The Bernstein problem for affine maximal hypersurfaces

By Neil S. Trudinger, Xu-jia Wang. 2000

193 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7d3280bf6c7a8f2ae95dbf8bf66820a31433b28c

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In Applied Mathematics

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Citations in the corpus

Extremal metrics on toric surfaces: a continuity method (2008, result): “The inspiration for these results comes from the work of Trudinger and Wang in [15], [16] and, particularly, [17], Remark 4.”
A Liouville theorem for solutions of the Monge–Ampère equation with periodic data (2004, background): “Trudinger and Wang proved [22] that the only open convex subset Ω of R which admits a convexC2 solution of det (D2u) = 1 in Ω with limx→∂Ω u(x) = ∞ is Ω = R.”
The affine Plateau problem (2004, methodology): “In two dimensions, the local strict convexity of affine maximal functions which vanish on the boundary of a convex domain Ω follows from our treatment of the affine Bernstein problem in [21] and the approximation in Section 6.”
On locally convex hypersurfaces with boundary (2002, methodology): “Using the Main Lemma above we proved in [17] that an affine complete locally convex hypersurface is also Euclidean complete, which implies, by virtue of our solution of the affine Bernstein problem in [18], that an affine complete, affine maximal surface in R(3) is an elliptic paraboloid.”
C1 regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two (2009, methodology): “We choose to present separately the proof for c(x, y) =− x · y: indeed the proof in this case is much shorter, and it might help the reader to follow the argument in the general case (see also for example [ 8 ,9], where similar techniques are employed).”
Affine complete locally convex hypersurfaces (2002, background): “Corollary 2 follows immediately since by Theorem A, M is Euclidean complete and so the result in [3] (or [13]) applies.”
Centro-Affine Invariants for Smooth Convex Bodies (2010, background): “2 ALINA STANCU1 to be an invaluable tool in convexity (see the monograph [21]), stochastic geometry, [4], [9] – [12], [41], differential geometry and differential equations [1], [3], [42], [49] – [53].”
Interior estimates for solutions of Abreu's equation (2004, methodology): “We get a lower bound on the determinant in all cases and, using the technique of Trudinger and Wang, an upper bound involving also a “modulus of convexity”.”
General affine surface areas (2009, background): “Numerous important results regarding equi-affine surface area were obtained in recent years (see, for example, [1], [2], [40], [43] – [46]).”
An extension to a theorem of Jörgens, Calabi, and Pogorelov (2003, methodology): “It was proven by Trudinger and Wang in [19] that the only open convex subset of Rn which admits a convex C2 solution of det(D2u) = 1 in with limx→∂ u(x) = ∞ is = Rn .”
Survey on Affine Spheres (2008, methodology): “The corresponding problem is open in higher dimension, but non-smooth viscosity solutions in dimension n ≥ 10 are presented in [70].”
Convex solutions to the mean curvature flow (2004, unknown): “See [25] for a brief discussion.”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “Another classical instance where the Monge-Ampère equation arises is affine geometry, more precisely in the “affine sphere problem” and the “affine maximal surfaces” problem (see for instance [28, 98, 30, 107, 108, 109]).”