arXiv 2001.098182020-01math.APmath.DG

一类特殊Lagrange位势方程的伪凸性

Pseudoconvexity for the Special Lagrangian Potential Equation

F. Reese Harvey, H. Blaine Lawson

备注：: Several things were added to the paper. More examples were given, some new results were proved, and the historical discussion was expanded

摘要：

本文给出了区域$Ω\subset {\bf R}^n$上函数$u$的特殊拉格朗日势方程，即对常数$θ\in (-n {π\over 2}, n {π\over 2})$的${\rm tr}\{\arctan(D^2 \,u) \} = θ$。对于$C^2$解，$Ω\times {\bf R}^n$中的$Du$图是一个特殊的拉格朗日子流形.关于这个方程的狄利克雷问题已经有很多了解，但是存在性结果依赖于显式计算相关的边界条件（或者，换句话说，计算相关势理论的伪凸性）。这篇论文就是这么做的，答案很有趣。结果延续到许多相关的方程-例如，那些通过采取$\sum_k \arctan\, λ_k^{\mathfrak g} = θ$其中${\mathfrak g} : {\rm Sym}^2({\bf R}^n)\to {\bf R}$是一个Garding-Dirichlet多项式，这是双曲的身份。一个特别的例子是变形的厄米-杨-米尔斯方程，它出现在镜像对称中。另一个例子是$\sum_j \arctan κ_j = θ$，其中$κ_1, ... , κ_n$是$Ω\times {\bf R}$中$u$的图的主曲率。我们还讨论了非齐次Dirichlet问题${\rm tr}\{\arctan(D^2_x \,u)\} = ψ(x)$，其中$ψ: \overlineΩ\to (-n {π\over 2}, n {π\over 2})$ .该方程的特点是$ψ$到拉格朗日子流形$L\equiv {\rm graph}(Du)$的拉回是$L$的切空间的相函数$θ$。在$L$上，它满足方程$\nabla ψ= -JH$，其中$H$是$L$的平均曲率向量场。

英文摘要：

The Special Lagrangian Potential Equation for a function $u$ on a domain $Ω\subset {\bf R}^n$ is given by ${\rm tr}\{\arctan(D^2 \,u) \} = θ$ for a contant $θ\in (-n {π\over 2}, n {π\over 2})$. For $C^2$ solutions the graph of $Du$ in $Ω\times {\bf R}^n$ 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 over to many related equations -- for example, those obtained by taking $\sum_k \arctan\, λ_k^{\mathfrak g} = θ$ where ${\mathfrak g} : {\rm Sym}^2({\bf R}^n)\to {\bf R}$ is a Garding-Dirichlet polynomial which is hyperbolic with respect to the identity. A particular example of this is the deformed Hermitian-Yang-Mills equation which appears in mirror symmetry. Another example is $\sum_j \arctan κ_j = θ$ where $κ_1, ... , κ_n$ are the principal curvatures of the graph of $u$ in $Ω\times {\bf R}$. We also discuss the inhomogeneous Dirichlet Problem ${\rm tr}\{\arctan(D^2_x \,u)\} = ψ(x)$ where $ψ: \overlineΩ\to (-n {π\over 2}, n {π\over 2})$. This equation has the feature that the pull-back of $ψ$ to the Lagrangian submanifold $L\equiv {\rm graph}(Du)$ is the phase function $θ$ of the tangent spaces of $L$. On $L$ it satisfies the equation $\nabla ψ= -JH$ where $H$ is the mean curvature vector field of $L$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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