By Li A.-M., et al.
During this monograph, the interaction among geometry and partial differential equations (PDEs) is of specific curiosity. It supplies a selfcontained creation to investigate within the final decade referring to international difficulties within the concept of submanifolds, resulting in a few forms of Monge-AmpÃ¨re equations. From the methodical perspective, it introduces the answer of definite Monge-AmpÃ¨re equations through geometric modeling strategies. the following geometric modeling capability the correct collection of a normalization and its brought about geometry on a hypersurface outlined by means of a neighborhood strongly convex international graph. For a greater realizing of the modeling concepts, the authors supply a selfcontained precis of relative hypersurface idea, they derive vital PDEs (e.g. affine spheres, affine maximal surfaces, and the affine consistent suggest curvature equation). pertaining to modeling thoughts, emphasis is on rigorously dependent proofs and exemplary comparisons among diversified modelings.
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Additional info for Affine Bernstein problems and Monge-Ampere equations
1) M is an affine hypersphere. 2) B = L1 · G. 3) B = L1 · id. , n. Definition and Remark. Assume that x is locally strongly convex; that means that the Blaschke metric G is (positive) definite. In this case the affine Weingarten operator B has n real eigenvalues λ1 , λ2 , · · ·, λn , the affine principal curvatures. Then: (i) The relation B = L1 · G is equivalent to the equality of the affine principal curvatures: λ1 = λ 2 = · · · = λ n . (ii) All affine principal curvatures are constant. (iii) An affine hypersphere is called an elliptic affine hypersphere if L1 > 0; it is called hyperbolic if L1 < 0; it is called parabolic if L1 = 0.
0, 1)). , −∂n f, 1), Y = q. Therefore one can easily construct (U (ca), Y (ca)) from an arbitrary normalization (U, Y ). Thus the Calabi geometry is gauge invariant. , we express them in terms of an arbitrary relative normalization. The traceless tensor field A and the characterization of quadrics. On a non-degenerate hypersurface, let U be an arbitrary conormal field; from U one can define the corresponding metric h and the projectively flat connection ∇∗ , and from this A and T . Define the symmetric (1,2) tensor field A as traceless part of A: A(v, w) := A(v, w) − n n+2 (T (v)w + T (w)v + h(v, w)T ); then (a) A is a gauge invariant; (b) A = A(e), thus A = 0 if and only if the hypersurface is a hyperquadric; (c) we calculate A 2 = A 2 − 3n2 n+2 T 2 .
Lemma. . The notion of Euclidean completeness on M is independent of the choice of a Euclidean metric on An+1 . Proof. Consider two inner products on V , denoted by , and , ; they n+1 define two Euclidean metrics on A . Let η1 , η2 , · · ·, ηn+1 and η¯1 , η¯2 , · · ·, η¯n+1 be orthonormal bases in V relative to , and , , respectively, related by ηα = Cαβ η¯β where C = (Cαβ ) ∈ GL(n + 1, R). The Euclidean structures of V induce Euclidean metrics on M ; we can write them in the form dx1 dx2 · 2 1 2 n+1 ds = dx , dx , · · ·, dx := (dx)τ · (dx); · · dxn+1 d¯ s2 = (C · dx)τ · C · dx = dxτ · C τ C · dx; here we use an obvious matrix notation, and C τ denotes the transposed matrix of C.