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 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-Ampère equations. From the methodical standpoint, it introduces the answer of convinced Monge-Ampère 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.

Show description

Read or Download Affine Bernstein Problems and Monge-Ampère Equations PDF

Similar geometry & topology books

Mathematics in Ancient and Medieval India

Heritage of arithmetic in historical and medieval India

Handbook of Geometric Analysis, Vol. 2 (Advanced Lectures in Mathematics No. 13)

Geometric research combines differential equations and differential geometry. a major element is to resolve geometric difficulties by way of learning differential equations. along with a few recognized linear differential operators akin to the Laplace operator, many differential equations bobbing up from differential geometry are nonlinear.

Vector Bundles and Complex Geometry

This quantity includes a number of papers from the convention on Vector Bundles held at Miraflores de los angeles Sierra, Madrid, Spain on June 16-20, 2008, which commemorated S. Ramanan on his seventieth birthday. the most components lined during this quantity are vector bundles, parabolic bundles, abelian forms, Hilbert schemes, touch constructions, index conception, Hodge idea, and geometric invariant thought.

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.

Download PDF sample

Rated 4.45 of 5 – based on 35 votes

Categories: Geometry Topology