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 supplies a selfcontained advent to investigate within the final decade referring to international difficulties within the thought of submanifolds, resulting in a few kinds of Monge-AmpÃ¨re equations. From the methodical standpoint, it introduces the answer of convinced Monge-AmpÃ¨re equations through geometric modeling ideas. the following geometric modeling capability the suitable number of a normalization and its precipitated 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 very important PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). pertaining to modeling recommendations, emphasis is on rigorously established proofs and exemplary comparisons among various modelings.

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**Additional info for Affine Bernstein Problems and Monge-Ampère Equations**

**Example text**

U fixes the tangent plane, thus, for any tangential frame, we have U, ei = 0. Exterior differentiation of the equation U, Y = 1 gives 0= 1 Ui , Y ω i + |H| n+2 i U, ei ωn+1 = Ui , Y i = 1, 2, · · ·, n. Ui , Y ω i . Hence = 0, Analogously an exterior differentiation of the equation U, ei = 0 implies 0= = U , ωij ej + ωin+1 en+1 Uj , ei ω j + ( Uj , ei + Gij ) ω j . This gives the assertion. 2) we defined the difference tensor A := ∇ − ∇. One easily verifies that also ∇∗ := ∇ − A defines another torsion free, Riccisymmetric connection.

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 . i The equations (e)-(h) between the linear differentiable forms ω i , ωij , ωn+1 are sufficient for the integration of the systems (a)-(d).

Different versions of fundamental theorems In relative geometry one can state different versions of a Fundamental Theorem, using different fundamental systems (∇∗ , h), or (∇, h), or (A, h), or even the conformal class C = {h} together with the projectively flat class P = {∇∗ }, [86]. Which version one will apply depends on the purpose. 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.

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