论文标题
来自具有边界的特殊Lagrangians的积极Lagrangians的大地测量学
Geodesics of positive Lagrangians from special Lagrangians with boundary
论文作者
论文摘要
Lagrangian submanifolds阳性空间中的测量学是完全非线性退化椭圆形PDE的解决方案。我们表明，在阳性Lagrangians空间中的一个大地段对应于一个特殊的Lagrangian圆柱体的一个参数家族，称为圆柱体变换。圆柱体的边界包含在地球末端的正lagrangian中。具有正拉格朗日边界条件的特殊拉格朗日方程是椭圆形的，溶液空间是平滑的歧管，在圆柱体的情况下，这是一个维度。通过解决每个圆柱体上的拉普拉斯操作员的差异问题，可以从其圆柱变换中恢复大地测量。 使用圆柱变换，我们显示了通过大地测量连接的阳性拉格朗日阳性球的空间。因此，我们获得了在任意维度不变的异构体中，在任意维度中获得强大溶液的第一个例子。实际上，我们获得的解决方案与有限的点相比平滑。
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.
