For the second installment of this series of posts (which started here) on the first Bourbaki seminar of 2015, we will discuss Gilles Carron talk entitled “New utilisation of the maximum principles in Geometry (after B. Andrews, S. Brendle, J. Clutterbuck)”. Here, besides the original works of Andrews-Clutterbuck and Brendle (quoted below), the main references are the video of Carron’s talk and his lecture notes (both in French).
Disclaimer. All errors, mistakes or misattributions are my entire responsibility.
Given a Riemannian -dimensional manifold , one can often study its Geometry by analyzing adequate smooth real functions on (such as scalar curvature). One of the techniques used to get some information about is the following observation (“baby maximum principle”): if has a local maximum at a point , then we dispose of
- a first order information: the gradient of at vanishes; and
- a second order information: the Hessian of at has a sign (namely, it is negative definite).
In order to extract more information from this technique, one can appeal to the so-called doubling of variables method: instead of studying , one investigates the local maxima of a “well-chosen” function on the double of variables (e.g., ). In this way, we have new constraints because the gradient and Hessian of depend on more variables than those of .
This idea of doubling the variables goes back to Kruzkov who used it to estimate the modulus of continuity of the derivative of solutions of a non-linear parabolic PDE (in one space dimension). In this post we shall see how this idea was ingeniously employed by Andrews and Clutterbuck (2011) and Brendle (2013) in two recent important works.
We start with the statement of Andrews-Clutterbuck theorem:
Theorem 1 (Andrews-Clutterbuck) Let be a convex domain of diameter . Consider the Schrödinger operator where is the Laplacian operator and is the operator induced by the multiplication by a convex function .Recall that the spectrum of with respect to Dirichlet condition on the boundary consists of a discrete set of eigenvalues of the form:
In this setting, the fundamental gap of is bounded from below by
Remark 1 This theorem is sharp: when and (by Fourier analysis). In other terms, Andrews-Clutterbuck theorem is an optimal comparison theorem between the fundamental gap of general Schrödinger operators with the one-dimensional case.
Next, we state Brendle’s theorem:
The sketches of proof of these results are presented in the next two Sections. For now, let us close this introductory section by explaining some of the motivations of these theorems.
1.1. The context of Andrews-Clutterbuck theorem
The interest of the fundamental gap comes from the fact that it helps in the description of the long-term behavior of non-negative non-trivial solutions of the heat equation
with on . More precisely, one has that
- is an adequate constant,
- is the ground state of , i.e., , on , on and is normalized so that , and
- denotes (as usual) a quantity bounded from above by for some constant and all .
The theorem of Andrews-Clutterbuck answers positively a conjecture of Yau and Ashbaugh-Benguria. This conjecture was based on a series of works in Mathematics and Physics: from the mathematical side, van den Berg observed during his study of the behavior of spectral functions in big convex domains (modeling Bose-Einstein condensation) that for the free Laplacian () on several convex domains. After that, Singer-Wong-Yau-Yau proved that
and Yu-Zhong improved this result by showing that
Furthermore, some particular cases of Andrews-Clutterbuck were previously known: for instance, Lavine proved the one-dimensional case , and other authors studied the cases of convex domains with some (axial and/or rotational) symmetries in higher dimensions.
1.2. The context of Brendle theorem
The theorem of Brendle answers affirmatively a Lawson’s conjecture.
Lawson arrived at this conjecture after proving (in this paper here) that every compact oriented surface without boundary can be minimally embedded in .
Remark 2 The analog of Lawson’s theorem is completely false in : using the maximum principle, one can show that there are no immersed compact minimal surfaces in .
Moreover, Lawson (in the same paper loc. cit.) showed that, if the genus of is not prime, then admits two non-isometric minimal embeddings in .
On the other hand, Lawson’s construction in the case of genus produces only the Clifford torus (up to isometries). Nevertheless, Lawson proved (in this paper here) that if is a minimal torus, then there exists a diffeomorphism taking to the Clifford torus : in other terms, there is no knotted minimal torus in !
In this context, Lawson was led to conjecture that this diffeomorphism could be taken to be an isometry, an assertion that was confirmed by Brendle.
2. Proof of Andrews-Clutterbuck theorem
One of the key points of Andrews-Clutterbuck argument is an improvement of a theorem of Brascamp-Lieb. More precisely, the Brascamp-Lieb theorem ensures, in the context of Theorem 1, the log-concavity of the the ground state of (i.e., the logarithm is a concave function). In this setting, a fundamental ingredient in Andrews-Clutterbuck proof of Theorem 1 is a quantitative statement about the log-concavity of .
Before discussing Andrews-Clutterbuck’s improvement of Brascamp-Lieb theorem, let us quickly review Korevaar’s proof of Brascamp-Lieb theorem as an excuse to introduce a first concrete instance of the doubling of variables method.
2.1. A sketch of Korevaar’s proof of Brascamp-Lieb theorem
We want to show that is log-concave. For this sake, we can assume that the domain and the potential are strictly convex. Indeed, this is so because and are convex, so that they can be approximated by strictly convex objects, and, furthermore, it can be shown that the ground state varies continuously under deformations of and .
By definition, is concave if and only if the function
on the double of variables is non-positive.
We divide the proof of the fact that for all into two parts.
First, we claim that . In fact:
- If with , then because (i.e., ) on and on . Here, we used that .
- If with , one exploits the strict convexity of to say that, near , the ground state “looks like” the distance to the boundary , so that is a concave function near .
Next, once we dispose of the fact that , the proof of the log-concavity of will be complete if we show that at any local maximum .
In this direction, we use the baby maximum principle. If is a local maximum of , then vanishes to the first order at , i.e., . Thus, if denoting by , we deduce from the definition of and the equation that
Now, a simple calculation reveals that the Laplacian of satisfies the equation
Since is strictly convex, this inequality implies that , and, a fortiori, , as we wanted to prove. This completes the sketch of Korevaar’s proof of Brascamp-Lieb theorem.
2.2. An improvement of Brascamp-Lieb’s theorem
This estimate provides new important informations beyond the statement of Brascamp-Lieb theorem: for example, when , the right-hand side of the inequality goes to (which is much better than simply knowing that it is non-positive).
The proof of this estimate is somewhat complicated: it involves a combination of the doubling of variables method, a comparison argument with the one-dimensional case and the study of parabolic PDEs.
For this reason, by following Carron’s talk, we will skip the proof of this estimate, and we will now discuss how this estimate can be used to get lower bound on the fundamental gap in Theorem 1. In this direction, we will follow an approach proposed by Lei Ni (which is not exactly the original argument of Andrews-Clutterbuck).
2.3. End of the (sketch of) proof of Theorem 1
We consider the eigenfunction with , , , and on .
The previous method of Singer-Wong-Yau-Yau consisted of studying first two derivatives of the function
at its local maximum points, extract an inequality, and obtain a (non-optimal) lower bound on by integration of this inequality (together with the fact that satisfies (4)).
The approach proposed by Andrews-Clutterbuck consists in studying the oscillations of , i.e., we compare with the one-dimensional model (on an interval) by means of the function:
where and . (This choice of avoids “boundary effects”).
We want to use the doubling of variables method, i.e., the baby maximum principle applied to . For this sake, we introduce the quantity:
One can show that the strict convexity of implies that is attained on or in the interior of .
Remark 3 We have not defined on , but a first order expansion says that it is natural to pose
In both cases, the study of the first two derivatives of at a maximum point and the improvement (3) of Brascamp-Lieb theorem imply that
For the sake of exposition, we show the validity of (5) only in the case that is attained at , i.e., for some : indeed, the other case is similar (in some sense) to this one.
If is attained at ,, then is a maximum for , so that
Of course, this inequality is a good starting point to study (via (4)), but it is only a partial information obtained by letting vary only along !
In other words, we got an extra (better) information on thanks to the doubling of variables method applied to !
Now, we notice that the Andrews-Clutterbuck improvement (3) of Brascamp-Lieb theorem says (among other things) that . By plugging this into the previous inequality, we conclude that
Since is a non-constant function (and ), we have from this inequality that
This proves (5) when is attained at , as desired.
3. Proof of Brendle theorem
Let be a minimal torus inside the round -sphere . Denote by a choice of unit normal to .
The second fundamental form is the Hessian at of the function (from to ) whose graph over is (locally) equal to . In particular, is a symmetric quadratic form, and, hence, can be diagonalized. The (real) eigenvalues of are called principal curvatures of at .
By definition, is minimal if and only if the trace of vanishes (for all ). In other words, the eigenvalues of are when is minimal.
For later use, we recall the following three facts:
- Lawson proved that when is a minimal torus in . (Of course, this result strongly uses that has genus , and, indeed, it is completely false for other genera)
- The minimality of imposes a constraint on known as Simon’s formula. In our setting, this means that the principal curvature verifies the following PDE:
- Lawson also showed that if is constant, then the minimal torus is isometric to Clifford’s torus .
The last item above says that our objective is very clear: in order to prove Brendle’s theorem 2, we have to show that is constant.
By Gauss-Bonnet theorem, we have that , i.e., equals to in average. From this point, a natural strategy would be to combine this information with Simons’ equation (and some maximum principles) to show that . Unfortunately, this idea does not work mainly because of the (negative) sign of the term in Simons formula.
At this point, Brendle introduces the function
(Note that, since , , and, thus, .)
The geometrical meaning of is the following. The quantity is the biggest radius such that stays outside a ball of radius tangent to at .
In other terms, stays outside of the oscullating balls of radius , and , and are mutually tangent at . From this fact, it is possible to check that
This means that the global information (curvature of oscullating balls) controls the local information (principal curvature).
in the sense of viscosity. For the sake of exposition, let us prove that satisfies this inequality when : the general case ( is a viscosity solution when is not smooth) follows by a simple modification of the argument below.
Up to changing our choice of unit normal , we can write . Let us apply the doubling of variables method by considering the function
Given a point , we have two possibilities: either or .
In the first case (), since , one has (from the baby maximum principle) that and . By plugging this into Simons formula (7), we deduce that
In the second case , we have that there exists , such that . Since , we get that . Geometrically, this condition means that stays outside the ball and is tangent to at and . In particular, this implies that the tangent planes and are symmetric with respect to the mediator hyperplane of the segment between and . By exploiting this symmetry, Brendle chose good coordinate systems on and leading him to the following inequality
with after seven pages of calculations in his paper! Since , we have the good sign to conclude from this estimate that
as it was claimed.
where is a constant.
We claim that this implies that is constant, so that the proof of Brendle theorem would be complete (by Lawson’s result quoted right after Simons formula (7)). Indeed, we have again two cases: either or .
In the first situation, since the oscullating balls with have the same principal curvature of at , a third order expansion of (the graph of) at reveals that for all , so that is constant.
In the second situation with , we look again at the inequality
showed above (under the assumption , which is our current situation at all since , ).
Because satisfies Simons formula (7) and with a constant, we get that the first term of the previous inequality vanishes. In particular, we deduce that
This means that for all (since and ), so that is constant, as it was claimed.
This completes the proof of Brendle’s theorem and, consequently, the discussion of this post.