Download e-book for kindle: Affine Bernstein Problems and Monge-Ampère Equations by An-Min Li, Ruiwei Xu, Udo Simon, Fang Jia

Geometry Topology

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 advent to investigate within the final decade referring to international difficulties within the thought of submanifolds, resulting in a few different types of Monge-Ampère equations. From the methodical viewpoint, it introduces the answer of convinced Monge-Ampère equations through geometric modeling recommendations. right here geometric modeling capacity the precise collection of a normalization and its caused geometry on a hypersurface outlined by means of an area strongly convex worldwide graph. For a greater knowing of the modeling options, the authors provide a selfcontained precis of relative hypersurface conception, they derive vital PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). bearing on modeling recommendations, emphasis is on rigorously based proofs and exemplary comparisons among assorted modelings.

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Proposition. Let M be a non-degenerate hypersurface in An+1 . 1) M is an affine hypersphere. 2) B = L1 · G. 3) B = L1 · id. , n. Definition and Remark. Assume that x is locally strongly convex; that means that the Blaschke metric G is (positive) definite. In this case the affine Weingarten operator B has n real eigenvalues λ1 , λ2 , · · ·, λn , the affine principal curvatures. Then: (i) The relation B = L1 · G is equivalent to the equality of the affine principal curvatures: λ1 = λ 2 = · · · = λ n .

According to our notation in Riemannian geometry, κ(h) is the normed relative scalar curvature of the relative metric h. 4 Classical version of the fundamental theorem Uniqueness Theorem. Let (x, U, Y ) and (x , U , Y ) be non-degenerate hypersurfaces with the same parameter manifold: x, x : M → An+1 . Assume that h = h and A = A. Then (x, U, Y ) and (x , U , Y ) are equivalent modulo a general affine transformation. Existence Theorem. 5in Local Relative Hypersurfaces ws-book975x65 39 such that the integrability conditions in the classical version are satisfied.

Affine Gauß maps and Euclidean structure. In the case rank B = n it is often convenient to consider the two hypersurfaces, defined from the affine Gauß maps, as follows: We consider a Euclidean inner product , : V × V → R on V and identify V and V ∗ as usual. The three relations U, Y = 1, U, dY = 0, dU, Y = 0 imply that both affine Gauß indicatrices are a polar pair, that means they correspond via an inversion at the unit sphere. , en ]. , en ]. Using the Euclidean structure of V , we can express the conormal in terms of the Euclidean unit normal µ of x: 1 U = |K| n+2 · µ.

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