Mergulho de superfície mínima e curvatura total de seu bordo

Abstract

Given a minimal submanifold Σn ⊂ Rm with compact boundary Γn−¹, in this work we analyze under which conditions on Γ we can conclude that Σ is embedded. We show that if a simple closed curve Γ ⊂ Rm, piecewise regular and with total curvature less than or equal to 4π is the boundary of a minimal surface Σ, then Σ is embedded (EKHOLM; WHITE ; WIENHOLTZ, 2002). Moreover, following the steps of (CHOE; GULLIVER, 2017), we present a possible way to obtain a generalization of this result to higher dimensions.