Academic literature on the topic 'Minimal Lagrangian Submanifolds'

In questa tesi studiamo alcune proprietà delle "Varietà Doppie" o D-Varietà. In particolare studiamo la teoria delle deformazioni di D-Strutture e di D-Strutture CR, e troviamo una condizione che è equivalente alla classica condizione di Maurer-Cartan che descrive l'integrabilità di deformazioni di D-Strutture. Successivamente prestiamo attenzione alla coomologia delle D-Varietà, provando che una versione D-complessa del del-delbar-Lemma non può essere vera per D-varietà compatte. Inoltre sono stabilite alcune proprietà di sottogruppi speciali della coomologia di de-Rham, ottenute studiando il loro comportamento sotto l'azione di deformazioni. Infine, un risultato riguardante le sottovarietà Lagrangiane minimali dovuto ad Harvey e Lawson riguardante le varietà D-Kahler Ricci-Piatte è generalizzato a una classe di varietà simplettiche quasi D-complesse.<br>We study some properties of Double Manifold, or D-Manifolds. In particular, we study of deformations of D-structures and of CR D-structures, and we found a condition which is equivalent to the classical Maurer-Cartan equation describing the integrability of the deformations. We also focus on the cohomological properties of D-Manifold, showing that a del-delbar-Lemma can not hold for any compact D-Manifold. We also state some properties of special subgroups of de-Rham cohomology, studing also their behaviour under small deformations. Finally, a result by Harvey and Lawson about the minimal Lagrangian Submanifold of a D-Kahler Ricci-flat manifold is generalized to the case of a special almost D-complex symplectic manifold.

博士<br>國立臺灣大學<br>數學研究所<br>107<br>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 $mathbb{C}^{m}$.