Let (M^n, g) denote a smooth and compact Riemannian manifold with no boundary equipped with a smooth Riemannian metric g. Courant’s nodal domain theorem asserts that for the Laplace-Beltrami operator on M, if we order the eigenvalues in increasing order with multiplicity, then the eigenfunction for the k-th eigenvalue has at most k connected components(the nodal domains) where the eigenfunction does not vanish. C. Fefferman and H. Donnelley proposed about 30 years ago a local version of this result on every ball in M. This local question is connected with the question of S.-T. Yau on the length of the zero set of the eigenfunctions. We propose to give answers to this question. This work is joint with A. Logunov E. Mallinikova and D. Mangoubi.
The so called Yamabe problem in Conformal Geometry consists in finding a metric conformal to a given one and which has constant scalar curvature. From the analytic point of view, this problem becomes a semilinear elliptic PDE with critical (for the Sobolev embedding) power non-linearity. If we study the problem in the Euclidean space, allowing the presence of nonzero-dimensional singularities can be transformed into reducing the non-linearity to a Sobolev-subcritical power. A quite recent notion of non-local curvature gives rise to a parallel study which weakens the geometric assumptions giving rise to a non-local semilinear elliptic PDE.
In this talk, we will focus on metrics which are singular along nonzero-dimensional singularities. In collaboration with Ao, Chan, Fontelos, González and Wei, we covered the construction of solutions which are singular along (zero and positive dimensional) smooth submanifolds in this fractional setting. This was done through the development of new methods coming from conformal geometry and Scattering theory for the study of non-local ODEs. Due to the limitations of the techniques we used, the particular case of “maximal” dimension for the singularity was not covered. In a recent work, in collaboration with H. Chan, we cover this specific dimension constructing and studying singular solutions of critical dimension.