Projects / Programmes
Interactions between complex analysis, minimal surfaces and PDEs
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P001 |
Natural sciences and mathematics |
Mathematics |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
complex analysis, minimal surfaces, partical differential equations, harmonic analysis, numerical analysis, invariant theory
Researchers (16)
Organisations (2)
Abstract
We shall investigate problems in complex analysis and geometry with applications to the theory of minimal surfaces, almost complex analysis, CR geometry, harmonic analysis, theory of partial differential equations, numerical analysis, and invariant theory of Lie grupoids and related structures.
A major goal is the analysis of the classical Calabi-Yau problem on the existence of metrically complete bounded conformal minimal immersions. F. Forstnerič and A. Alarcon started research on this subject in a series of recent papers (see sect. 12). We shall attack the following problem: Given a compact bordered Riemann surface M, construct a continuous map f:M→Rn which is a complete conformal minimal immersion in M∂M and whose boundary f(∂M) consists of Jordan curves. We shall investigate the asymptotic boundary behavior with emphasis on the existence of proper maps into certain classes of domains, and the topological structure of the space of conformal minimal immersions M→Rn.
We shall develop the theory of minimal hulls of compact sets K in Rn - the smallest set containing all minimal surfaces with boundary in K. We expect to characterize the minimal hull of K by sequences of conformal minimal discs whose boundaries converge to K in the measure theoretic sense, and by Green currents.
Closely related is the problem of P. Yang concerning the existence of bounded complete complex submanifolds of Cn. Recently J. Globevnik found an optimal solution by constructing holomorphic functions on the ball of Cn all of whose level sets are complete complex hypersurfaces (Annals of Math., in print). We shall investigate the topological and holomorphic type of complete complex submanifolds and the existence of complete proper holomorphic immersions and embeddings from lower to higher dimensional balls.
We will study the existence of proper J-holomorphic maps from the disc into almost complex Stein manifolds of dimension at least 3 and approximation of non-holomorphic maps with small anti-complex derivative. We will also study boundary properties and deformation theory for extremal and stationary J-holomorphic discs.
We shall study the structure of CR singular points of immersed real manifolds in complex manifols. We will show that the vanishing of the first Pontryagon class and the Euler characteristic in a necessary and sufficient condition for the existence of CR regular immersions of real orientable closed 4-manifolds into C3. For the existence of CR regular embeddings we need in addition that the 4-manifold is spin.
We shall extend the reach of the heat-flow method associated with the Nazarov-Treil Bellman function to prove i) the holomorphic functional calculus for non-symmetric Ornstein-Uhlenbeck operators and ii) optimal bilinear embedding theorem for elliptic differential operators in divergence form with complex coefficients.
We shall investigate the principles of superposition of the periodic solutions of the focusing nonlinear Schrödinger equation. Our goal is to calculate and classify the traveling solutions and combine families of traveling solutions with different frequencies into quasiperiodic solutions.
We shall investigate invariants of topological and Lie grupoids, Lie algebroids and Hopf algebroids. The results will be applied in the theory of foliations and orbifolds. We shall develop new effective methods for calculation (co) homology groups of topological grupoids.
Significance for science
The line of investigation described in the research proposal will bring major original new results in the proposed topics of investigation. In particular, the proposed research in the first mentioned subject will provide a solid background for the modern theory of conformal minimal immersions and the conformal Calabi-Yau problem. Forstnerič's recent collaboration with the group from the University of Granada in Spain represents a discovery of a major common theme of two so far separate schools of complex analysis and minimal surface theory. The subject of minimal hulls and minimal plurisubharmonic functions has attracted considerable interest at the recent invited lecture of F. Forstnerič in Oberwolfach in January 2015. The results on the subject of almost holomorphic discs will also be of basic importance for the future investigations; here U. Kuzman started a collaboration with Florian Bertrand at the American University of Beirut. Concerning the harmonic analysis part, it is known that both functional calculus and the Kato problem partially bear their origins in PDEs and in turn arise from real-life problems of mathematical physics, fluid mechanics and elasticity. The attention that both of the above questions have gotten in the last 50 years testifies to their significance. Successful work on problems which have not only been open for a long time, but also much studied, helps maintain Slovenia's contact with the edge of today's science. The relevance of Dragičević's recent work with Carbonaro represents a discovery of a major common point of two hitherto separated schools of harmonic analysis in Europe.
We expect that the results obtained under this project will be published in high level international mathematical journals and will provide important framework for work in this area for years to come. They will also be presented to the international mathematical audience in lectures at conferences and PhD schools, they will be applied in our teaching at the University of Ljubljana, and will provide a rich source of problems and directions that could be addressed by a new generation of researchers.
Significance for the country
Mathematics is the basic universal language of science and, as such, is present in every field, from natural and technical sciences to medicine, economics and social scienes. One of the main direct impacts of the proposed research project is on education of students and young researchers at the University of Ljubljana. educating generations of students and also new resarchers. Our main achievements are regularly incorporated into the teaching process and are presented in the Seminar for Complex Analysis which has been running wiothout a break for the last 40 years.
Several parts of the project, in particular the problems in PDEs and geometric interpolation, also have direct applications to various problems in technical fields as can be seen from the description of the research project. The proposed research problems in the fields of complex and harmonic analysis are of fundamental nature and are important for the intrinsic development of these fields. In addition, results and new techniques from these two fields regularly find their way into other areas of mathematics, in particular to numerical analysis whose practical applicability has already been mention. Finally, the theory of partial differential equations is one of the most important basic fields of mathematics which has plenty of applications in diverse fields of science and technology.
Most important scientific results
Interim report,
final report
Most important socioeconomically and culturally relevant results
Interim report,
final report
