Minimal Conformally Flat Hypersurfaces

Author(s)

do Rei Filho, C. ^[1] ; Tojeiro, R. ^[2]

Affiliation

^[1] Univ Fed Alagoas, Inst Matemat, BR-57072900 Maceio, AL - Brazil

^[2] Univ Sao Paulo, Dept Matemat, BR-13560970 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: JOURNAL OF GEOMETRIC ANALYSIS; v. 29, n. 3, p. 2931-2956, JUL 2019.
• Web of Science Citations: 0

Abstract

We study conformally flat hypersurfaces f : M3. Q4(c) with three distinct principal curvatures and constant mean curvature H in a space form with constant sectional curvature c. First we extend a theorem due to Defever when c = 0 and show that there is no such hypersurface if H = 0. Our main results are for the minimal case H = 0. If c = 0, we prove that if f : M3. Q4(c) is a minimal conformally flat hypersurface with three distinct principal curvatures then f (M3) is an open subset of a generalized cone over a Clifford torus in an umbilical hypersurface Q3( c). Q4(c), c > 0, with c = c if c > 0. For c = 0, we show that, besides the cone over the Clifford torus in S3. R4, there exists precisely a one-parameter family of (congruence classes of) minimal isometric immersions f : M3. R4 with three distinct principal curvatures of simply connected conformally flat Riemannian manifolds. (AU)

• FAPESP's process: 16/23746-6 - Algebraic, topological and analytical techniques in differential geometry and geometric analysis
• Grantee: Paolo Piccione
• Support Opportunities: Research Projects - Thematic Grants