The uniqueness of Enneper’s surfaces and Chern–Ricci functions on minimal surfaces

ABSTRACT

We use two flat structures constructed by Chern and Ricci to build harmonic functions on negatively curved minimal surfaces in . Our main goal is to establish two new uniqueness results that a minimal surface has constant first Chern–Ricci function if and only if it is Enneper’s surface and that a minimal surface has constant second Chern–Ricci function if and only if it is a member of the associate families of catenoids and helicoids.

KEYWORDS:

• Enneper’s surface
• flat structures
• minimal surfaces
• Ricci condition

AMS SUBJECT CLASSIFICATIONS:

• 53A10
• 49Q05

Notes

No potential conflict of interest was reported by the authors.