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Geometry and analysis of the first eigenvalue functional

Final Report Summary - GAFEF (Geometry and analysis of the first eigenvalue functional)

The project is devoted to the study of the first eigenvalue problem for the Laplace operator in geometric settings. More precisely, we view the Laplace eigenvalue as a functional defined on Riemannian metrics whose volume is normalised to be equal to one. In certain geometric settings, for example when we consider conformal metrics only, the first eigenvalue functional is known to be bounded. The purpose of this project is to study the corresponding maximal (and, more generally, extremal) metrics for this functional. The prime addressed questions are related to the existence of maximal metrics, regularity theory, and concentration-compactness properties. Below we outline the main results in more detail.

The principal results are concerned with the study of the first eigenvalue within conformal classes on Riemannian surfaces (see Work package (WP) 1, annex 1 of the grant agreement). The researcher has developed an approach to the outlined problems via the direct method of calculus of variations. This includes a collection of new results in the following two directions: first, analysis of the first eigenvalue on the so-called weakly conformal metrics, regarded as Radon measures on a surface; second, the regularity theory for such extremal weakly conformal metrics. The problems considered are highly non-trivial, and many results obtained are interesting on their own. The progress made leads to a number of new research directions as well as, due to the interdisciplinary nature of the project, establishes interesting links with other open questions. For example, the study of weakly conformal metrics as Radon measures has direct bearing with (and applications to) the theory of Alexandrov surfaces. As another sample result, we mention a new concentration-compactness phenomenon for sequences of extremal metrics. It describes the structure of the space of conformal extremal metrics for the first eigenvalue functional. Many of these results have been received by experts with interest and enthusiasm.

In addition, a number of related questions in other geometric settings (see WPs 2, 3) have been also studied. These include the study of the first Laplace eigenvalue on metrics with a certain curvature bound on higher dimensional manifolds and metrics with cohomologous Kaehler form on Kaehler manifolds. The results obtained describe new geometric properties of the corresponding extremal metrics. For example, there has been found an interesting relationship between eigenvalue extremal Kaehler metrics and certain projective embeddings. The developed ideas have also led to new results in the theory of harmonic maps. The study made in WPs 2, 3 establishes a strong ground for fellow's future independent research.

The achieved results and developed methods are likely to interest experts working on spectral geometry, and more broadly, on geometric problems in spectral theory and calculus of variations. Their novelty and scientific impact will contribute to strengthening fellow's independent position and establishing new collaborative links beneficial both for the fellow and for the scientific activities of the host. In addition, we believe that their dissemination amongst the corresponding part of mathematical community will contribute to the European scientific excellence and competitiveness.