By An-min Li

During this monograph, the interaction among geometry and partial differential equations (PDEs) is of specific curiosity. It offers a selfcontained creation to investigate within the final decade pertaining to worldwide difficulties within the idea of submanifolds, resulting in a few varieties of Monge-Ampère equations.

From the methodical standpoint, it introduces the answer of sure Monge-Ampère equations through geometric modeling strategies. right here geometric modeling capacity the best collection of a normalization and its brought on geometry on a hypersurface outlined through an area strongly convex international graph. For a greater knowing of the modeling options, the authors supply a selfcontained precis of relative hypersurface thought, they derive very important PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). touching on modeling strategies, emphasis is on rigorously established proofs and exemplary comparisons among various modelings.

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**Extra info for Affine Berstein Problems and Monge-Ampere Equations**

**Example text**

B = 0). Proof. 2). For (ii) see [11]. Characterization of quadrics Theorem. (i) Any hyperquadric is an affine hypersphere. The quadric has a center if L1 = 0. (ii) A non-degenerate hypersurface x is a quadric if and only if the cubic form A vanishes identically on M . 4 in [58] (there we consider only locally strongly convex hypersurfaces). (b) in [56]. 4 below we will generalize the foregoing Theorem. 5in ws-book975x65 Chapter 3 Local Relative Hypersurfaces E. M¨ uller was the first to extend the development of a unimodular hypersurface theory to a so called relative hypersurface theory.

By a direct calculation we get ∆= 2 Gij ∂x∂i ∂xj − ∂ρ f ij ∂x j 2 ρ2 ∂ ∂xi + 1 ρ where (f ij ) denotes the inverse matrix of (fij ) and fij = entiation of the equation f ik fkj = δji one finds ∂f ik ∂xi fkj i,k f ik =− ∂fkj ∂xi = ∂f ij ∂ ∂xi ∂xj , ∂2f ∂xi ∂xj . 3) Taking the differ- n+2 ∂ρ ρ ∂xj . i,k It follows that ∂f ik ∂xi = ∂ρ f jk ∂x j . 3) and obtain ∆= 1 ρ 2 f ij ∂x∂i ∂xj + n ρ2 ∂ρ f ij ∂x j ∂ ∂xi . 5in ws-book975x65 Local Equiaffine Hypersurfaces 29 where e∗n+1 is in the affine normal direction.

It follows that ain+1 = ∂ f ji ∂x j ln ρ, and hence e∗n+1 = en+1 + ∂ f ji ∂x j ln ρ · ei . Therefore 1 1 1 Y = H n+2 e∗n+1 = H n+2 ∂ n+2 e f ji ∂x n+1 , j ln ρ · ei + H 2 f . where H = det ∂x∂i ∂x j Let x denote the position vector of the hypersurface M . We have dx = wα e∗α , de∗α = wαβ e∗β . The wα , wαβ are the Maurer-Cartan forms of the unimodular affine group. We compute i wn+1 = dain+1 − ain+1 d ln ρ f ki ∂x∂ k ln ρ − f ki ∂x∂ k ln ρ · ∂ ∂xj = ∂ ∂xj ln ρ wj . Therefore the affine Weingarten tensor is lk ∂ ∂ ∂ ∂ f lk ∂x − ∂x i l ln ρ + f ∂xl ln ρ ∂xi ln ρ fkj Bij = =− 2 1 ∂ ρ ρ ∂xi ∂xj + 2 ∂ρ ∂ρ ρ2 ∂xi ∂xj f kl ∂ρ ∂fij ρ ∂xl ∂xk .