The Hermitian Laplace Operator on Nearly Kähler Manifolds

Communications in Mathematical Physics (Impact Factor: 2.09). 02/2010; 294(1). DOI: 10.1007/s00220-009-0903-4
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The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6-dimensional homogeneous nearly Kähler manifolds. It turns out that the nearly Kähler structure is rigid except for the flag manifold F(1,2)=SU_3/T^2, which carries an 8-dimensional moduli space of infinitesimal nearly Kähler deformations, modeled on the Lie algebra su_3 of the isometry group.

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Available from: Andrei Moroianu,
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    • "In this article we shall show that nearly parallel G 2 -manifolds are also in another respect very similar to nearly Kähler manifolds: the description of infinitesimal deformations. In [16] the space of infinitesimal nearly Kähler deformations is identified with the space of primitive co-closed (1, 1)-eigenforms of the Laplace operator for the eigenvalue 2scal /5, [19] contains a similar description of the space of infinitesimal Einstein deformations. This space turns out to be the sum of three such eigenspaces. "
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    ABSTRACT: We study the infinitesimal deformations of a proper nearly parallel G_2-structure and prove that they are characterized by a certain first order differential equation. In particular we show that the space of infinitesimal deformations modulo the group of diffeomorphisms is isomorphic to a subspace of co-closed $\Lambda^3_{27}$-eigenforms of the Laplace operator for the eigenvalue 8 scal/21. We give a similar description for the space of infinitesimal Einstein deformations of a fixed nearly parallel G_2-structure. Moreover we show that there are no deformations on the squashed S^7 and on SO(5)/SO(3), but that there are infinitesimal deformations on the Aloff-Wallach manifold N(1,1) = SU(3)/U(1).
    Asian Journal of Mathematics 01/2011; 16(4). DOI:10.4310/AJM.2012.v16.n4.a6 · 0.53 Impact Factor
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    ABSTRACT: We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) G_2 manifolds. In the AC case, we show that if the rate of convergence nu to the cone at infinity is generic in a precise sense and lies in the interval (-4, -5/2), then the moduli space is smooth and we compute its dimension in terms of topological and analytic data. For generic rates nu < -4 in the AC case, and for generic positive rates of convergence to the cones at the singular points in the CS case, the deformation theory is in general obstructed. We describe the obstruction spaces explicitly in terms of the spectrum of the Laplacian on the link of the cones on the ends, and compute the virtual dimension of the moduli space. We also present several applications of these results, including: the local uniqueness of the Bryant--Salamon AC G_2 manifolds; the smoothness of the CS moduli space if the singularities are modeled on particular G_2 cones; and the proof of existence of a "good gauge" needed for desingularization of CS G_2 manifolds. Finally, we discuss some open problems for future study.
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