Calabi-Yau domains in three manifolds

TL;DR

This paper constructs specific domains within 3-manifolds that prohibit the existence of certain properly immersed surfaces with bounded mean curvature, advancing understanding of geometric constraints in 3-manifolds.

Contribution

It introduces a method to embed one-manifolds in 3-manifolds to prevent the existence of particular immersed surfaces with bounded mean curvature.

Findings

Existence of domains preventing certain immersed surfaces

Construction of smooth embedded one-manifolds in 3-manifolds

Implications for geometric analysis in 3-manifolds

Abstract

We prove that given any compact Riemannian 3-manifold with boundary M, there exists a smooth properly embedded one-manifold G, included in M, each of whose components is a simple closed curve and such that the domain D=Int(M)-G does not admit any properly immersed open surfaces with at least one annular end, bounded mean curvature, compact boundary (possibly empty) and a complete induced Riemannian metric.