Download PDF by Li A.-M., et al.: Affine Bernstein problems and Monge-Ampere equations

By Li A.-M., et al.

ISBN-10: 9812814167

ISBN-13: 9789812814166

During this monograph, the interaction among geometry and partial differential equations (PDEs) is of specific curiosity. It offers a selfcontained advent to analyze within the final decade touching on international difficulties within the thought of submanifolds, resulting in a few forms of Monge-Ampère equations. From the methodical viewpoint, it introduces the answer of yes Monge-Ampère equations through geometric modeling thoughts. the following geometric modeling capacity the suitable selection of a normalization and its triggered geometry on a hypersurface outlined via an area strongly convex worldwide graph. For a greater realizing of the modeling recommendations, 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). referring to modeling thoughts, emphasis is on conscientiously based proofs and exemplary comparisons among various modelings.

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Additional info for Affine Bernstein problems and Monge-Ampere equations

Example text

5) implies that on any hyperovaloid there exist points such that the Weingarten form is (positive) definite. 5in ws-book975x65 Local Equiaffine Hypersurfaces 25 give a linear system of first order PDEs for a local frame {e1 , · · ·, en , en+1 }. The coefficients define linear forms that are related to the two connections ∇, ∇, the quadratic form G, the cubic form A and the Weingarten operator: lik ωkn+1 = − i ωn+1 =− ωin+1 = lki ω k , ωij = Γjik ω k , −1 hij ω j , where Gij = |H| n+2 hij . From the integration theory for such linear systems we know that there exists at most one solution {e1 , · · ·, en , en+1 } for given coefficients and given initial values.

Relative structure equations in covariant local notation. We rewrite the structure equations in terms of h-covariant differentiation and with affinely invariant coefficients as follows: Gauß equation for x : Akij xk + hij Y. x,ij = k Yj = − Weingarten equation : U,ij = − Gauß equation for U : Sji xi . i Akij Uk − Bij U, ∗ . where the relative Weingarten form satisfies (n − 1)Bij = (n − 1)hik Sjk = Rij The relative support function. Let b be a fixed vector in V and U a relative conormal of x. The function Λ : M → R defined by Λ(p) := U, b − x(p) , p ∈ M, is called the relative support function of (x, U, Y ) with respect to the fixed point b ∈ Rn+1 .

Thus, roughly speaking, the coefficients must contain all geometric information about the hypersurface x. The existence of a solution of the system depends on the fact that the coefficients satisfy integrability conditions. We are going to clarify this. Choose a local equiaffine frame field {x; e1 , · · ·, en , en+1 } over M such that en+1 = Y , Gij = δij . Theorem. The integrability conditions of the system (a) dx = ω i ei , (b) dei = ωij ej + ωin+1 en+1 , (c) den+1 = (d) ωin+1 = ω i , read (e) i ωn+1 ei , n+1 ωn+1 =0 ωii = 0, (f ) dω i = ω j ∧ ωji , (g) dωij = j , ωik ∧ ωkj + ωin+1 ∧ ωn+1 ωin+1 = ω i , j i (h) dωn+1 = ωn+1 ∧ ωji .

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Affine Bernstein problems and Monge-Ampere equations by Li A.-M., et al.


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