Embeddedness of proper minimal submanifolds in homogeneous spaces

TL;DR

This paper establishes new embeddedness results for minimal submanifolds in homogeneous spaces, including curvature bounds, boundary conditions, and asymptotic properties, extending classical minimal surface theory.

Contribution

It introduces novel embeddedness criteria for minimal submanifolds based on boundary geometry, total curvature, and M{"o}bius volume in various homogeneous spaces.

Findings

Total curvature of certain Jordan curves is bounded by 2mπ.

Minimal surfaces bounded by specific curves are embedded.

Proper minimal submanifolds with certain boundary conditions are embedded.

Abstract

We prove the three embeddedness results as follows. $({\rm i})$ Let $\Gamma_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of $\Gamma_{2m+1}<2m\pi$. In particular, the total curvature of $\Gamma_5<4\pi$ and thus any minimal surface $\Sigma \subset \mathbb{R}^n$ bounded by $\Gamma_5$ is embedded. Let $\Gamma_5$ be a piecewise geodesic Jordan curve with $5$ vertices in $\mathbb{H}^n$. Then any minimal surface $\Sigma \subset \mathbb{H}^n$ bounded by $\Gamma_5$ is embedded. If $\Gamma_5$ is in a geodesic ball of radius $\frac{\pi}{4}$ in $\mathbb{S}^n_+$, then $\Sigma \subset \mathbb{S}^n_+$ is also embedded. As a consequence, $\Gamma_5$ is an unknot in $\mathbb{R}^3$, $\mathbb{H}^3$ and $\mathbb{S}^3_+$. $({\rm ii})$ Let $\Sigma$ be an $m$-dimensional proper minimal submanifold in $\mathbb{H}^n$ with the ideal boundary $\partial_{\infty} \Sigma = \Gamma$ in the infinite sphere $\mathbb{S}^{n-1}=\partial_\infty \mathbb{H}^n$. If the M{\"o}bius volume of $\Gamma$ $\widetilde{\vol}(\Gamma) < 2\vol(\mathbb{S}^{m-1})$, then $\Sigma$ is embedded. If $\widetilde{\vol}(\Gamma) = 2\vol(\mathbb{S}^{m-1})$, then $\Sigma$ is embedded unless it is a cone. $({\rm iii})$ Let $\Sigma$ be a proper minimal surface in $\hr$. If $\Sigma$ is vertically regular at infinity and has two ends, then $\Sigma$ is embedded.