subset \mathbb R^4$ such that the unique area minimizing surface spanned by $\Gamma$ has infinite topology. Furthermore the metric is almost K\"ahler and the area minimizing surface is calibrated.