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

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 provides a selfcontained creation to analyze within the final decade pertaining to worldwide difficulties within the conception of submanifolds, resulting in a few varieties of Monge-Ampère equations. From the methodical viewpoint, it introduces the answer of sure Monge-Ampère equations through geometric modeling innovations. the following geometric modeling ability the best selection of a normalization and its brought about geometry on a hypersurface outlined through a neighborhood strongly convex international graph. For a greater knowing of the modeling ideas, the authors provide a selfcontained precis of relative hypersurface idea, they derive very important PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). pertaining to modeling concepts, emphasis is on rigorously established proofs and exemplary comparisons among assorted modelings.

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The transversal field Y := Y (e) in this normalization historically is called the affine normal field. Nowadays the unimodular geometry is often called Blaschke geometry; this terminology should honour Blaschke’s many contributions to this field (without ignoring important contributions by other authors). The geometry induced from the Blaschke normalization (U (e), Y := Y (e)) was sketched in Chapter 2, in particular we have G(v, w) = h(e)(v, w). As stated before, this geometry is invariant under the unimodular transformation group (including parallel translations).

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. Then there exists a relative hypersurface (x, U, Y ) such that h is the relative metric and A the relative cubic form.

4) states the Weingarten equation for Y . 2 Affine shape operator and affine extrinsic curvature For a hypersurface in Euclidean space the Weingarten equation for the unit normal implicitly defines the Euclidean shape or Weingarten operator; from this we get the extrinsic curvature functions. 4) states that dY (v) is tangential to x(M ) for any v ∈ Tp M . This situation suggests to search for an affine analogue of the Euclidean Weingarten operator. Let x; e1 , ···, en , en+1 be an equiaffine frame on M .

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