Injectivity of minimal immersions and homeomorphic extensions to space

TL;DR

This paper explores conditions ensuring injectivity of conformal immersions of Riemannian manifolds into Euclidean space, deriving criteria for minimal surfaces and extending classical results with geometric fibrations.

Contribution

It introduces new injectivity criteria for conformal immersions, including a sharp curvature-dependent condition for minimal disks to be embedded, and constructs geometric fibrations analogous to Ahlfors-Weill extensions.

Findings

Derived a Becker type injectivity condition.

Established a sharp curvature and diameter criterion for minimal disks.

Identified extremal configurations only on a catenoid.

Abstract

We study a recent general criterion for the injectivity of the conformal immersion of a Riemannian manifold into higher dimensional Euclidean space, and show how it gives rise to important conditions for Weierstrass-Ennerper lifts defined in the unit disk $\mathbb{D}$ endowed with a conformal metric. Among the corollaries, we obtain a Becker type condition and a sharp condition depending on the Gaussian curvature and the diameter for an immersed geodesically convex minimal disk in $\mathbb{R}^3$ to be embedded. Extremal configurations for the criteria are also determined, and can only occur on a catenoid. For non-extremal configurations, we establish fibrations of space by circles in domain and range that give a geometric analogue of the Ahlfors-Weill extension.