By Li A.-M., et al.

During this monograph, the interaction among geometry and partial differential equations (PDEs) is of specific curiosity. It offers a selfcontained advent to investigate within the final decade referring to worldwide difficulties within the thought of submanifolds, resulting in a few different types of Monge-AmpÃ¨re equations. From the methodical perspective, it introduces the answer of yes Monge-AmpÃ¨re equations through geometric modeling thoughts. right here geometric modeling potential the proper number of a normalization and its brought on geometry on a hypersurface outlined by way of a neighborhood strongly convex worldwide graph. For a greater realizing of the modeling thoughts, the authors supply 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). bearing on modeling strategies, emphasis is on rigorously established proofs and exemplary comparisons among diverse modelings.

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Thus one can write down structure equations of Gauß type for any conormal field U : ¯ v dU (w) = dU (∇∗ w) + 1 Ric∗ (v, w) (−U ). ∇ v n−1 U : M → V ∗ is called the conormal indicatrix of (x, U, z). One verifies: Properties of the coefficients. , they have the same unparametrized geodesics; the class P = {∇∗ } is projectively flat. 2 Fundamental theorem for non-degenerate hypersurfaces Uniqueness Theorem. Let (x, U, z) and (x , U , z ) be non-degenerate hypersurfaces with the same parameter manifold: x, x : M → An+1 .

Properties of the coefficients. Let (x, U, Y ) be a relative hypersurface. Then: (i) The induced connection ∇ is torsion free and Ricci-symmetric. (ii) The relative shape operator S is h-self-adjoint and satisfies (n − 1)S (v, w) := (n − 1)h(Sv, w) = Ric∗ (v, w). Its trace gives the relative mean curvature nL1 := tr S. (iii) The triple (∇, h, ∇∗ ) is conjugate, that means it satisfies the following generalization of the Ricci Lemma in Riemannian geometry: u h(v, w) = h(∇u v, w) + h(v, ∇∗u w). (iv) The Levi-Civita connection ∇(h) of the non-degenerate relative metric h satisfies ∇(h) = 12 (∇ + ∇∗ ).

A) similarly locally defines a proper affine sphere. 8, namely that both classes of affine spheres are very large. 3 The Pick invariant on affine hyperspheres We recall a well known inequality for the Laplacian of the Pick invariant on affine hyperspheres. For n = 2 it first was obtained by W. Blaschke [9]. For higher dimensional affine spheres it was obtained by E. Calabi [19] in the case of parabolic affine hyperspheres, and for arbitrary affine hyperspheres by R. Schneider [82] (with a minor misprint of a constant) and also by Cheng and Yau [25].