Interior estimates for solutions of Abreu’s equation
نویسنده
چکیده
Here (u) denotes the inverse of the Hessian matrix uij = ∂u ∂xi∂xj . We call this PDE Abreu’s equation since the expression S(u) appears in [1], in the study of the differential geometry of toric varieties. In this paper we will be primarily interested in the case when A is a constant. Solutions to Equation 1 then correspond to certain Kahler metrics of constant scalar curvature, as we will recall in more detail in Section 5 below. Our purpose is to derive a priori estimates for solutions of Abreu’s equation, which can be applied to existence questions for such constant scalar curvature metrics, on the lines of [6]. However in the present paper we will keep the differential geometry in the background, and concentrate on the PDE aspects of the equation. Abreu’s equation can be fitted, as a limiting case, into a class of equations considered by Trudinger and Wang [12],[13]. These authors study the EulerLagrange equation of the functional
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