Affine Berstein Problems and Monge-Ampere Equations by An-min Li

By An-min Li

During this monograph, the interaction among geometry and partial differential equations (PDEs) is of specific curiosity. It provides a selfcontained creation to analyze within the final decade referring to worldwide difficulties within the idea of submanifolds, resulting in a few varieties of Monge-Ampère equations.

From the methodical standpoint, it introduces the answer of yes Monge-Ampère equations through geometric modeling thoughts. the following geometric modeling capability the perfect selection of a normalization and its triggered geometry on a hypersurface outlined by means of an area strongly convex international graph. For a greater realizing of the modeling recommendations, the authors supply a selfcontained precis of relative hypersurface conception, they derive very important PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). touching on modeling suggestions, emphasis is on conscientiously based proofs and exemplary comparisons among assorted modelings.

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2 Proper affine hyperspheres Let x be an elliptic or hyperbolic affine hypersphere and assume that x locally is given as a graph of a strictly convex C ∞ -function on a domain Ω ⊂ Rn : xn+1 = f x1 , · · ·, xn , x1 , · · ·, xn ∈ Ω. 4: F : Ω → Rn where (x1 , · · ·, xn ) → (ξ1 , · · ·, ξn ) and ξi := ∂i f = ∂f ∂xi , i = 1, 2, · · ·, n. 5in ws-book975x65 The Theorem of J¨ orgens-Calabi-Pogorelov 49 When Ω is convex, F : Ω → F (Ω) is a diffeomorphism. Then the hypersurface can be represented in terms of (ξ1 , ξ2 , · · ·, ξn ) as follows: x = x1 , · · ·, xn , f x1 , · · ·, xn = ∂u ∂ξ1 , · · ·, ∂u ∂ξn , −u+ ξi ∂u ∂ξi .

1 23 Properties of the Fubini-Pick form Lemma. 2) hik ωjk − hkj ωik . Aijk = Ajik = Aikj . 3) (iii) The cubic form A is invariant under unimodular transformations. (iv) By definition the difference tensor A measures the deviation of the two connections ∇ and ∇. 2. 2; in the covariant form below there appear A and G as coefficients, both are equiaffinely invariant tensor fields. The simplest scalar invariant of the metric and the cubic form is defined by J := 1 n(n−1) Gil Gjm Gkr Aijk Almr = 1 n(n−1) A 2, where the tensor norm · is taken with respect to the Blaschke metric G.

They give information about relations between the invariants that appear in the relative structure equations. We are going to express these conditions in terms of the quadratic and cubic forms h, S , and A . Locally write Sij := Sij . We express the integrability conditions in terms of the metric h and the cubic form A, in analogy to the classical approach in Blaschke’s unimodular theory. 3 Classical version of the integrability conditions In covariant form the integrability conditions read: Aijk,l − Aijl,k = Rijkl = 1 2 (hik Sjl + hjk Sil − hil Sjk − hjl Sik ) , m (Am il Amjk − Aik Amjl ) + 1 2 (hik Sjl + hjl Sik − hil Sjk − hjk Sil ) , Sjl Alik − Slk Alij .

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