Pseudoconvexity for the special Lagrangian potential equation
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Calculus of Variations
Pseudoconvexity for the special Lagrangian potential equation F. Reese Harvey2 · H. Blaine Lawson Jr.1 Received: 1 August 2019 / Accepted: 31 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The Special Lagrangian Potential Equation for a function u on a domain ⊂ Rn is given by tr{arctan(D 2 u)} = θ for a contant θ ∈ (−n π2 , n π2 ). For C 2 solutions the graph of Du in × Rn is a special Lagrangian submanfold. Much has been understood about the Dirichlet problem for this equation, but the existence result relies on explicitly computing the associated boundary conditions (or, otherwise said, computing the pseudo-convexity for the associated potential theory). This is done in this paper, and the answer is interesting.The result carries g over to many related equations—for example, those obtained by taking k arctan λk = θ 2 n where g : Sym (R ) → R is a Gårding-Dirichlet polynomial which is hyperbolic with respect to the identity. A particular example of this is the deformedHermitian–Yang–Mills equation which appears in mirror symmetry. Another example is j arctan κ j = θ where κ1 , . . . , κn are the principal curvatures of the graph of u in × R. We also discuss the inhomogeneous Dirichlet Problem tr{arctan(Dx2 u)} = ψ(x) where ψ : → (−n π2 , n π2 ). This equation has the feature that the pull-back of ψ to the Lagrangian submanifold L ≡ graph(Du) is the phase function θ of the tangent spaces of L. On L it satisfies the equation ∇ψ = −J H where H is the mean curvature vector field of L. Mathematics Subject Classification 35G30 · 53C38 · 14J33
Communicated by A. Neves. Partially supported by the NSF.
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H. Blaine Lawson Jr. [email protected] F. Reese Harvey [email protected]
1
Stony Brook University, Stony Brook, New York, USA
2
Rice University, Houston, Texas, USA 0123456789().: V,-vol
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F. R. Harvey, H. B. Lawson
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . → 2 Geometric conditions for strict F θ -boundary convexity . . . . . . . . . . . . . 3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The special Lagrangian potential equation . . . . . . . . . . . . . . . . . . 3.2 The asymptotic interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Some general results on the pure second-order Dirichlet problem . . . . . . 4 Computing the asymptotic interior of Fθ . . . . . . . . . . . . . . . . . . . . . 4.1 Phases (or values) of the special Langrangian potential operator . . . . . . . σ 4.2 Describing the branches k and k n−1 in terms of roots and critical points 4.3 Second proof of Proposition 4.5 part (i) . . . . . . . . . . . . . . . . . . . 5 The refined Dirichlet problem for Fθ . . . . . . . . . . . . . . . . . . . . . . . 6 The inhomogeneous Dirichlet problem for the SL potential operator . . . . . . . 7 A generalized version of
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