Halfspace Theorem in Differential Geometry

Question

I'm sutdying the Classical Halfspace Theorem, it says that :"There is no non-planar, complete, minimal surface properly immersed in a halfspace of $\mathbb{R}^3$". But in this theorem, I don't know the definition of "complete". By the way, I would like to understand what does Theorem mean via some simple examples.

Answer 1

For a Riemannian manifold without boundary $(M,g)$, completeness means one of the following equivalent conditions:

• all (unitary parametrized) geodesics are defined on $\mathbb{R}$
• the metric space $(M,d_g)$ is complete (all Cauchy sequences converge)
• the compact subsets are the closed and bounded subsets

This is called the Hopf-Rinow theorem. Here, complete is to be understood as "every surface endowed with the induced Riemannian metric that is complete".

The half space theorem says that if a surface in $\mathbb{R}^3$ is non planar (thus, "curved"), complete (thus, geodesics do not end somewhere: there is no "missing point" or "boundary"), properly immersed (thus, the surface "extends to infinity"), and if it is minimal, then it goes "in every direction": there is no half-space in which you can enclose the surface. It has to go out of it at some time. Equivalently, it intersects every affine hyperplane.