# zbMATH — the first resource for mathematics

Second variation and stabilities of minimal Lagrangian submanifolds in Kähler manifolds. (English) Zbl 0721.53060
Let M be a Kähler manifold, $$C_ 1(M)$$ be the first Chern class and L be a minimal Lagrangian submanifold of M. First, by using a second variation formula, the author proves: a) If $$C_ 1(M)\leq 0$$, then L is stable. b) If $$C_ 1(M)>0$$, and L is stable then $$H^ 1(L,{\mathbb{R}})=\{0\}$$. Then, in case M is an Einstein-Kähler manifold, he defines both local and global Hamiltonian stability of L. Finally he finds a general criterion of the local Hamiltonian stability and shows that such a criterion is satisfied by several minimal Lagrangian submanifolds.
Reviewer: A.Bejancu (Iaşi)

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text:
##### References:
 [1] [BGM] Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété Riemannienne. (Lecture Notes in Math., Vol. 194) Berlin-Heidelberg-New York: Springer 1971 · Zbl 0223.53034 [2] [D] Dazord, P.: Sur la géométrie des sous-fibrés et des feuilletages lagrangiens. Ann. Sci Éc. Norm Super, IV Ser.13, 465-480 (1981) · Zbl 0491.58015 [3] [F] Floer, A.: Morse theory for lagrangian intersections. J. Differ. Geom.28, 513-547 (1988) · Zbl 0674.57027 [4] [G] Givental, A.: A letter to A. Weinstein (1988) [5] [GS] Guillemin, V., Sternberg, S.: Geometric Asymptotics. (Math. Surveys, Vol. 14). Providence, R.I.: Am. Math. Soc. 1977 · Zbl 0364.53011 [6] [He] Helgason, S.: Differential geometry, Lie Groups and Symmetric Spaces. New York: Academic Press 1978 · Zbl 0451.53038 [7] [Hs] Hsiang, W.Y.: On compact homogenous minimal submanifolds. Proc. Natl. Acad. Sci. USA56, 5-6 (1966) · Zbl 0178.55904 · doi:10.1073/pnas.56.1.5 [8] [K] Kleiner, B.: Private discussion [9] [KN] Kobayashi, S., Nomizu, K.: Foundations of differential Geometry, Vol II. New York: Wiley-Interscience 1969 · Zbl 0175.48504 [10] [L] Le, H.V.: Stability of minimal ?-lagrangian submanifolds, first Chern form and Maslov-Trofimov index. Preprint [11] [LS] Lawson, B., Simons, J.: On stable currents and their applications to global problems in real and complex geometry. Ann Math.98, 427-450 (1973) · Zbl 0283.53049 · doi:10.2307/1970913 [12] [O] Oh, Y.-G.: Floer cohomology of symmetric lagrangian submanifolds. MSRI preprint (1989) [13] [S] Simons, J.: Minimal varieties in Riemannian manifolds. Ann Math.88, 82-105 (1968) · Zbl 0181.49702 · doi:10.2307/1970556 [14] [T] Takeuchi, M.: Stability of certain minimal submanifolds of compact Hermitian symmetric spaces. Tohoku Math. J.36, 293-314 (1984) · Zbl 0539.53043 · doi:10.2748/tmj/1178228853 [15] [TK] Takeuchi, M., Kobayashi, S.: Minimal embeddings ofR-symmetric spaces. J. Differ. Geom.2, 203-215 (1968) [16] [W] Weinstein, A.: Connections of Berry and Hannay type for moving lagrangian submanifolds. Preprint (1988) · Zbl 0713.58015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.