On the total curvature of minimal annuli in $R^3$ and Nitsche's conjecture

TL;DR

This paper proves a generalized version of Nitsche's conjecture, demonstrating that certain minimal annuli in three-dimensional space have finite total curvature, and extends this result to all properly embedded minimal surfaces with multiple ends.

Contribution

The authors provide a proof of the generalized Nitsche's conjecture and establish that minimal annuli with specific boundary conditions have finite total curvature, extending to surfaces with multiple ends.

Findings

Minimal annuli with boundary in a horizontal plane and intersecting all horizontal planes in simple closed curves have finite total curvature.

Properly embedded minimal surfaces with more than one end in R^3 have finite total curvature.

The proof confirms a long-standing conjecture in the theory of minimal surfaces.

Abstract

We present a proof of the generalized Nitsche's conjecture proposed by W.H.Meeks III and H. Rosenberg: For $t\ge 0$, let $P_t$ denote the horizontal plane of height $t$ over the $x_1,x_2$ plane. Suppose that $M \subset R^3$ is a minimal annulus with the boundary contains in $P_0$ and that $M$ intersects every $P_t$ in a simple closed curve. Then $M$ has finite total curvature. As a consequence, we show that every properly embedded minimal surface of finite topology in $R^3$ with more than one end has finite total curvature.