Using the local picture of degeneration sequences minimal surfaces developed by Chodosh et al. (Invent Math 209(3):617–664, [1]) we show that in any closed Riemannian 3-manifold (M, g), genus an embedded CMC surface can be bounded only terms its index and area, independently value mean curvature. We also if M has finite fundamental group, area non-minimal term a lower bound for

We establish a connection between the function space BMO and theory of quasisymmetric mappings on \emph{spaces homogeneous type} $\widetilde{X} :=(X,\rho,\mu)$. The is that logarithm generalised Jacobian an $\eta$-quasisymmetric mapping $f: \widetilde{X} \rightarrow \widetilde{X}$ always in $\rm{BMO}(\widetilde{X})$. In course proving this result, we first show $\widetilde{X}$, reverse-H\"{o}ld...

in this paper, a new definition of fuzzy bounded sets and totallyfuzzy bounded sets is introduced and properties of such sets are studied. thena relation between totally fuzzy bounded sets and n-compactness is discussed.finally, a geometric characterization for fuzzy totally bounded sets in i- topologicalvector spaces is derived.