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On eigenvalues of Laplacian in kahler manifolds

Grant number: 16/10009-3
Support Opportunities:Scholarships abroad - Research
Effective date (Start): September 01, 2016
Effective date (End): August 31, 2017
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Marcus Antônio Mendonça Marrocos
Grantee:Marcus Antônio Mendonça Marrocos
Host Investigator: Gerard Besson
Host Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil
Research place: Institut Fourier, France  


The spectrum of the Laplace operator plays a central role in Riemannian Geometry. From qualitative point of view, one of the most important results in this direction is due to Uhlenbeck [20]. It is known that the following properties are generic on the set of metrics: a) The eigenspaces are all of dimension 1, b) 0 is not a critical point for any eigenfunction, c) All eigenfunctions are Morse functions, which is, their critical points are non-degenerate. If a Riemannian manifold possesses big isometry group one cannot expect that the eigevalues are simple, since the eigenspaces are represented by the action of the isometry group. Thus, the best we can hope for is that on the set of symmetric metrics with respect to a fixed group G the eigenspaces are generically irreducible representations with respect to the group action. On any compact Kahler manifold M there exists a canonical action of a super Lie algebra Aon the space of differential forms which commutes with the Laplace - Hodge operator. Such an algebra is determined by the Lefschetz operator, a de Rham complex and its dual see [1] ou [23]. Therefore, the eigenvalues of the Laplace - Hodge operator acting on the space of differential forms cannot all be simple. The objective of the present project is twofold. Firstly, to establish which of the properties), b) and c) are generic on the set of Kahler metrics. Secondly, to prove that on the set of Kahler metrics the eigenspaces of the Laplace - Hodge operator acting on the space of differential forms are generically irreducible representations of the action of the super-algebra A. (AU)

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Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
MARROCOS, MARCUS A. M.; GOMES, JOSE N. V.. Generic Spectrum of Warped Products and G-Manifolds. JOURNAL OF GEOMETRIC ANALYSIS, v. 29, n. 4, p. 3124-3134, . (16/10009-3)
GOMES, V, JOSE N.; MARROCOS, MARCUS A. M.. On eigenvalue generic properties of the Laplace-Neumann operator. JOURNAL OF GEOMETRY AND PHYSICS, v. 135, p. 21-31, . (16/10009-3)

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