Associate and conjugate minimal immersions in MxR

TL;DR

This paper investigates the existence, uniqueness, and geometric properties of associate and conjugate minimal immersions in MxR, extending classical results using harmonic map theory and analyzing minimal vertical graphs.

Contribution

It introduces new results on associate minimal surfaces in MxR, including conditions for their existence, uniqueness, and geometric behavior, generalizing classical theorems.

Findings

Associate surfaces of vertical graphs on convex domains are also graphs.

The paper establishes existence and uniqueness results for associate and conjugate minimal immersions.

It applies harmonic map theory to minimal surface problems in MxR.

Abstract

We consider minimal immersions in MxR. We study existence and uniqueness of associate and conjugate isometric immersions to a given minimal surface. We use the theory of univalent harmonic map between surfaces. Then we study the geometry of associate minimal vertical graphs. We prove that an associate surface of a vertical graph on a convex domain is a graph. In the classical theory it is a theorem of R. Krust.