By Li A.-M., et al.
During this monograph, the interaction among geometry and partial differential equations (PDEs) is of specific curiosity. It provides a selfcontained advent to analyze within the final decade referring to worldwide difficulties within the conception of submanifolds, resulting in a few sorts of Monge-AmpÃ¨re equations. From the methodical perspective, it introduces the answer of definite Monge-AmpÃ¨re equations through geometric modeling thoughts. the following geometric modeling potential the suitable collection of a normalization and its brought on geometry on a hypersurface outlined by way of an area strongly convex worldwide graph. For a greater knowing of the modeling thoughts, the authors supply a selfcontained precis of relative hypersurface concept, they derive very important PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). referring to modeling innovations, emphasis is on conscientiously established proofs and exemplary comparisons among diverse modelings.
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Additional resources for Affine Bernstein problems and Monge-Ampere equations
3 The Pick invariant on affine hyperspheres We recall a well known inequality for the Laplacian of the Pick invariant on affine hyperspheres. For n = 2 it first was obtained by W. Blaschke . For higher dimensional affine spheres it was obtained by E. Calabi  in the case of parabolic affine hyperspheres, and for arbitrary affine hyperspheres by R. Schneider  (with a minor misprint of a constant) and also by Cheng and Yau . U. Simon calculated ∆J for arbitrary non-degenerate hypersurfaces and applied his formula to get some new characterization of ellipsoids (cf.
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).
The integrability conditions of the classical Blaschke version, based on the fundamental system (A, h), have a very complicated form; this is a disadvantage. But this version is useful for the application of subtle tools from Riemannian geometry, like maximum principles or the Laplacian Comparison Theorem. , are modifications of the version using (∇∗ , h). These versions lead to a much better understanding of the theory, based on the results in . 4 Gauge Invariance and Relative Geometry To investigate the geometry of a given non-degenerate hypersurface, we have different possibilities for an appropriate choice of a normalization; even within the distinguished class of relative normalizations there are infinitely many possibilities.
Affine Bernstein problems and Monge-Ampere equations by Li A.-M., et al.