2026-01-022026-01-022025ALENCAR, Hilário; GARCIA, Ronaldo; SILVA NETO, Gregório. A proof of a conjecture of W. Hsiang on invariant cmc hypersurfaces with a singularity at the origin. Bulletin of the Brazilian Mathematical Society, Berlin, v. 56, e13, 2025. DOI: 10.1007/s00574-024-00437-0. Disponível em: https://link.springer.com/article/10.1007/s00574-024-00437-0. Acesso em: 12 dez. 2025.1678-7544e- 1678-7714https://link.springer.com/article/10.1007/s00574-024-00437-0In differential geometry, curvature-based functionals, such as the total Gaussian curvature, the Willmore energy, and the total geodesic torsion, play a central role in both theoretical investigations and practical applications. In this paper, we study geometric properties of the extremal curves for the next functional where ds is the arc element on S and are the principal curvatures. First, we establish that a necessary and sufficient condition for a surface to be a Dupin cyclide is that its lines of curvature and the extremal curves of functional intersect at a constant angle. Secondly, we demonstrate that the extremal curves of the functional are invariant under inversion. Finally, we show that the determination of functional extremal curves of for any cone, general cylinder, and surfaces of revolution can be reduced to quadratures.engAcesso RestritoArtigo10.1007/s00574-024-00437-0