Description
Laminations by Riemann surfaces arise naturally from differential equations. Locally, they are very simple objects, as they are complex disks (called plaques) parametrized by a transversal topological space. However, globally they can get really complicated. Foliations by Riemann surfaces are a special case of laminations. The project aims to address important problems in the theory of holomorphic foliations and laminations. The three parts in which the project is divided cover the analytical, dynamical and topological aspects of the theory, and the interplay between them. The specific problems chosen, the proof of the embeddability of certain foliations, the generality of analytical rigidity, and the characterization of the transverse homonymy, are very interesting and progress in any of them would be a very valuable contribution in the area. The scientific community is expected to benefit from the project. The project promoter, Complutense University of Madrid, has long experience in the study of foliations and laminations by Riemann surfaces and the donor partner University of Oslo, is a recognized expert in the area, with important results published.
Summary of project results
The Theory of foliations is a meeting point for different parts of Mathematics. It can be seen, for instance, as the geometrical qualitative point of view of differential equations. In this sense, foliations by Riemann surfaces in the plane were considered by Ilhyashenko in order to obtain one of the major breakthroughs regarding the 16th problem of Hilbert’s list. This problem is still unsolved after more than a hundred years, and developing the theory of foliations might shed some light in the problem,, as already happened in the past. Besides, tangentially to the field of foliations by Rieamann surfaces lies the theory of CR-manifolds. Partners believe that further comprehension of foliations with complex leaves will lead to solutions to these problems. This project was a continuation of a previous one. In the former project, Prof. Wold proposed the problem of solving d-bar equation on foliations by complex manifolds. At the beginning of this second stay, partners found that one of the arguments used in the proof (some estimates on the transversal regularity, concretely), were not rigourous enough, hence they had to find better estimates and a very through description of the transversal behaviour of local solutions, Latter on, they also found that certain estimates needed to be independent of the power of the bundle in order to obtain the embedding described and they had to arrange the statements and proofs of several lemmas in order to get this desired independency. Moreover, they introduced the use of pseudo-singular Hermitian metrics and as a by-product they proved that the space of tangentially holomorphic C* smooth sections is infinite dimensional for certain tensor power N8k) depending on k. Partners prepared a joint paper to be published, and several seminars and meetings were held. As a complementary result, a Norwegian-Spanish Workshop on Complex Variables was organized, funded by another call of NILS programme. The research area in which the project has taken place is a meeting point for many different fields in mathematics such as Geometry, Dynamical Systems and Analysis. The results obtained might be applied in several directions within these areas. Special attention deserve the development of singular metrics with parameters on which Fusheng Deng, a researcher in the University of Oslo has shown large interest.
Summary of bilateral results
The partnership has brought the opportunity to strengthen the existing relationships between the groups in Spain and Norway started thanks to Carlos Perez’s PhD thesis which was supervised by a Norwegian researcher and by a Spanish one. Thanks to this opportunity the research group based in Madrid has widen its potential collaboration scope. This situation applies reciprocally to the Oslo based research group that hosted the researcher. As any collaboration in Pure Mathematics, the development of this research was carried out thanks to many face to face discussions, supporting arguments in the blackboard and analyzing together the steps they needed to follow to overcome the difficulties that arised. This face to face interaction was crucial for the success of the collaboration.