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.