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You can see the book by Li-Simon-Zhao:"Global Affine differetial geometry of hypersurfaces. Berlin: Walter de Gruyter, 1993." I think yours definition of the constant centro-affine curvatire curvature surfaces means the ellptic affine spheres. Let M be a surface with constant centro-affine curvatire curvature 1, locally given by the graph $(x, f(x))$, then by the Legendre transformation relative to $f$, the equation is (with center at the orgin) $$\det(D^2 u)=(u)^{-4}.$$ The answer can be found in this book, due to Blaschke, Deicke, Calabi.
You can see the book by Li-Simon-Zhao:"Global Affine differetial geometry of hypersurfaces. Berlin: Walter de Gruyter, 1993." I think yours definition of the constant centro-affine curvatire surfaces means the hyperbolic ellptic affine spheres. Let M be a surface with constant centro-affine curvatire surface curvature 1, locally given by the graph $(x, f(x))$, then by the Legendre transformation relative to $f$, the equation is (with center at the orgin) $$\det(D^2 u)=(-u)^{-4}.$$u)=(u)^{-4}.$$The answer can be found in this book(proved by Minkowski integral formulas, and hold with more larger class)due to Blaschke, Deicke, Calabi. 1 You can see the book by Li-Simon-Zhao:"Global Affine differetial geometry of hypersurfaces. Berlin: Walter de Gruyter, 1993." I think yours definition of the constant centro-affine curvatire surfaces means the hyperbolic affine spheres. Let M be a surface with constant centro-affine curvatire surface 1, locally given by the graph (x, f(x)), then by the Legendre transformation relative to f, the equation is$$\det(D^2 u)=(-u)^{-4}.