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

Abstract

In this paper, we discuss complete minimal immersions in ${\mathbb{R}}^N$($N\ge 4$) 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 $\ne 4$, 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 $g\ge 1$ and $g+2\le d\le 2g+1$. These examples include a family of embedded minimal tori with 3 embedded planar ends.

19 pages