Research project PN-III-P3-3-1-PM-RO-FR-2019-0234, Nr. 1BM/2019
a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI
Director: Professor Cezar Oniciuc
Biharmonic and constant mean curvature submanifolds
Abstract: The theory of minimal submanifolds has been a major research theme in Differential Geometry for more than fifty years. Originally stated as the problem of finding optimal shapes in space, it evolved into a central question of contemporary mathematics, as the numerous difficulties, both theoretical and conceptual, led to the creation of new objects and forced new approaches to geometry. In particular, the advent of General Relativity and its insistence on a geometrical setting, brought a new meaning to research on the natural objects of Riemannian Geometry. A generalisation of minimal submanifolds was introduced in 1964 by Eells and Sampson under the name of harmonic maps, which gave a new impetus on a wider scale, with a very clear link to the classical case, since isometric immersions are minimal if and only if they are harmonic. While minimal submanifolds remain a very substantial area of research activity, they are now seen as part, though certainly a central one, of the wider concept of Constant Mean Curvature (CMC) submanifolds, i.e. the length of mean curvature vector being a constant instead of merely zero. This expansion has also been witnessed with the theory of harmonic maps where a higher order problem was introduced to encompass the original question of Eells and Sampson, by measuring the failure of harmonicity. Such maps, called biharmonic, are solutions to a system of fourth-order non-linear elliptic partial differential equations and, interestingly, are linked to CMC submanifolds, when maps are isometric immersions, but not as straightforwardly as harmonic maps and minimal submanifolds are. Exploring this more intricate relationship is one of the major stimuli of our recent work, especially the case of surfaces in spheres. In particular, this provides an alternative approach to CMC submanifolds and, in the better cases, a classification could be within reach.
O1. The classification of biharmonic and biconservative hypersurfaces in space forms.
O2. The construction of non-CMC complete biconservative surfaces in space forms; their uniqueness and the posibility to factorize to a torus.
O3. Finding of constructive solutions to the metrizability problem.
O4. Finding new characterizations of Finsler spaces of constant curvature.
O5. The study of biharmonic and biconservative surfaces in complex space forms.
ISI accepted papers
1. D. Fetcu, E. Loubeau, C. Oniciuc, Bochner-Simons Formulas and the Rigidity of Biharmonic Submanifolds, J. Geom. Anal. (2019).
2. G. Cretu, New classes of projectively related Finsler metrics of constant flag curvature, Int. J. Geom. Methods Mod. Phys. 17 (2020), no. 5, 2050068, 22 pp.
1. S. Nistor, C. Oniciuc, On the uniqueness of complete biconservative surfaces in 3-dimensional space forms, Preprint 2019.
2. H. Bibi, E. Loubeau, C. Oniciuc, Unique continuation property for biharmonic hypersurfaces in spheres , Preprint 2020.
Work in progress
1. H. Bibi, B.-Y. Chen, D. Fetcu, Biconservative surfaces in complex space forms.