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Let x : M m → S n be a submanifold in the n-dimensional sphere S n without umbilics. Two basic invariants of x under the Möbius transformation group in S n are a 1-form Φ called the Möbius form and a symmetric (0, 2) tensor A called the Blaschke tensor. x is said to be Möbius isotropic in S n if Φ ≡ 0 and A = λdx • dx for some smooth function λ. An interesting property for a Möbius isotropic submanifold is that its conformal Gauss map is harmonic. The main result in this paper is the classification of Möbius isotropic submanifolds in S n . We show that (i) if λ > 0, then x is Möbius equivalent to a minimal submanifold with constant scalar curvature in S n ; (ii) if λ = 0, then x is Möbius equivalent to the preimage of a stereographic projection of a minimal submanifold with constant scalar curvature in the n-dimensional Euclidean space R n ; (iii) if λ < 0, then x is Möbius equivalent to the image of the standard conformal map τ : H n → S n + of a minimal submanifold with constant scalar curvature in the n-dimensional hyperbolic space H n . This result shows that one can use Möbius differential geometry to unify the three different classes of minimal submanifolds with constant scalar curvature in S n , R n and H n .

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Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann surface by Riemann surface with elementary branch points and prescribed ramification type over a special point.

ABSTRACT. We study a generalized Abreu Equation in n-dimensional polytopes and prove some differential inequalities for homogeneous toric bundles.