Geodesics of positive Lagrangians from special Lagrangians with boundary
Geodesics of positive Lagrangians from special Lagrangians with boundary
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来自有边界的特殊拉格朗日量的正拉格朗日量测地线
DOI：
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发表时间：
2020
期刊：
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通讯作者：
Amitai M. Yuval
中科院分区：
文献类型：
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作者：
Jake Solomon;Amitai M. Yuval
正拉格朗日子流形空间中的测地线是完全非线性退化椭圆偏微分方程的解。我们表明，测地线段在空间中的正拉格朗日对应于一个参数家庭的特殊拉格朗日缸，称为圆柱变换。圆柱体的边界包含在测地线两端的正拉格朗日量中。具有正拉格朗日边界条件的特殊拉格朗日方程是椭圆型的，其解空间是一个光滑流形，而在柱体情况下是一维的。通过求解每个柱面上的拉普拉斯算子的狄利克雷问题，可以从测地线的柱面变换中恢复测地线。
利用柱面变换，我们证明了由测地线连接的正拉格朗日球面对空间是开的。这样，我们得到了任意维测地线方程在等距下非不变的强解的第一个例子。事实上，我们得到的解是光滑的，远离有限的点集。
Geodesics in the space of positive Lagrangian submanifolds are solutions of a fully non-linear degenerate elliptic PDE. We show that a geodesic segment in the space of positive Lagrangians corresponds to a one parameter family of special Lagrangian cylinders, called the cylindrical transform. The boundaries of the cylinders are contained in the positive Lagrangians at the ends of the geodesic. The special Lagrangian equation with positive Lagrangian boundary conditions is elliptic and the solution space is a smooth manifold, which is one dimensional in the case of cylinders. A geodesic can be recovered from its cylindrical transform by solving the Dirichlet problem for the Laplace operator on each cylinder.
Using the cylindrical transform, we show the space of pairs of positive Lagrangian spheres connected by a geodesic is open. Thus, we obtain the first examples of strong solutions to the geodesic equation in arbitrary dimension not invariant under isometries. In fact, the solutions we obtain are smooth away from a finite set of points.