極小拉格朗日子流形與拉格朗日均曲率流孤立子的穩定性

Abstract

在文章中我們將會探討加權體積泛函的極值點在假設為拉格朗日子流形的條件下的線性穩定性和動態穩定性條件。其中，線性穩定性關心的是加權體積泛函的二階變分的正則性，而動態穩定性關心的是極值點附近的點在加權體積泛函的負梯度流之下的長時間存在性和收斂性。 在線性穩定性方面，我們藉由推廣 Chen [Che81] 和 Oh [Oh90] 的體積二階變分式到有加權的情況，從而得到一般 f-極小拉格朗日子流形的線性穩定條件。經由選取特定的加權函數 f，我們便由此得到拉格朗日均曲率流孤立子的穩定性條件。特別地，所有拉格朗日擴張孤立子和平移孤立子都是線性穩定的。我們也觀察到拉格日平移孤立子可以看作是加權體積意義下的校準子流形，所以可看作是特殊拉格朗日子流形的一種推廣。 我們也研究了某些複歐式空間中的極小拉格朗日子流形與拉格朗日擴張孤立子的動態穩定性。由於這些子流形必定非緊緻，我們假定這些子流形是漸進錐狀然後證明了拉格朗日均曲率流在此條件下的短時間存在性。再進一步假設 SO(m)-對稱性的情形下我們可以得到一個長時間存在性和收斂性的定理，亦即滿足這些條件的極小拉格朗日子流形與拉格朗日擴張孤立子是動態穩定的。

Stability provides important information about critical points of some functionals. In this thesis, the class of functionals we are interested in are the $f$-volume functionals defined on the space of Lagrangian submanifolds in a K'ahler manifold $X$, where $f$ is a function on $X$. The critical points for the $f$-volume functional are called the $f$-minimal Lagrangian submanifolds, which are generalizations of minimal Lagrangian submanifolds and soliton solutions for Lagrangian mean curvature flow. We study two different notions of stability with respect to the $f$-volume functional, namely the linear stability and dynamic stability. The linear stability concerning the positivity of second variation of $f$-volume functional at an $f$-minimal Lagrangian submanifold. We derive a second variation formula for $f$-minimal Lagrangian submanifolds, which is a generalization of the second variation formula by Chen [Che81] and Oh [Oh90]. Using this we obtain stability criterions for $f$-minimal Lagrangian submanifolds in gradient K'ahler--Ricci solitons. In particular, we show that expanding and translating solitons for Lagrangian mean curvature flow are $f$-stable. The dynamic stability on the other hand regarding the existence and convergence of the negative gradient flow of the $f$-volume functional, the generalized Lagrangian mean curvature flow, starting from an initial data nearby a critical point. Since the examples of $f$-minimal Lagrangians we are most interested in are complete noncompact, we first prove a short-time existence for asymptotically conical Lagrangian mean curvature flow. Then we give some long-time existence and convergence results for equivariant, almost-calibrated, asymptotically conical Lagrangian mean curvature flow in $mathbb{C}^{m}$.