In this paper we study surfaces in$R$^{3}that arise as limit shapes in random surface models related to planar dimers. These limit shapes are$surface tension minimizers$, that is, they minimize a functional of the form ∫$σ$(∇$h$)$dx$$dy$among all Lipschitz functions$h$taking given values on the boundary of the domain. The surface tension$σ$has singularities and is not strictly convex, which leads to formation of$facets$and$edges$in the limit shapes.

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The main goal of this paper is to present an alternative, real variable proof of the$T$(1)-theorem for the Cauchy integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of the$T$(1)-theorem. An example shows that the$L$^{∞}-BMO estimate for the Cauchy integral does not follow from$L$^{2}boundedness when the underlying measure is not doubling.

In [$Rong$, F., Quasi-parabolic analytic transformations of$C$^{$n$},$J. Math. Anal. Appl.$$343$(2008), 99–109], we showed the existence of “parabolic curves” for certain quasi-parabolic analytic transformations of$C$^{$n$}. Under some extra assumptions, we show the existence of “parabolic manifolds” for such transformations.

Following our previous work \cite{HLY}, we introduce the notions of partial seed homomorphisms and partial ideal rooted cluster morphisms. Related to the theory of Green's equivalences, the isomorphism classes of sub-rooted cluster algebras of a rooted cluster algebra are corresponded one-by-one to the regular $\mathcal D$-classes of the semigroup consisting of partial seed endomorphisms of the initial seed. Moreover, for a rooted cluster algebra from a Riemannian surface, they are also corresponded to the isomorphism classes of the so-called paunched surfaces.