In the last 5 months I was really happy to see announcements of solutions to important problems and conjectures in several fields of Mathematics. For instance,
- (1) Ian Agol announced the proof of the virtual Haken conjecture (see the blog of D. Calegari for three posts on this subject) in -manifold topology,
- (2) Fernando Codá Marques and André Neves announced the proof of the Willmore conjecture (see this journal article of F. Morgan for an informal explanation of this 47-years old conjecture),
- (3) Alex Eskin, Maxim Kontsevich and Anton Zorich recently completed the proof of a formula for the sum of the Lyapunov exponents of the Kontsevich-Zorich cocycle (announced 15 years ago in this paper of M. Kontsevich), and
- (4) Alex Eskin and Maryam Mirzakhani recently announced the proof of a Ratner type theorem (classification of invariant measures) in the (non-homogenous) setting of the moduli spaces of Abelian differentials.
In particular, since I’ve already met Fernando Codá Marques, André Neves, Anton Zorich and Alex Eskin before in several occasions, I would like to take the opportunity (and liberty) to congratulate them for these quite impressive works! (indeed, besides presenting excellent results, the articles in the second, third and fourth items above have 95, 106 and 152 pages resp. )
My current plan is to dedicate some posts in this blog to expose a few points of the articles mentioned in the three last items above. More concretely, the rest of today’s post is entirely dedicated to the presentation of the solution of the Willmore conjecture by Fernando and André (here I’ll call them by their first names as I think they will not mind ). Then, in (a?) future post(s?), I will discuss the article of Alex, Anton and M. Kontsevich on the sum of Lyapunov exponents of the Kontsevich-Zorich cocycle. Finally, I will complete this “series” with some post(s?) about the Ratner type theorem of Alex and M. Mirzakhani based on some notes I took from the course Alex gave in Luminy/Marseille a few weeks ago.
Before proceeding further, let me make some disclaimers and comments. Firstly, the task of writing today’s post was substantially simplified by the fact that Fernando and André wrote an excellent outline of their arguments and ideas in Section 2 of their article. In particular, today I’ll essentially follow their outline, but please beware that maybe I introduced some mistakes in this process (and, of course, such mistakes are my responsibility only). Secondly, the article of item (3) will be the subject of a talk by Pascal Hubert at Séminaire Bourbaki in October 2012. In particular, he will prepare a serious set of notes for his talk, so that my posts around this topic should be seen as an informal preparation to Pascal’s notes. Finally, concerning the article in item (4) above, Alex Eskin made available in his webpage (see here) an excellent set of notes around his minicourse: indeed, the set of notes by Alex are so pleasant to read that I’ve immediately changed my initial plan of posting my original notes here and I’ll content myself to make a few comments around the techniques used by Alex and M. Mirzakhani, e.g., I’ll try to post my notes from the minicourse (also in Luminy/Marseille) taught by J.-F. Quint around the so-called “exponential drift” idea (from his celebrated paper with Y. Benoist).
After these comments, let’s start (below the fold) the discussion of the Willmore conjecture and its solution by Fernando and André!
Let be a closed (i.e., compact and boundaryless) immersed surface of . Among the most simple geometric invariants one can associate to , we arguably have the Gaussian curvature and the mean curvature (at each point ) obtained from the determinant and the trace of the so-called second fundamental form (a linear operator/matrix whose eigenvalues are the principal curvatures of at ). For a excellent exposition of these concepts see, e.g., M. do Carmo’s classical book “Differential geometry of curves and surfaces”.
In this language, the Willmore energy is
where is the natural area form of .
The Willmore energy naturally appears in some physical (study of elastic shells) and biological (study of cell membranes) contexts, where it is sometimes called bending energy.
In the mathematical context, it was known to be invariant under conformal transformations of by e.g. W. Blaschke, and a very natural geometric (variational) problem consists in its minimization within a given class of immersed closed surfaces .
Here, it is possible to show that for any immersed closed surface
where is the } of radius . Moreover, if and only if is a round sphere. In other words, the round spheres of minimize the Willmore energy in the class of all closed immersed surfaces. Actually, the proof of this fact is not hard:
- Denoting the principal curvatures by , and recalling that the mean curvature is , and the Gaussian curvature is , we get from the abstract equality that
On the other hand, by the Gauss-Bonnet theorem, . Therefore, assuming that is topologically a sphere, i.e., it has genus and Euler characteristic , we can conclude that
and the equality holds if and only if
that is, at all points of , i.e., all points are umbilical, and this last property allows to show precisely when is a round sphere.
- In general (i.e., when has higher genus ), the calculation above is not useful because , that is, the estimate relying on Gauss-Bonnet theorem is weaker than the trivial bound . However, one can overcome this by using the Chern-Lashof inequality
In fact, denoting by and , we have
so that . Now, we can use to the region to get that
In any event, once one realizes that the minimization of the Willmore energy in the class of all immersed compact surfaces of is not difficult, one can follow T. Willmore and ask about the problem of minimizing the Willmore energy in the class of immersed torii of :
Willmore conjecture (1965). for any imersed torus .
The equality is attained by the torus of revolution obtained by rotating a circle of radius with center at distance from the axis of revolution:
See, e.g., P. Nylander’s post here for nice pictures of this torus.
More generally, the Willmore conjecture can be seen as a question about immersed closed surfaces on by means of the stereographical projection : indeed, given , by putting , one has that
where is the mean curvature of . So, the minimization of is equivalent to the minimization of . For this reason, we will also call the Willmore energy of .
Even though this reformulation is not sophisticated, it is very interesting given that the Willmore conjecture is a variational (minimization) problem. For instance, if one consider a sequence of surfaces converging to the infimum of the Willmore energy, and one try to construct a surface minimizing the Willmore energy by taking some “limit” along a subsequence of , it is better to know that than because is compact but is not.
Concerning some previously known results towards Willmore conjecture, one has:
- (a) it was shown by P. Li and S.-T. Yau that if an immersion covers a points at least times, then . In particular, if is an immersed non-embedded surface (i.e., at some ), then . In other words, it suffices to estimate for embedded ;
- (b) L. Simon proved the existence of a torus minimizing the Willmore energy (and later this result was generalized by M. Bauer and E. Kuwert for higher genus cases);
- (c) critical points of the Willmore functional are called Willmore surfaces and the Euler-Lagrangeequation of this functional is
where is the Laplacian on and is the Gaussian curvature of . From this equation (sometimes called Willmore equation), we see that minimal, i.e., , surfaces of (such as the ones constructed by B. Lawson in this paper here) are Willmore surfaces, but they are not the sole ones (see, e.g., the articles of R. Bryant, U. Pinkall). For more information on the analytical aspects of the Willmore equation, see, e.g., this article of T. Rivière.
Moreover, if and only if is conformal to the Clifford torus .
By item (a) above, this profound result implies the Willmore conjecture.
Remark 1 Here, in th order of approximation, the “philosophy” behind this reduction is the fact that the Willmore energy
coincides with the area functional when is minimal (i.e., ), and the problem of studying the area of minimal surfaces is a classical subject in Differential Geometry.
Roughly speaking, given a general embedded closed surface of genus , they consider its natural “min-max homotopy class” and the corresponding width . Informally, we can describe general homotopy classes and their widths in the case of -dimensional surfaces inside a compact -dimensional manifold as follows.
Given , let be the unit cube of and let be a (continuous) map defined on such that is a compact boundaryless surface of for each . We say that two such maps and are homotopic (relatively to ) if we can find a (continuous) map defined on such that
- is a compact boundaryless surface of for all ;
- and for all ;
- for all , .
Below we illustrate the concept of homotopy with two pictures in a low-dimensional setting, that is, (instead of a -dimensional manifold) and the maps take values in the space of curves (instead of surfaces). For the first picture, we took , so that and we drew the images of under and by the blue and red curves resp., while for the second picture we took , so that , the images of , resp. , is the family of blue, resp. red, curves, and the image of the boundary are the two trivial (degenerate) “curves” corresponding to the north and south poles of .
The homotopy class of is the set of all homotopic to , and the width of is
Remark 2 The formal definition of homotopy classes (and their widths) in the article of Fernando and André rely on some concepts from Geometric Measure Theory such as (integral) currents, varifolds, and flat metric and the flat topology (based on the notion of mass) in the space of integral currents. In particular, the maps above associate an integral current without boundary to each , and is assumed to be continuous in the flat topology. Similarly, the homotopy is continuous in the mass topology. See Sections 4 and 7 of the article of Fernando and André for more details.
Remark 3 In some sense, the homotopy class of the previous example can be though as an element of the “homotopy group” where is the space of surfaces in and is the trivial surface (of area zero).
In the case of a closed embedded surface of of genus , Fernando and André associate a certain -dimensional min-max homotopy class (i.e., they consider the homotopy class of a certain related to and defined on ) and they show the following min-max result:
Theorem 2 (F. C. Marques and A. Neves) Let be a closed embedded surface of genus . Then, there exists a minimal embedded surface of genus (maybe not connected and possibly with multiplicities) such that
where is the min-max homotopy class of .
As we already mentioned, this permits to reduce the proof of Theorem 1 for a general to the study of the area of a minimal . Here, Fernando and André show that:
Moreover, the equality holds if and only if is isometric to the Clifford torus .
Of course, by combining Theorems 2 and 3, the inequality in the statement of Theorem 1 “essentially” follows. Indeed, we said “essentially” because if the minimal surface of genus were known to be connected and of multiplicity , then Theorem 1 would really follow. Of course, this is not true in general, but, as we’ll see by the end of this post, it is easy to conclude Theorem 1 from Theorems 2 and 3. Also, the fact that characterizes the conformal class of the Clifford torus is not a direct consequence of Theorems 2 and 3 as stated above, but nevertheless we’ll see that it follows from the arguments used in the proofs of these results anyway.
In any event, after these highlights of the main results obtained by Fernando and André, we pass to the description (compare with Section 2 of the article of Fernando and André) of the general strategy of the proofs of Theorems 1, 2 and 3.
2. Strategy of the proof of Theorem 2
A large portion of the article of Fernando and André is devoted to the careful construction of a min-max theory à la Almgren-Pitts in order to realize the width of certain homotopy classes by the area of minimal surfaces. In the next subsection, a prototypical result of this min-max theory is stated.
2.1. Prototype of min-max theorem
Let be a compact -dimensional manifold and let be a homotopy class of a (continuous) map associating a compact surface of to each .
Then, there exists a smooth embedded minimal surface (maybe not connected and with multiplicities) such that
Furthermore, given with
we can select , , with
In rough terms, the condition in the previous theorem is setup as a sort of barrier at the boundary. In this language, the min-max theorem above says that the variational problem of realizing the width of a homotopy class by the area of a minimal surface can be solved by taking the limit of an appropriate sequence of surfaces whose areas approximates because the “barrier condition” prohibits to “escape” through the boundary (and so must converge).
Of course, the statement above is an approximation of the min-max theorem proved by Fernando and André. For a more precise result, see Theorem 8.5 (and also Proposition 8.4) in their article. As we already mentioned, the proof of the min-max theorem (or, more precisely, the proof of Proposition 8.4) is quite long and technical and it occupies a significant portion of the article (namely, Sections 13, 14, 15 and Appendix C). In particular, a detailed discussion of this part of the article is out of the scope of this post, and we will take this result for granted in what follows.
Remark 4 As it is mentioned by Fernando and André in the article, by analogy with Morse theory, the minimal surface in the min-max theorem above is expected to have index (because it comes from a -parameter family of surfaces). In general, this seems a delicate issue to verify, but, nevertheless, we will see (during the proof of Theorem 3) that this is the case for the min-max homotopy class of the Clifford torus (where ).
Once we dispose of the min-max theorem, we continue the outline of the proof of Theorem 2 by sketching (in the next three subsections) the construction of the min-max homotopy class associated to a smooth compact embedded surface .
2.2. Canonical family
Let be the open unit ball of . For each , let be the conformal map
Given smooth compact embedded surface, we have the following canonical family of surfaces
Here, is the oriented distance between and the surface in the canonical (round) metric of , that is, by choosing an orientation of , we can write where is the “interior” of (a connected component of ) and is the “exterior” of (the other connected component of ), and we put
Remark 5 Since the diameter of is , we have that and are trivial (zero area) surfaces for all .
Because is conformal, we have that for every . On the other hand, as it was shown by A. Ros, one has
where is the traceless part of the second fundamental form of . In fact, the proof of this estimate is not difficult: it is a direct (one-page) computation with that can be found in page 15 (cf. Theorem 3.4) of the article of Fernando and André.
In view of this estimate and the statement of Theorem 2, it is tempting to try to apply the Min-Max Theorem 4 to the canonical family , . However, this doesn’t quite work: is not compact, and any attempt to “reasonably” extend the canonical family to is not continuous, e.g., in the flat topology (see Remark 2), so that such extended canonical families don’t define reasonable homotopy classes. For instance, while and still make sense for , e.g., or as depending on whether or , the same is not true for in the sense that the limit of as depend on the “angle of convergence”: more precisely, denoting by the unit normal vector to at , if
approaches (i.e., , and as ), then
where . See the picture below for an illustration (in low-dimensions) of a sequence approaching a point in a totally geodesic sphere (equator) .
A detailed discussion of these facts about the behavior of as is performed in Section 5 of the article of Fernando and André (see Proposition 5.3, Lemma 5.4 and Appendix B).
In any case, we just saw that the main obstacle to “reasonably” extend the canonical family to is the set . Here, Fernando and André propose to overcome this difficulty by “blowing up” , the topic of our next subsection.
2.3. A blow-up procedure
Fix small and let be the tubular neighborhood of of radius in , i.e.,
Then, one considers a continuous map such that:
- is a homeomorphism between to , and
- sends to by projection to the nearest point.
More concretely, one can define to be the identity map on and where is a smooth bump function, equals to on , is strictly increasing on , and equals to on .
Below we give a pictorial description of when is a meridian (totally geodesic sphere), where the reader can (hopefully) visualize what is the role of the blow-up map .
Using , Fernando and André modify the initial canonical family by introducing the family for . Here, since we removed a tubular neighborhood of , one sees from the features of that the problem of dependence on the “angle of convergence” for is now solved for : morally, the selection of and the definition of serves to stop before reaching and to choose a “definite angle of convergence”.
In particular, it is not hard to convince oneself that the new family admits a “reasonable” — continuous in the flat topology on the space of integral currents without boundary, see Remark 2 — extension to . Then, one can further extend to so that is constant in the radial direction, i.e., .
In this way, Fernando and André show (see Theorem 5.1 of their article) that one gets a “reasonable” family defined on such that
- for all ;
- is a geodesic sphere of when ;
- for each , there exists an unique such that is a totally geodesic sphere of , i.e.,
for some ; for example, if , then and depending on whether belongs to the interior or the exterior of .
Actually, in the last item, can take two values in principle, but, by considering oriented surfaces, one has , and thus is also unique (as well as ).
Later on, we’ll see that this equality is a “barrier at boundary” condition in the sense discussed in Subsection 2, but, for now we’ll use the family to complete the definition of the min-max family associated to in the next subsection.
2.4. Min-max family associated to
Choose a homeomorphism (so that is a homeomorphism between and ) and an extension of (for a concrete choice of , see Section 6 of the article of Fernando and André). Using and , we define the min-max family associated to by the following reparametrization of :
From the definitions, we have that the min-max family verify the following properties (see Theorem 6.3 of the article of Fernando and André):
- (a) for each , is a totally geodesic sphere if and only if ;
- (b) by Equation (1)(A. Ros’ estimate) above, one has
- (c) by Equation (2)(“barrier at the boundary” condition), one has
Informally, is an element of the “homotopy group” where is the space of surfaces of and is the space of geodesic spheres of . More formally, Fernando and André discuss in Sections 7 and 8 of their article (see, in particular, Theorem 8.1) the fact that the min-max family associated to “naturally” determines a homotopy class called min-max homotopy class associated to .
At this stage, one would like to prove Theorem 2 by applying the Min-Max Theorem 4 to the min-max homotopy class associated to a compact surface of genus . Of course, before doing so, we need, by item (c) above, to check the “barrier” condition
2.5. Degree of
Recall that during the blow-up procedure (see Subsection 2), we introduced a map with .
The main topological ingredient towards the estimate of is the following result (see Theorem 3.7 of the article of Fernando and André):
Below we sketch the calculation of the degree of referring to the original article for the details.
Proof: It is possible to check that:
- if ;
- if ;
- with if .
From this point, the continuity of is checked by direct inspection.
On the other hand, for the computation of the degree of , we have to calculate, by definition, the following quantity
where is the volume form of .
In this direction, one uses that is a (orientation-preserving) diffeomorphism from to , and from to , so that
Since has zero measure (volume) in and , we get that
and it remains only to calculate the integral .
Consider , . It is not hard to check that is an orientation-preserving diffeomorphism: for instance, by fixing a positive basis of , one has . So, by putting ,
By selecting a positive orthonormal basis of diagonalizing the second fundamental form, we have (by definition) that . After some calculations, Fernando and André find that
Because , we obtain
Since and , we get
Plugging this equation in the formula for the degree of derived above, we conclude that
This completes the sketch of proof of the theorem.
This topological theorem has the following nice homological consequence. As we mentioned in Subsection 2 (see item (a)), the min-max family satisfies for all , where denotes the set of all unoriented totally geodesic spheres of and is the surface without a choice of orientation. Note that is naturally isomorphic to the real projective space by associating to each geodesic sphere the line passing by its center.
In the next subsection, we will use this fact to control the width of the min-max homotopy classes and complete the proof of Theorem 2.
2.6. The width of min-max homotopy classes
In this subsection we will consider a compact surface of genus , so that, by Equation (4),
and we want to get that the width of the min-max homotopy class associated to satisfies . Compare with Theorem 9.1. Of course, once we proved this, the proof of Theorem 2 is complete because (in view of the discussion in Subsection 2) is precisely the “barrier” condition one needs to apply the Min-Max Theorem 4.
The proof of goes by contradiction: suppose that and consider a sequence with
By the definition of the homotopy classes, one also has for all and .
Fix small and let be the set of such that is “-far” (in an appropriate sense in the space of currents) from the set of totally geodesic spheres of . Since is a trivial (zero area) surface for every , we have that . Let denote the connected component of containing . For sake of simplicity of the exposition, we will assume that and are compact manifolds with boundary.
We start by showing that for all large. Assume that this is not the case. Then, one can find a sequence of continuous paths with and . The maps defined on are mutually homotopic and their homotopy class is very similar to the homotopy class of Example 1. In particular, the width of is .
Thus, we have
Since and are trivial surfaces, we can apply the Min-Max Theorem 4to produce a sequence such that converges to a minimal surface of of area . From our discussion in the introduction, must be a totally geodesic sphere, and this is a contradiction with the definition of , i.e., is “-far” from the set of totally geodesic spheres.
Now, since for all large, we have that
Consider the -dimensional submanifold . By definition of , for any , is “-close” (i.e., not “-far”) from the set of totally geodesic spheres. On the hand, trivial surfaces are “-far” from , and the image of under (or equivalently ) consists of trivial (zero area) surfaces. Putting this together with the set inclusion of the previous paragraph, we deduce that
In resume, is a -dimensional submanifold separating from . Furthermore, by item (a) in Subsection 2, for all , is a totally geodesic sphere if and only if . Hence, given , we can select small so that
We claim that is homologous to in . Indeed, define . Since has no boundary, one has
In particular, since , we conclude that is homologous to in . Hence, is also homologous to in , as it was claimed.
Finally, we consider the map associating to the unoriented totally geodesic sphere . Because on (by definition of the homotopy class ) and (see above), we have that is “-close” to . Using this and the fact that is “-close” to for (see the definition of above), Fernando and André construct a continuous map from the -dimensional manifold to approximating on such that on (see Lemma 9.10 of their article). In homology, this implies that
However, we just saw that is homologous to . Hence, by the topological result in Equation 5 above,
Of course this is a contradiction showing that the width of min-max homotopy classes of surfaces of genus verify .
3. Strategy of the proof of Theorem 3
Before starting our considerations, we introduce the notion of (stability) index of a compact surface . We denote by the Laplace-Beltrami operator, the second fundamental form and the Jacobi operator of . The index of is the number of negative eigenvalues of . For a discussion of the index and second variation formulas for surfaces, see e.g. this post of Danny Calegari.
As it is explained in Appendix A of the article of Fernando and André, one can always find a minimal surface of genus of least area among all minimal surfaces of with genus . Because the Clifford torus has area , we have that . Consider the min-max family associated to and denote by the corresponding min-max homotopy class.
By a theorem of F. Urbano, it suffices to show that to conclude that is isometric to the Clifford torus (see also Remark 4 above). In other words, the proof of Theorem 3 is reduced to the proof of the estimate . We argue by contradiction. Suppose that and consider the canonical family , associated to (see Subsection 2). Recall that the canonical family satisfies
Also, since is a minimal surface, . So,
Moreover, since is a minimal surface, the function has a global isolated maximum at . Because is -dimensional and we’re assuming that , one can (slightly) perturb near to produce a new family with
See Section 10 of the article of Fernando and André for more details.
Now we consider the min-max family and the min-max homotopy class associated to . Since the (small) perturbation of is performed near , one has that
- coincides with on
- by Theorem 2, (as has genus )
- furthermore, there exists a minimal surface such that
Because the area of any minimal surface of is at least, and round spheres are precisely the minimal surface of with area , it follows from the estimate above that is a connected minimal surface of genus and multiplicity . On the other hand, was chosen to minimize the area among all minimal surfaces of of genus , so that we get . Of course, this is a contradiction with the estimate above, and thus as desired. Now that the sketch of proof of Theorem 3 is complete, we close today’s post with the sketch of the proof of Theorem 1.
4. Strategy of the proof of Theorem 1
Given an embedded compact surface of with genus . Our goal is to prove that . Because , we can assume (without loss of generality) that . Denote by the min-max family and the min-max homotopy class associated to . By Theorem 2, we have that
where is a minimal surface (maybe not connected and possibly with multiplicities). By the same arguments of the last paragraph of the previous section, the fact implies that is a connected minimal surface with genus and multiplicity . By Theorem 3, we have . Thus, we get
and the first part of the statement of Theorem 1is proved.
Finally, concerning the second (“rigidity”) part of the statement of Theorem 1, given a compact surface of genus with , one considers the conformal transformations introduced in Subsection 2 and one starts the argument by proving that there exists with . Here, the proof of this fact goes by contradiction: by supposing that for all , one can approach the boundary along a subsequence to produce a totally geodesic sphere contained in the minimal surface constructed above, but this is not possible because has genus . Then, once we know that for some , we have by Theorem 3 that is isometric to the Clifford torus, that is, is conformal to the Clifford torus.