The Fundamental Gap of Horoconvex Domains

Malik Tuerkoen and I have just uploaded our paper “Spectral Gap Estimates on Conformally Flat Manifolds” to the Arxiv. In this paper, we derive lower bounds on the fundamental gap for domains which satisfy a particular type of convexity in a conformally flat Riemannian manifold. The primary motivation for this is that it can be used to prove a conjecture that was put forth by Xuan Hien Nguyen, Alina Stancu, Guofang Wei.

I’ve written several posts before on the fundamental gap problem, so let me link to those to give some background on the problem.

  1. The mathematics of vibrations and Why is the sound of a single string harmonious?
    These are two videos which introduce the spectrum of the Laplacian in terms of acoustics and discuss the motivation for studying the fundamental gap.
  2. Modulus of Concavity and Fundamental Gap Estimates on Surfaces
    In this post, I gave some background on the fundamental gap problem and what can be said about the gap for surfaces.
  3. Log-Concavity and Conformal Gauges
    This post discusses how one can use conformal gauge theory to study the fundamental gap problem and that choosing a conformal gauge carefully allows one to prove estimates which were previously out of reach.

With that out of the way, let me discuss the conjecture and what is involved in the proof.

Horoconvex Domains and their Fundamental Gaps

For convex domains hyperbolic space, Theodora Bourni, Julie Clutterbuck, Xuan Hien Nguyen, Alina Stancu, Guofang Wei and Valentina-Mira Wheeler constructed convex domains where the fundamental gap is arbitrarily small. They included the following picture which I found really clearly explains the underlying reason for this phenomenon. It takes a bit of background to explain, but essentially the domain can look like a dumbbell with two regions connected by a narrow neck, and for these we know the gap can be very small.

Convex domains in hyperbolic space can have “necks”

In other words, there is no lower bound on the fundamental gap for convex domains in terms of the diameter. From this, the natural next step is to consider domains which satisfy stronger convexity assumptions to see if it is possible to control the fundamental gap for these domains. In order to explain the specific conjecture, I need to first discuss how to strengthen the notion of convexity in hyperbolic geometry.

Horocycles and horoconvex domains

In hyperbolic geometry, a horocycle is a curve whose geodesic curvature is identically 1 and which converges in both directions to a single point in the ideal boundary of hyperbolic space. This is a fairly complicated definition, so it can be helpful to see a picture.

Here, the black circle represents the Poincaré disk model of hyperbolic geometry and the red circles are two horocycles which converge to the points \infty and \infty'. In other words, in the disk model of hyperbolic geometry, horocycles are simply circles which are tangent to the boundary circle.

Note that both horocycles in the picture pass through points P and Q. In general, given any two points in hyperbolic space, there are two horocycles which pass between them. We say that a domain is horoconvex if given any two points in the domain, both of the horocycles segments between them are contained entirely within the domain.

I haven’t checked rigorously if this domain is actually horoconvex. Murphy’s law says it probably isn’t…

Here, the green domain is horoconvex because it contains all the horocycles connecting points in the domain. This is equivalent to the geodesic curvature of the boundary being at least 1, so horoconvexity is a strengthening of convexity (which is equivalent to the boundary curvature being non-negative).

The horoconvexity conjecture

In late 2020, Hien, Alina and Guofang studied the fundamental gap for domains which are horoconvex. They found that as the diameter gets large, the gap is at most C_n/D^3 where C_n is some constant depending on the dimension of the space and D is the diameter of the domain. This decays faster than 3 \pi^2/D^2 , which is the sharp lower bound for the fundamental gap of convex domains in Euclidean space. As such, even for horoconvex domains, we cannot expect to prove the same type of fundamental gap estimates that we have in flat space.

However, it didn’t seem like the gap of horoconvex domains could be arbitrarily small unless they were very large, so they suspected it was possible to obtain a lower bound on the gap in terms of the diameter. Therefore, they posed the following conjecture.

[W]e conjecture that, for all horoconvex convex domains \Omega \subset \mathbb{H}^n, we have \lambda_2(\Omega) - \lambda_1(\Omega) \geq c(n,D) for some function c(n, D) depending on the dimension and diameter…

Insights from Conformal Gauges

Although we had written a number of papers on the fundamental gap, until last November we had no way to attack this problem. In particular, an essential step to proving fundamental gap estimates is to show that the first eigenfunction is log-concave. Our strategy to doing so is to apply a maximum principle to show that the Hessian of the log of the principle eigenfunction can never touch a barrier function. From this, we are able to show that the function is log-concave using a continuity argument where we deform the domain/geometry so that the Hessian of the logarithm of eigenfunction is strongly negative (i.e., lies entirely on one side of the barrier).

However, this argument depends in a crucial way on the geometry having positive curvature. Any negative curvature would introduce terms that we could not control. In fact, when the curvature is negative it is possible to make the principle eigenfunction arbitrarily log-convex and the fundamental gap arbitrarily small. And since I didn’t have any ideas for the problem, I wasn’t really thinking about it too much.

Last year, I went back to our first paper and noticed that the argument to establish log-concavity would go through if the geometry was spherical but there was a non-constant weighting function applied to the eigenfunction. Soumyajit Saha, Malik and I started to write a paper to explain this generalized approach. Initially, I thought about this weighting function in terms of composite membranes, but Malik mentioned our work to Guofang, who immediately recognized that the weighting function could be understood in terms of a conformal deformation of the metric. And so instead of studying geometries with negative or variable curvature (where our problem was very difficult), when the geometry was conformally flat we could transform the problem into spherical geometry, where the analysis is tractable. And since hyperbolic geometry is conformally equivalent to spherical geometry, this would give a breakthrough to studying the gap of horoconvex domains.

Local and non-local arguments

Once we realized this, we changed the initial paper to focus on the idea of how conformal gauges can be used to establish log-concavity. Although this paper introduced a key idea needed to study horoconvex domains, we weren’t able to resolve the full conjecture here because used a local (i.e., one-point) maximum principle. The advantage of the local approach is that the computations involved are significantly easier and the underlying approach is much simpler to explain. Since the point of the paper was to introduce our idea, we felt like it was better to err on the side of trying to make things readable (relatively speaking, that is).

However, the disadvantage of local arguments is that they do not detect subtle geometric information, so there are situations where they fail but a more sophisticated non-local approach works. Also, local methods cannot be used to prove sharp bounds on the gap. In practice, this meant that the first paper was able to prove the horoconvexity conjecture when the diameter was sufficiently small and the dimension was two. However, the argument did not work for larger domains or higher dimensions, so there was a natural opportunity to write a follow-up.

Finding the two-point argument

While we were writing the first paper, Malik started to apply the conformal approach to non-local calculations, which would be able to prove more subtle log-concavity estimates. And instead of simply generalizing the one-point maximum principle to a two-point maximum principle, he decided to use a different approach based on stochastic calculus and coupled diffusions which was introduced by Fuzhou Gong, Huaiqian Li, and Dejun Luo. Last year, Gunhee Cho, Guofang and Guang Yang generalized this argument to spherical geometry, but there was still a lot of technical work and difficult computations needed to make these computations work under conformal changes.

However, Malik developed estimates that could handle essentially any conformal factor in any dimension, which is what we needed to finish the proof of the horoconvexity theorem. In this case, we wouldn’t be able to prove that the eigenfunction was log-concave, but we could put bounds on its log-convexity. And this was be sufficient to bound the fundamental gap from below, using some previously known results about Neumann eigenvalues in spaces with Bakry-Émery Ricci lower bounds.

There were some technical details about how the log-concavity estimates depend on the domain, and it took us a while to figure out exactly what geometric data we needed. But once we had what we needed, we could use the disk model of hyperbolic geometry (where horoconvex domains are always convex) to compute the bounds induced by this model. Doing so yields vanishingly small quantity, which decays at a rate that is more than doubly-exponential in the diameter. In particular, when the diameter is much larger than the dimension, we have that

\Gamma(\Omega) > \frac{\pi^2(N-1)^2 D^2} {16} \exp\left[ - (N-1) (D^2) \left(1+ 2 \exp(2D) \right)^2 \right]

This is much, much smaller than C_n/D^3, but the important thing is that it is non-zero and only depends on the dimension and the diameter.

A follow-up question

So now the natural question is how fast the gap actually decays. We know that it is somewhere between C_n /D^3 and some absurd doubly-exponential quantity. Our argument holds for a wider class of domains, not simply horoconvex ones, so our bounds are almost certainly not sharp. We have no idea what the sharp rate is, and it would be very interesting to determine what this might be. More precisely, fixing the dimension n , we want to determine (or at least get better bounds) for the quantity \inf_{ \Omega \textrm{ horocvx, diam } \Omega \leq D} (\lambda_2(\Omega) - \lambda_1(\Omega) as a function of D.

Other conformally flat manifolds and sharpened bounds

And apart from hyperbolic space, this approach works for convex domains other conformally flat manifolds, such as \mathbb{S}^1 \times \mathbb{S}^n . Here, we must assume that the domain is convex in terms of the universal affine cover, which is a different notion of convexity compared to convexity with respect to the Riemannian geometry. The bounds in this case are also a bit less explicit since they involve the principle eigenvalue as well as the diameter and the dimension.

We were also able to go back and sharpen many of the bounds that we had found using local methods. In the first paper, there are several theorems with technical hypotheses that end in the conclusion that the fundamental gap is greater than some quantity which is slightly less than \pi^2/D^2 . For the most part, this paper supplants these with another version whose hypotheses are even more technical but result in the gap being slightly less than 3\pi^2/D^2 .

Final thoughts

First off, I should emphasize that the vast majority of this paper (including the hardest parts of the proof) was done by Malik. It was very impressive to see him learn coupled diffusions so quickly and apply them in a non-trivial way. I figured out a few technical issues and did some of the computations in bounding the gap, but I want to make sure that he gets appropriate credit for his work. He easily could have written this as a single author paper and the results would have been very similar.

Overall, it’s nice to have an answer to this question, especially since it builds off our previous work in a natural way. Over the past few years, we’ve developed a productive working group and it’s satisfying to see how our understanding of the gap has developed and matured. We’re still a long ways from proving some of the most difficult questions in the area, but it feels like we are making steady progress. And my hope is that these ideas can be applied to other problems as well, which is something we’ve been thinking about recently.

Image credits

The cover image is Circle Limit III by M. C. Escher, which depicts a tiling of the disk model of hyperbolic geometry. Note that the curves traced out in the figure are geodesics, and not horocycles. The photograph of this work was taken by Peter E and made available under a CC BY-NC-SA 2.0 DEED license.

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