The maximum principle at infinity for minimal surfaces in flat three manifolds
William H. Meeks, Harold Rosenberg · Commentarii Mathematici Helvetici · 1990
Maximum principles are used as basic analytic tools for studying properties of functions defined on domains in R n and satisfying certain equations (e.g. elliptic). In general these maximum principles play a fundamental role in analysis on complete Riemannian manifolds, especially in the study of variational problems. For example, the well-known maximum principle for harmonic functions has had both a simplifying and unifying effect on the fields of harmonic and complex analysis. H. Hopf [18] gave an important general maximum principle for second order linear elliptic partial differential equations. The Hopf maximum principle easily yields a maximum principle for solutions of the minimal surface equation. In this context the principle states that if D c R 2 is a smooth connected domain and f~ ,f2 are two smooth functions on D that satisfy the minimal surface equation, then the difference fl -f2 cannot have an interior maximum or minimum unless the difference is constant. The maximum principle for minimal graphs gives rise to the following geometric result for minimal surfaces in Riemannian three-manifolds: IfM~ and ME are minimal surfaces in a Riemannian three-manifold that intersect at a common interior point p and M1 is on one side of M2 near p, then M~ intersects M2 in an open surface containing