四維空間辛幾何可說是最簡單的四維辛流形，然吾人對其上的拉格朗日輪胎面的辛合痕類的了解仍十分有限‧截至目前僅知存在有一族的特殊辛合痕類‧ 本計畫乃為探計以下兩個有關拉格朗日輪胎面的辛合痕類問題‧其一為是否存在新的辛合痕類‧其二為已知的特殊合痕類可否經由在基本的辛合痕類上的手術產生‧ 對於第一個問題本計畫將研究並決定一組新近建構的拉格朗日輪胎面是否代表新的辛合痕類，對於第二個問題本計畫將(1)深入探討Luttinger手術對辛合痕類的影響，(2)著手建構一種帶邊的廣義Dehn twist，並研究其對辛合痕類之作用‧ 本計畫的結果不單將對四維辛空間中拉格朗日輪胎面的分類問題做出新的貢獻，亦將有助於對一般四維辛流形中拉格朗日曲面的理解‧ ; Symplectic 4-space is arguably the simplest symplectic 4-manifold, yet the classification of embedded Lagrangian tori in it up to symplectic isotopy is far from being complete. Up to now there has been found only one type (up to scaling) of special Lagrangian tori which are not symplectically isotopic to any of the Clifford tori, and these special tori are monotone. This project focuses on the following two problems concerning the symplectic isotopy of Lagrangian tori in R4: One is the existence of new isotopy classes of Lagrangian tori in R4, and the other is to find a surgery that will produce a currently known special torus from a Clifford torus. For the first question the PI will study an infinite family of potentially new Lagrangian tori produced by Dehn twists. On the second question the PI will investigate the Luttinger’s surgery and study the construction of a relative version of generalized Dehn twist and examine its effect on Lagrangian tori. The results of this project will not only make new progress on the classification of Lagrangian tori in R4, but also be applied to study Lagrangian surfaces in general symplectic 4-manifolds. ; 研究期間 9708 ~ 9807
