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 offers a selfcontained creation to analyze within the final decade relating worldwide difficulties within the concept of submanifolds, resulting in a few forms of Monge-Ampère equations. From the methodical standpoint, it introduces the answer of yes Monge-Ampère equations through geometric modeling thoughts. right here geometric modeling capability the precise collection of a normalization and its prompted geometry on a hypersurface outlined via an area strongly convex international graph. For a greater figuring out of the modeling concepts, the authors provide a selfcontained precis of relative hypersurface idea, they derive vital PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). touching on modeling innovations, emphasis is on conscientiously established proofs and exemplary comparisons among diversified modelings.

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We recall the notion of the conormal line bundle along M and call any nowhere vanishing section of this bundle a conormal field on M . We are going to search for a conormal field that is invariant under unimodular transformations. First let us recall some elementary facts from multilinear algebra. Let η1 , η2 , · · ·, ηn+1 be a basis of V . Then there exists a canonical isomorphism from the vector space of exterior n-forms to V ∗ , which is given by n i: n V →V∗ where i (α) (v) = a, α∈ V, a∈R if and only if Thus we can identify n α ∧ v = a · η1 ∧ η2 ∧ · · · ∧ ηn+1 .

7) where L1 is the affine mean curvature as before, and Rik denote the local components of the Ricci tensor of (M, G) . 6) we obtain, by another contraction, the so called Equiaffine Theorema Egregium. 8) where κ= 1 n(n−1) Gik Gjl Rijkl . 9) According to our notation in Riemannian geometry R = n(n − 1)κ is the scalar curvature and κ the normed scalar curvature of the metric G. Corollary. The form B can be expressed in terms of G, A and their derivatives: Bjk = (κ − J) Gjk − 2 n Aljk, l . 76. , en+1 }.

Obviously the parabolic affine hyperspheres are exactly the improper affine hyperspheres. (iv) For an elliptic affine hypersphere, the center is on the concave side of x(M ). For a hyperbolic affine hypersphere the center is on the convex side of x(M ). For a parabolic affine hypersphere we may consider the center to be at infinity. (v) For a hypersurface in Euclidean space with Euclidean Weingarten operator S and mean curvature H the equation S = H · id implies H = const; if H = 0 we have a hyperplane, if H = 0 we have a sphere with curvature equal to H > 0.

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