Calibrations associated to Monge-Ampère equations

Warren, Micah

doi:10.1090/s0002-9947-10-05109-3

Cited by 47 publications

(44 citation statements)

References 17 publications

(19 reference statements)

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“…By Theorem 1.1, we reprove the existence result on the second boundary value problem for special Lagrangian equations which belongs to S.Brendle and M.Warren. By [1] and [6], we present the similar special Lagrangian diffeomorphism problem in the following. The two convex domains Ω,Ω ⊂ R n with smooth boundary are fixed.…”

Section: Lemma 52 (U-estimates)mentioning

confidence: 99%

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On the Second Boundary Value Problem for a Class of Fully Nonlinear Flows I

Huang

2017

International Mathematics Research Notices

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Abstract. In this paper, a class of fully nonlinear flows with nonlinear Neumann type boundary condition is considered. This problem was solved partly by the first author under the assumption that the flow is the parabolic type special Lagrangian equation in R 2n . We show that the convexity is preserved for solutions of the fully nonlinear parabolic equations and prove the long time existence and convergence of the flow. In particular, we can prescribe the second boundary value problems for a family of special Lagrangian graphs in Euclidean and pseudoEuclidean space.

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Section: Lemma 52 (U-estimates)mentioning

confidence: 99%

“…In calibrated geometry, we adapt the details of the extremal Lagrangian surfaces in (R n × R n , g τ ) which were firstly obtained by M. Warren [6] and arise as solutions to a family of special Lagrangian equations: Here we consider the following fully nonlinear elliptic equations with second boundary condition:…”

Section: Introductionmentioning

confidence: 99%

On the Second Boundary Value Problem for a Class of Fully Nonlinear Flows I

Huang

2017

International Mathematics Research Notices

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“…This reveals another unexpected connection between optimal transportation and symplectic (or para-Kähler) geometry. When ρ andρ are given by the Euclidean volumes on two convex domains, and c(x,x) = |x −x| 2 /2, this is related to the work of Wolfson [55] and Warren [54] on special Lagrangian submanifolds, where a pseudo-Riemannian metric of signature (n, n) also appears [54]. We defer the details of this development to the appendix below.…”

Section: Perspective and Conclusionmentioning

confidence: 99%

Continuity, curvature, and the general covariance of optimal transportation

Kim¹,

McCann²

2010

J. Eur. Math. Soc.

117

261

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Abstract. Let M andM be n-dimensional manifolds equipped with suitable Borel probability measures ρ andρ. For subdomains M andM of R n , Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C 4 (M ×M) to guarantee smoothness of the optimal map pushing ρ forward toρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M ×M. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry.Using the pseudo-Riemannian structure, we extend Ma, Trudinger and Wang's conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principle characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and M of R n . This maximum principle plays a key role in Loeper's Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered.

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“…Here we consider the minimal Lagrangian diffeomorphism problem [1] which is equivalent to the following fully nonlinear elliptic equations with second boundary condition (cf. [2], [3] and [10]):…”

Section: Introductionmentioning

confidence: 99%

“…Our motivation of studying equation (1.1) is that they have geometric meanings which were studied by Warren. To illustrate this, let us recall the definition of Lagrangian and special Lagrangian graph as in [9] or [10]. The graph Σ = {(x, f (x)) :…”

Section: Introductionmentioning

confidence: 99%

On the second boundary value problem for a class of fully nonlinear flows II

Chen

Huang

2018

Arch. Math.

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This article is a continuation of earlier work [R.L. Huang and Y.H. Ye, On the second boundary value problem for a class of fully nonlinear flows I, to appear in International Mathematics Research Notices], where the long time existence and convergence were given on some general parabolic type special Lagrangian equations. The long time existence and convergence of the flow had been obtained in all cases. In particular, we can prescribe the second boundary value problems for a family of special Lagrangian graphs.(1.1) F τ (D 2 u) = c, x ∈ Ω,

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Calibrations associated to Monge-Ampère equations

Abstract: Abstract. We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to MongeAmpère equations.

Cited by 47 publications

References 17 publications

On the Second Boundary Value Problem for a Class of Fully Nonlinear Flows I

On the Second Boundary Value Problem for a Class of Fully Nonlinear Flows I

Continuity, curvature, and the general covariance of optimal transportation

On the second boundary value problem for a class of fully nonlinear flows II

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