特殊拉格朗日球面存在性問題之探討

Abstract

在 Seidel 的博士論文 [Sei97] 中，他與他的指導教授 Donaldson 證明，若一緊緻凱勒流形 (compact Kahler manifold) 擁有一個尋常退化 (ordinary degeneration)，則此凱勒流形內存在拉格朗日球面 (Lagrangian sphere)。這個結果引發以下的延伸問題：如果此凱勒流形為一卡拉比 -丘流形 (Calabi-Yau manifold)，我們是否能夠在其中找出一個特殊拉格朗日球面 (special Lagrangian sphere)？透過文獻回顧，我們將探討特殊拉格朗日子流形 (special Lagrangian submanifolds) 的基本知識，以及球面的切叢 (the cotangent bundle of sphere) 上的瑞奇平坦度量 (Ricci-flat metrics)。在論文的最後,我們透過均曲率流 (mean curvature flow) 來探討一維的情形。

In his PhD thesis[Sei97], Paul Seidel and his advisor Simon K. Donaldson gave two proofs showing that a vanishing cycle in a Kahler manifold admitting an ordinary degener- ation can be chosen to be Lagrangian. This gives rise to the question whether the vanishing cycle is special Lagrangian if the manifold is Calabi-Yau. We investigate this problem by reviewing the geometric aspect of special Lagrangian manifolds and the Ricci-flat met- rics on the noncompact local model, namely the cotangent bundle of sphere. Finally, we approach this problem in dimension one through mean curvature flow.