论文标题
特殊拉格朗日电位方程式的假关联性
Pseudoconvexity for the Special Lagrangian Potential Equation
论文作者
论文摘要
在域上$ u $ $ u $的特殊拉格朗日电位方程式由$ {\ rm tr} \ {\ artctan（d^2 \，u）=θ$给出了$ {\ rm tr} \ {\ rm tr} \ {\ rm tr} \ {\ artctan（d^2 \} =θ$，用于contant $θ\ in（intant $θ\ in）（ - 对于$ c^2 $ solutions $ du $ in $ω\ times {\ bf r}^n $的图是一个特殊的拉格朗日submanfold。关于该方程的Dirichlet问题已经了解了很多，但是存在结果依赖于明确计算相关的边界条件（或者否则说为相关潜在理论计算伪符号）。这是在本文中完成的，答案很有趣。该结果延续到许多相关方程式中 - 例如，通过取$ \ sum_k \ arctan \，λ_k^{\ mathfrak g} =θ$其中$ {\ mathfrak g}：{\ rm sym}相对于身份的双曲线。一个特定的例子是镜像中出现的变形遗传性阳性米尔斯方程。另一个示例是$ \ sum_j \arctanκ_j=θ$其中$κ__1，...，κ_n$是$ u $ in $ω\ times \ times {\ bf r} $的主要曲率。 我们还讨论了不均匀的dirichlet问题$ {\ rm tr} \ {\ artctan（d^2_x \，u）\} =ψ（x）$ where $ψ：\overlineΩ\ to（-n {π\ off 2}，n {fover 2}，n {π\ for 2}）$。该方程式的功能是，$ψ$的下拉背包对拉格朗日submanifold $ l \ equiv {\ rm graph}（du）$是$ l $的切线空间的相位函数$θ$。在$ l $上，它满足等式$ \nablaψ= -jh $其中$ h $是$ l $的平均曲率向量场。
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 $L$.
