Periodic Reporting for period 2 - CHANGE (CHallenges in ANalysis and GEometry, between mean and scalar curvature)
Periodo di rendicontazione: 2022-09-01 al 2024-02-29
The concept of shape is one of the big themes permeating the whole history of Mathematics. Among the very many ways it has been formalized, since the ancient times and through the centuries, the notion of curvature is one of the most successful and significant ones. Starting with pioneering work by Gauss and Riemann in the first half of the 19th century, this concept has been developed at increasing levels of abstraction, and has then emerged one hundred years ago in the context of Einstein's description of gravitational forces within his general theory of relativity. The language of differential geometry has proven to be a fundamental ingredient in the most diverse physical theories, when trying to describe the mysteries of the world we live in.
Along this journey lots of questions have been raised and novel techniques have been developed to answer them. In particular, over the last fifty years we have witnessed the massive use of analytic techniques, based on the analysis of partial differential equations, to attack fundamental problems in geometry, and the interplay between these two worlds, their mutual contaminations, have dramatically come out as one of the main trends in our discipline. Among the very many milestones that exemplify this phenomenon, we mention the proof of the positive mass theorem by Schoen and Yau, the proof of the Poincaré conjecture by Perelman and, more recently, the proof of the Willmore conjecture by Marques and Neves. The whole field which we now call geometric analysis is more vital than ever, new intriguing questions keep arising and breakthroughs happen more often that has ever been the case in the past.
In this project, we focus on two broad themes that have on the hand strong connections with some of the most exciting recent developments in the field, and on the other display the potential of opening new avenues in the decades to come. First, we aim at discovering patterns in the mysterious landscape that is emerging from the solution of Yau’s conjecture about the existence of infinitely many minimal hypersurfaces in Riemannian manifolds. Second, we wish to shed some light on the world of positive scalar curvature manifolds, which sits somewhere inbetween the rigid world of convex objects and the flexible world of geometric topology. The methodology we employ relies on a combination of elliptic and parabolic techniques.
Our goals are to provide significant contributions to these themes, and to attract highly promising young researchers so to favor the development of geometric analysis in the European context.
Along this journey lots of questions have been raised and novel techniques have been developed to answer them. In particular, over the last fifty years we have witnessed the massive use of analytic techniques, based on the analysis of partial differential equations, to attack fundamental problems in geometry, and the interplay between these two worlds, their mutual contaminations, have dramatically come out as one of the main trends in our discipline. Among the very many milestones that exemplify this phenomenon, we mention the proof of the positive mass theorem by Schoen and Yau, the proof of the Poincaré conjecture by Perelman and, more recently, the proof of the Willmore conjecture by Marques and Neves. The whole field which we now call geometric analysis is more vital than ever, new intriguing questions keep arising and breakthroughs happen more often that has ever been the case in the past.
In this project, we focus on two broad themes that have on the hand strong connections with some of the most exciting recent developments in the field, and on the other display the potential of opening new avenues in the decades to come. First, we aim at discovering patterns in the mysterious landscape that is emerging from the solution of Yau’s conjecture about the existence of infinitely many minimal hypersurfaces in Riemannian manifolds. Second, we wish to shed some light on the world of positive scalar curvature manifolds, which sits somewhere inbetween the rigid world of convex objects and the flexible world of geometric topology. The methodology we employ relies on a combination of elliptic and parabolic techniques.
Our goals are to provide significant contributions to these themes, and to attract highly promising young researchers so to favor the development of geometric analysis in the European context.
The initial period of activity of the CHANGE project has been extremely intense and productive, and has already led to some exciting new developments in the field. The solid background and great talent of the firstly hired junior group members (one PhD and one postdoc) has allowed for a relatively short setup phase, thus getting quite quickly to the core of some of the objectives of the project.
On the one hand, the activity of the group has been devoted to the study of minimal surfaces, both in terms of developing methodologies to construct new examples (with peculiar properties) and in designing - or possibly refining - techniques to investigate the Morse index as a tool for decoding the rich landscape that emerged in the last decade in connection with the solution of Yau's conjecture on the abundance of minimal surfaces, both in the closed and in the free boundary case. Somewhat more specifically, we have profitably worked on: i) min-max constructions with topological control, including in particular a proof of the lower semicontinuity of the first Betti number in min-max constructions (which discloses compensation phenomena between the number of handles and the number of boundary components); ii) gluing and/or desingularization methods *at* the free boundary; for what concerns effective methods to produce Morse index estimates we have studied and significantly extended both iii) partitioning methods in the spirit of Montiel-Ros, and iv) covering methods along the lines of Song's work.
On the other hand, the PI has brought to completion the project - developed in collaboration with Chao Li (NYU) - showing that on any compact 3-manifold with boundary the space of positive scalar curvature metrics with minimal boundary is either empty or contractible, thereby deriving the corresponding result for scalar-flat asymptotically flat 3-manifolds with horizon boundary (and, more generally, for maximal vacuum initial data sets for the Einstein equations in general relativity). In order to further investigate related problems, with a view towards the next phase(s) of the CHANGE project, the same team has obtained a novel desingularization theorem, that allows to smoothly attach two given manifolds with corners by suitably gluing a pair of isometric faces, with control on both the scalar curvature of the resulting space and the mean curvature of its boundary. This extends from the closed to the bordered case a famous result by P. Miao, and we expect our it to become a standard tool in the field. Such a desingularization technique result has also been employed by us to define a suitable, weak notion of concordance for PSC metrics on manifolds with boundary.
During this period, there have already appeared two seminal papers, which in our opinion clearly convey the main message and spirit of the project: (Carlotto-Franz-Schulz, on Memoirs of the American Mathematical Society) and (Carlotto-Li, on Comm. Pure Appl. Math.), respectively for what pertains to the first and second main research directions of the project. In addition, the group's activity on the themes above is reflected in twelve more research articles (not accounting for surveys, reports etc...) that are all available on the general-purpose arXiv server and have either appeared in print on high-level mathematical journals or are currently (i.e. at the moment this report is being written) under review after submission for publication. On top of such primary research output, significant efforts have been spent in various forms of dissemination, including invited talks at various events (ranging from the 8th European Congress of Mathematics, to multiple workshops and research seminar at a number of institutions worldwide) and organisation of project-related research meetings, culminating in particular in the June 2024 forthcoming international conferences to happen in Zurich, for which we can already count on 24 speakers of the highest caliber.
On the one hand, the activity of the group has been devoted to the study of minimal surfaces, both in terms of developing methodologies to construct new examples (with peculiar properties) and in designing - or possibly refining - techniques to investigate the Morse index as a tool for decoding the rich landscape that emerged in the last decade in connection with the solution of Yau's conjecture on the abundance of minimal surfaces, both in the closed and in the free boundary case. Somewhat more specifically, we have profitably worked on: i) min-max constructions with topological control, including in particular a proof of the lower semicontinuity of the first Betti number in min-max constructions (which discloses compensation phenomena between the number of handles and the number of boundary components); ii) gluing and/or desingularization methods *at* the free boundary; for what concerns effective methods to produce Morse index estimates we have studied and significantly extended both iii) partitioning methods in the spirit of Montiel-Ros, and iv) covering methods along the lines of Song's work.
On the other hand, the PI has brought to completion the project - developed in collaboration with Chao Li (NYU) - showing that on any compact 3-manifold with boundary the space of positive scalar curvature metrics with minimal boundary is either empty or contractible, thereby deriving the corresponding result for scalar-flat asymptotically flat 3-manifolds with horizon boundary (and, more generally, for maximal vacuum initial data sets for the Einstein equations in general relativity). In order to further investigate related problems, with a view towards the next phase(s) of the CHANGE project, the same team has obtained a novel desingularization theorem, that allows to smoothly attach two given manifolds with corners by suitably gluing a pair of isometric faces, with control on both the scalar curvature of the resulting space and the mean curvature of its boundary. This extends from the closed to the bordered case a famous result by P. Miao, and we expect our it to become a standard tool in the field. Such a desingularization technique result has also been employed by us to define a suitable, weak notion of concordance for PSC metrics on manifolds with boundary.
During this period, there have already appeared two seminal papers, which in our opinion clearly convey the main message and spirit of the project: (Carlotto-Franz-Schulz, on Memoirs of the American Mathematical Society) and (Carlotto-Li, on Comm. Pure Appl. Math.), respectively for what pertains to the first and second main research directions of the project. In addition, the group's activity on the themes above is reflected in twelve more research articles (not accounting for surveys, reports etc...) that are all available on the general-purpose arXiv server and have either appeared in print on high-level mathematical journals or are currently (i.e. at the moment this report is being written) under review after submission for publication. On top of such primary research output, significant efforts have been spent in various forms of dissemination, including invited talks at various events (ranging from the 8th European Congress of Mathematics, to multiple workshops and research seminar at a number of institutions worldwide) and organisation of project-related research meetings, culminating in particular in the June 2024 forthcoming international conferences to happen in Zurich, for which we can already count on 24 speakers of the highest caliber.
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.
[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.