Final Report Summary - RSC AND RMCF (Rigidity of Scalar Curvature and Regularity for Mean Curvature Flow)
The project had 4 objectives belonging to two different fields of Mathematics, study of minimal surfaces (Objective 1 and 2) and study of Geometric flows (Objective 3 and 4).
Minimal surfaces are equilibrium configurations and they appear naturally in Nature, e.g. soap films.
Minimal surfaces come in two types: those for which small perturbations increase their area (stable equilibrium) and those which admit perturbations that decrease their area (unstable equilibrium). The first type has been heavily studied in Geometry because they can be obtained by minimizing area among all surfaces that cannot be squashed to a point (think of a non-contractible curve on a doughnut) and their applications to Geometry have been immense.
I became interested in the properties of unstable minimal surfaces because they seemed interesting and very little was known. Even their existence was a highly non-trivial question.
Objective 1 wanted to understand basic properties the most simple unstable minimal surface that exist on 3-spheres that are positively curved. Jointly with Fernando Marques we answered that and found that the area of that unstable minimal surface is bounded in a optimal way.
Objective 2 was to answer a question of S. T. Yau who conjectured that any 3-manifold has an infinite number of unstable minimal surfaces. Jointly with Fernando Marques we answered that positively for positively curved spaces. We are now actively trying to understand their properties and how they look like.
Curiously, the methodology used during this part of the project allowed us to answer the following very down-to-earth question asked by Willmore in 1965: what is the least bended toroidal shape in space?
If one asks this question for spherical shapes, common sense dictates that the answer should be the round sphere. For toroidal shapes there is no obvious candidate and Willmore conjectured that it should be the Clifford torus.
I solved this problem jointly with Fernando Marques by understanding it in terms of unstable minimal surfaces in the 3-sphere. Namely we found that the Clifford torus, when seen as a surface in the 3-sphere, has less area than the surface of highest area of any given non-trivial five-parameter family of surfaces.
This was very exciting and had a serious impact for the following reasons: (1) an old question that resisted attempts from famous mathematicians, like S. T. Yau or Leon Simon, had now been solved; (2) using the method of the proof in other contexts we solved two other problems that had been open for many years as well; (3) our proof indicates that the study of unstable minimal surfaces is a promising new direction in Geometry: there is a whole brave new world out there. Let's go and explore it!
The second part of the project was to study optimal deformations of complicated surfaces in four dimensional ambient manifolds into those that have optimal shapes.
Unfortunately, the subject is quite technical and this cannot be performed so easily as there are various types of singularities developing in this process. Objective 3 consisted in understanding the simplest type of those singularity models (self-expanders asymptotic to two planes) and show that they are unique. I solved it with Jason Lotay.
Objective 4 was to make sure that singularities are isolated and then be able to continue the flow past those singularities. Unfortunately this was not completed as it seems to be beyond reach.
As it stands, the socio-economic implications of cutting edge research in pure mathematics are hard to see in the medium term. That said, the problem of finding shapes in equilibrium position is relevant in Biophysics because it explains the shape that human blood red cells assume and has applications in the design of medical drugs.
For instance, the Biophysics community arrived independently at the Willmore conjecture by observing it experimentally in toroidal vesicles.
My perspective on wider applications of pure maths is that reasonable and natural questions in Geometry, like the Willmore conjecture, are destined to appear in different contexts in other branches of science and the answer, theoretical or experimental, will be of great interest to both communities.
Minimal surfaces are equilibrium configurations and they appear naturally in Nature, e.g. soap films.
Minimal surfaces come in two types: those for which small perturbations increase their area (stable equilibrium) and those which admit perturbations that decrease their area (unstable equilibrium). The first type has been heavily studied in Geometry because they can be obtained by minimizing area among all surfaces that cannot be squashed to a point (think of a non-contractible curve on a doughnut) and their applications to Geometry have been immense.
I became interested in the properties of unstable minimal surfaces because they seemed interesting and very little was known. Even their existence was a highly non-trivial question.
Objective 1 wanted to understand basic properties the most simple unstable minimal surface that exist on 3-spheres that are positively curved. Jointly with Fernando Marques we answered that and found that the area of that unstable minimal surface is bounded in a optimal way.
Objective 2 was to answer a question of S. T. Yau who conjectured that any 3-manifold has an infinite number of unstable minimal surfaces. Jointly with Fernando Marques we answered that positively for positively curved spaces. We are now actively trying to understand their properties and how they look like.
Curiously, the methodology used during this part of the project allowed us to answer the following very down-to-earth question asked by Willmore in 1965: what is the least bended toroidal shape in space?
If one asks this question for spherical shapes, common sense dictates that the answer should be the round sphere. For toroidal shapes there is no obvious candidate and Willmore conjectured that it should be the Clifford torus.
I solved this problem jointly with Fernando Marques by understanding it in terms of unstable minimal surfaces in the 3-sphere. Namely we found that the Clifford torus, when seen as a surface in the 3-sphere, has less area than the surface of highest area of any given non-trivial five-parameter family of surfaces.
This was very exciting and had a serious impact for the following reasons: (1) an old question that resisted attempts from famous mathematicians, like S. T. Yau or Leon Simon, had now been solved; (2) using the method of the proof in other contexts we solved two other problems that had been open for many years as well; (3) our proof indicates that the study of unstable minimal surfaces is a promising new direction in Geometry: there is a whole brave new world out there. Let's go and explore it!
The second part of the project was to study optimal deformations of complicated surfaces in four dimensional ambient manifolds into those that have optimal shapes.
Unfortunately, the subject is quite technical and this cannot be performed so easily as there are various types of singularities developing in this process. Objective 3 consisted in understanding the simplest type of those singularity models (self-expanders asymptotic to two planes) and show that they are unique. I solved it with Jason Lotay.
Objective 4 was to make sure that singularities are isolated and then be able to continue the flow past those singularities. Unfortunately this was not completed as it seems to be beyond reach.
As it stands, the socio-economic implications of cutting edge research in pure mathematics are hard to see in the medium term. That said, the problem of finding shapes in equilibrium position is relevant in Biophysics because it explains the shape that human blood red cells assume and has applications in the design of medical drugs.
For instance, the Biophysics community arrived independently at the Willmore conjecture by observing it experimentally in toroidal vesicles.
My perspective on wider applications of pure maths is that reasonable and natural questions in Geometry, like the Willmore conjecture, are destined to appear in different contexts in other branches of science and the answer, theoretical or experimental, will be of great interest to both communities.