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

ISBN-10: 9812814167

ISBN-13: 9789812814166

During this monograph, the interaction among geometry and partial differential equations (PDEs) is of specific curiosity. It provides a selfcontained advent to investigate within the final decade touching on international difficulties within the idea of submanifolds, resulting in a few kinds of Monge-Ampère equations. From the methodical viewpoint, it introduces the answer of definite Monge-Ampère equations through geometric modeling innovations. the following geometric modeling capability the best collection of a normalization and its precipitated geometry on a hypersurface outlined by means of a neighborhood strongly convex international graph. For a greater figuring out of the modeling innovations, the authors provide a selfcontained precis of relative hypersurface thought, they derive vital PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). relating modeling thoughts, emphasis is on rigorously based proofs and exemplary comparisons among various modelings.

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As the centroaffine support function satisfies Λ(c) = 1, we see that h(c) = Λ−1 · h for any relative normalization. From this the centroaffine metric and its intrinsic geometry are gauge invariant; one can finally prove that this is true for all centroaffine invariants. 4 is a gauge invariant geometry. , 0, 1)). , −∂n f, 1), Y = q. Therefore one can easily construct (U (ca), Y (ca)) from an arbitrary normalization (U, Y ). Thus the Calabi geometry is gauge invariant. , we express them in terms of an arbitrary relative normalization.

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 . Assume that h = h and ∇ ∗ = ∇∗ . Then (x, U, z) and (x , U , z ) are equivalent modulo a general affine transformation. Existence Theorem. On a connected, simply connected differentiable manifold M there are given: (i) a conformal class C = {h} of semi-Riemannian metrics; (ii) a projectively flat class P = {∇∗ } of torsion free, Ricci-symmetric connections; (iii) there exists a pair (∇∗ , h) such that they satisfy Codazzi equations.

Proof. 1. Terminology. In the terminology of moving frames the integrability conditions (e)-(h) are called structure equations, which means that they are necessary and sufficient for the existence of the hypersurface structure. 5in ws-book975x65 Affine Bernstein Problems and Monge-Amp` ere Equations equations, as their coefficients contain all information on the geometry of the hypersurface. To avoid any misunderstanding, we will use the terminology integrability conditions for the system (e)-(h) and structure equations for the equations of Gauß and Weingarten.

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

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