Affine Bernstein Problems and Monge-Ampère Equations by An-Min Li, Ruiwei Xu, Udo Simon, Fang Jia

By An-Min Li, Ruiwei Xu, Udo Simon, Fang Jia

During this monograph, the interaction among geometry and partial differential equations (PDEs) is of specific curiosity. It supplies a selfcontained advent to investigate within the final decade touching on international difficulties within the concept of submanifolds, resulting in a few different types of Monge-Ampère equations. From the methodical perspective, it introduces the answer of convinced Monge-Ampère equations through geometric modeling ideas. the following geometric modeling ability the proper number of a normalization and its triggered geometry on a hypersurface outlined by way of a neighborhood strongly convex international graph. For a greater figuring out of the modeling options, the authors supply a selfcontained precis of relative hypersurface concept, they derive very important PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). referring to modeling recommendations, emphasis is on rigorously established proofs and exemplary comparisons among varied modelings.

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3. 7 n+2 2n gradh(c) ln Λ(e). Different versions of fundamental theorems In relative geometry one can state different versions of a Fundamental Theorem, using different fundamental systems (∇∗ , h), or (∇, h), or (A, h), or even the conformal class C = {h} together with the projectively flat class P = {∇∗ }, [86]. Which version one will apply depends on the purpose. The integrability conditions of the classical Blaschke version, based on the fundamental system (A, h), have a very complicated form; this is a disadvantage.

I) When x : M → An+1 is locally strongly convex, from the above calculation one can easily see that Y always points to the concave side of x(M ). (ii) The geometric meaning of the apolarity condition is the following: Both, the Levi-Civita and the induced connection, have symmetric Ricci tensors. Thus both connections ∇ and ∇ admit parallel volume forms; in case of the Levi-Civita connection it is the Riemannian volume form. Now the apolarity condition, written in the form Gij Γkij = Gij Γkij , also implies that both volume forms coincide (modulo a non-zero constant factor).

N ) and u(ξ) the Blaschke metric is given by 2 u Gij = ρ ∂ξ∂i ∂ξ , j and ∂2u ∂ξi ∂ξj ∂2f ∂xi ∂xj is the inverse matrix of ∂2u ∂ξi ∂ξj ρ = det . 91). By a similar calculation as above we get ∆= 1 ρ 2 uij ∂ξ∂i ∂ξj − 2 ρ2 ∂ρ uij ∂ξ j ∂ ∂ξi . 5in ws-book975x65 Affine Bernstein Problems and Monge-Amp` ere Equations Affine Spheres and Quadrics As before we consider non-degenerate hypersurfaces with unimodular normalization. 1 in [58]. Definition. A non-degenerate hypersurface x in An+1 is called an affine hypersphere if the affine normal line bundle has one of the following two properties: (i) All affine normal lines meet at one point c0 ∈ An+1 ; in this case x is called a proper affine hypersphere with center c0 .

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