Geometrial and analytical aspects of constant mean curvature immersions
| 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 |
Abstract 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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