Diameter estimates for surfaces in conformally flat spaces (Q6564492)

\textit{L. Simon} [Commun. Anal. Geom. 1, No. 2, 281--326 (1993; Zbl 0848.58012)] obtained an upper bound for the extrinsic diameter involving the \(L^2\)-norm of the mean curvature and the \(L^1\)-norm of the second fundamental form for closed surfaces and for surfaces with boundary. \textit{P. Topping} [Comment. Math. Helv. 83, No. 3, 539--546 (2008; Zbl 1154.53007)] proved an upper bound for the intrinsic diameter of closed manifolds immersed in \({\mathbb R}^n\).\N\textit{J.-Y. Wu} and \textit{Y. Zheng} [Proc. Am. Math. Soc. 139, No. 11, 4097--4104 (2011; Zbl 1230.53054)] obtained an analogous result in the Riemannian setting, adapting Topping's proof to the more general framework.\N\textit{T. Miura} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 23, No. 4, 1707--1721 (2022; Zbl 1506.49025)] gave a construction to get a diameter bound for surfaces with boundary immersed in \({\mathbb R}^n\), where the idea was to double the surface, glue the two copies together and apply Topping's diameter bound to the closed surface.\N\NThe paper under review deals with an upper bound for the intrinsic diameter of a surface in a conformally flat \(3\)-dimensional Riemannian manifold with application of the inequality to minimal surfaces in the 3-sphere and in hyperbolic space.