Liouville-type properties for embedded minimal surfaces
The following properties of a noncompact Riemann surface \(M\) without boundary are considered {\parindent=8mm \begin{itemize}\item[(1)]\(M\) admits a nonconstant positive superharmonic function (equivalently, \(M\) is transient for Brownian motion); \item[(2)] \(M\) admits a nonconstant positive harmonic function; \item[(3)] \(M\) admits a nonconstant bounded harmonic function. \end{itemize}} A noncompact Riemann surface \(M\) without boundary is said to be recurrent for Brownian motion if it does not satisfy property (1). Clearly \((3)\Rightarrow(2)\Rightarrow(1)\). In this paper, it is determined which of the above three properties is satisfied by a complete embedded minimal surface \(M\subset\mathbb{R}^3\), in terms of its topology and geometry: For example, the following theorem is proved: Theorem 1.4. Let \(M\subset\mathbb{R}^3\) be a nonflat \(k\)-periodic complete embedded minimal surface such that \(M/G\) has finite topology. Then, every positive harmonic function on \(M\) is constant. \(M\) is said to be \(k\)-periodic if its group of isometries \(G\) has rank \(k=0,1,2,3\). These results are applied to study the question of existence of complete embedded minimal surfaces which are \(a\)-stable for some \(a> 0\).
scientific article; zbMATH DE number 5121964