Variational problems and PDEs in affine differential geometry

Abstract

Abstract. This paper is part of the autumn school on "Variational problems and higher order PDEs for affine hypersurfaces". We discuss variational problems in equiaffine differential geometry, centroaffine differential geometry and relative differential geometry, which have been studied by Blaschke We first derive the Euler-Lagrange equations in these settings; these equations are complicated, strongly nonlinear fourth order PDEs. We consider classes of solutions satisfying these equations together with completeness conditions. We also formulate Bernstein problems and give partial solutions. Contents

Improper affine spheres and δ-invariants

“…This is written as D X f * (Y ) = f * (∇ X Y ) + h(X, Y )ξ, where h is a symmetric tensor of type (0, 2). If h is non-degenerate, then h can be considered as semi-Riemannian metric on M , called the Calabi metric [3]. Let∇ denote Levi-Civita connection of (M, h) and K be the difference tensor ∇ −∇ on M .…”