Minimal surfaces in $\mathbb{R}^4$ like the Lagrangian catenoid

TL;DR

This paper investigates the existence and nonexistence of complete minimal surfaces with finite total curvature and embedded planar ends in higher-dimensional Euclidean spaces, providing new examples and classifications especially in $ ext{R}^4$.

Contribution

It establishes nonexistence results for certain genus and end configurations and constructs new embedded minimal surfaces, including spheres and tori, in $ ext{R}^4$ with specified properties.

Findings

Nonexistence of genus 1 surfaces with 2 ends

Existence of embedded minimal spheres with 3 ends in $ ext{R}^4$

Construction of genus $g$ surfaces with $d$ ends satisfying $g+2 \,\leq\, d \leq 2g+1$

Abstract

In this paper, we discuss complete minimal immersions in $\mathbb{R}^N$($N\geq4$) with finite total curvature and embedded planar ends. First, we prove nonexistence for the following cases: (1) genus 1 with 2 embedded planar ends, (2) genus $\neq4$, hyperelliptic with 2 embedded planar ends like the Lagrangian catenoid. Then we show the existence of embedded minimal spheres in $\mathbb{R}^4$ with 3 embedded planar ends. Moreover, we construct genus $g$ examples in $\mathbb{R}^4$ with $d$ embedded planar ends such that $g\geq 1$ and $g+2\leq d\leq 2g+1$. These examples include a family of embedded minimal tori with 3 embedded planar ends.