Concerning part I of the project, that is the study of minimal subvarieties, there are two findings that appear to a large extend unexpected and beyond the state of the art:
[1]: In our article "Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group" (now accepted for publication as a 120-pages research monograph on Memoirs of the American Mathematical Society) we showed that if one does not restrict to ``low topological complexity'' then the topological uniqueness question for free boundary minimal surfaces in the Euclidean ball can be answered in the strongest negative terms: ``The topology and symmetry group of a free boundary minimal surface in the Euclidean unit ball do not determine the surface uniquely.'' This follows from showing that for any sufficiently large integer g there exist in the unit ball of the three-dimensional Euclidean space two distinct, properly embedded, free boundary minimal surfaces having genus g, three boundary components and symmetry group coinciding with the antiprismatic group of order 4(g+1).
One important motivation, lying behind the precise formulation of this theorem, is the recent result by Kapouleas and Wiygul, asserting (on the contrary) the uniqueness of each Lawson surface, in the round three-dimensional sphere, given its topology and symmetry group. Thus, our statement above should indeed be viewed in that perspective, and contrasted with such a theorem in the context of the comparative study of closed minimal surfaces in S^3 and of free boundary minimal surfaces in B^3, showing an interesting broken symmetry.
There is then a second, completely independent, reason why such a construction has strikingly unexpected implications: indeed, one of the two families above is a sequence of free boundary minimal surfaces in the Euclidean ball whose Morse indices do not seem to exhibit any periodicity or regular pattern at all. Thereby, there emerges a landscape that is much more complex than previously expected, significantly downgrading our expectations of any ``'universal laws'' for the Morse index in whichever setting, yet thereby making this line of investigation extremely exciting and with plenty of potential for more discoveries.
[2] In the recent article ``Topological control for min-max free boundary minimal surfaces'' Franz and Schulz rigorously proved the existence of an index 5 free boundary minimal surface in B^3. This is particularly interesting because we now have a pair of properly embedded free boundary minimal surface (henceforth: FBMS) in B^3 having indices two consecutive integers (the critical catenoid having index =4); this phenomenon has so far not been observed in the much studied situation of closed minimal surfaces in the three-dimensional round sphere, and would correspond to the equally shocking discovery of a new index 6 minimal surface therein. We note that the far-reaching results about topological control in min-max minimal surfaces obtained by the authors also imply that such an index 5 FBMS has either genus one and connected boundary or is an exotic free boundary minimal annulus, with significant numerical evidence supporting the first case. In fact, we are close to showing that such an un-identified FBMS should a posteriori coincide with the first element M_1 of the sequence constructed by the PI, Franz and Schulz in settling Schoen's conjecture, which we did in our paper ``Free boundary minimal surfaces with connected boundary and arbitrary genus'' now published in Cambridge Journal of Mathematics.
In addition, a somewhat new trend which is - a posteriori - the highlight of our latest article ``Disc stackings and their Morse index'' is the comparative analysis of variational vs. gluing/desingularization methods based on the analysis of the Morse index of the minimal surfaces under consideration. For instance, it follows from our work that any possible realization of free boundary minimal stackings in B^3 via an equivariant min-max method would need to employ sweepouts with an arbitrarily large number of parameters and, in addition, that it is only for N=2 and N=3 layers that free boundary minimal disc stackings are achievable by means of one-dimensional mountain pass schemes.
Along this research line, in the years to come we wish to obtain a full analogous of Savo's celebrated estimate for FBMS in the unit Euclidean ball of any dimension, to reach an exact computation of an affine index law for an infinite family of examples and to contrast it with numerical results displaying erratic behaviour for other families, which we indeed expect to be a ``generic'' behaviour in a sense that we aim at making precise. Also, we shall keep working on the famous Schoen-Marques-Neves conjecture about the linear lower bound for the Morse index in terms of the first Betti number of a closed minimal hypersurface in any manifold of positive Ricci curvature, and shall properly phrase and resolve the analogous question for relative cycles (namely: for free boundary minimal hypersurfaces in compact manifolds with boundary).
Concerning part II of the project, that is the study of positive scalar curvature on (smooth or possibly singular) manifolds with boundary:
[3] we were able to design an equivariant counterpart of the Bamler-Kleiner far-reaching machinery of singular Ricci flows, and used it (in conjunction with a variety of other tools employed to understand the topology of certain classes of infinite-dimensional Fréchet manifolds) to address the central problem of studying the homotopy type of spaces of Riemannian metrics (on a given background manifold) singled out by curvature conditions;
[4] we put together a solid methodology to perform gluing of manifold with corners attaching a pair of isometric faces, by blending ideas and methods of Miao with the more recent ones by Bär-Hanke. This is arguably a delicate problem, where the level of interplay between interior and boundary curvature conditions is more subtle and challenging that in our previous projects. For what concerns the second period of the CHANGE project we will thoroughly investigate the high-dimensional scenario (that is: the situation for spaces having dimension four or higher), with special focus on obstructions to (suitably defined notions of) isotopy and concordance; here we wish to remark that the definition of a (good) weak notion of concordance for PSC metrics on manifolds with boundary precisely builds on the developments [4] indicated above. Moreover, we will employ these tools to obtain significant advances on filling problems (under curvature constraints) as well as towards a deeper understanding of notions of positive scalar curvature for interesting classes of singular spaces. The development of part II of the project will crucially benefit from the selection and hiring of two new members of the group.
