Geometry of affine spheres and projectivelyminimal surfaces

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While affine spheres are classically known to be projectively minimal, projectively minimal surfaces make a very large class.To find intermediate classes between affine spheres and projectively minimal surfaces is the problem considered here. We characterized the condition for centroaffinely minimal surfaces to be projectively minimal and the form of the associated system of differential equations; we gave a procedure of how to construct Marcus transformations of coincidence surfaces and centroaffinely minimal surfaces of codimension two. Relatedresults include a study of reducibility of the monodromy group of certain hypergeometric systems of differential equations, an application of the affine volume to mathematical statistics, and a construction of discrete surfaces.