Let $\Sigma$ be a complete Riemannian manifold with the volume doubling property and uniform Neumann-Poincar$\mathrm{\acute{e}}$ inequality. We show that any positive minimal graphic function on is constant.

In this paper we discuss various aspects of non-compact models of CFT of the type: ∏NL j=1{N = 2 Liouville theory}j ⊗ ∏NM i=1{N = 2 minimal model}i and ∏NL j=1{SL(2;R)/U(1) supercoset}j⊗ ∏NM i=1 {N = 2 minimal model}i. These models are related to each other by T-duality. Such string vacua are expected to describe non-compact Calabi-Yau compactifications, typically ALE fibrations over (weighted)...