On the number of minimal surfaces with a given boundary

TL;DR

This paper extends White's parity theorem for minimal surfaces to more general 3-manifolds, demonstrating the existence of multiple minimal surfaces with given boundaries and symmetries, and applying these results to construct embedded genus-g helicoids.

Contribution

It generalizes the parity theorem for minimal surfaces to broader classes of 3-manifolds, including weakly mean convex and piecewise smooth boundary cases.

Findings

Parity theorem holds for strictly mean convex 3-manifolds homeomorphic to a ball.

The theorem extends to weakly mean convex manifolds with piecewise smooth boundary.

Application to proving existence of embedded genus-g helicoids in ext{S}^2 imes ext{R}.

Abstract

We generalize the following result of White: Suppose $N$ is a compact, strictly convex domain in $\RR^3$ with smooth boundary. Let $\Sigma$ be a compact 2-manifold with boundary. Then a generic smooth curve $\Gamma\cong \partial\Sigma$ in $\partial N$ bounds an odd or even number of embedded minimal surfaces diffeomorphic to $\Sigma$ according to whether $\Sigma$ is or is not a union of disks. First, we prove that the parity theorem holds for any compact riemannian 3-manifold $N$ such that $N$ is strictly mean convex, $N$ is homeomorphic to a ball, $\partial N$ is smooth, and $N$ contains no closed minimal surfaces. We then further relax the hypotheses by allowing $N$ to be weakly mean convex and to have piecewise smooth boundary. We extend the parity theorem yet further by showing that, under an additional hypothesis, it remains true for minimal surfaces with prescribed symmetries. The parity theorems are used in an essential way to prove the existence of embedded genus-$g$ helicoids in $\SS^2\times \RR$. We give a very brief outline of this application. (The full argument will appear elsewhere.)