January 27, 2019

On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Posted by Simon Willerton guest post by Heiko Gimperlein and Magnus Goffeng

This fourth and final blog post concludes our discussion of the magnitude function of Euclidean domains. The previous post and this one discuss the following prototypical open problems that connect magnitude to other fields of mathematics:

• Topology: In what sense (i.e. in which topology) is the magnitude function continuous in the Euclidean domain?
• Geometry: Can one “magnitude the shape of a drum”?
• Analysis: Is convexity detected by the poles of the magnitude function?

Below the fold we address the geometry and analysis of magnitude. Topological problems, examples and counterexamples were discussed in our third blog post.

This blog post is a continuation of the third entry in this series of blog posts. You might want to have a look at if first for some further motivation on the questions to come. We continue the numbering of problems from our third blog post.

Some geometric problems

At the heart of the Leinster-Willerton conjecture lies the statement that the magnitude function captures much of the geometry of a metric space. One might ask: how much?

From our geometric formulas for the first three expansion coefficients $c_j(X)$, it turns out that the first two appear in the Weyl law and all three appear in the heat trace expansion. Both describe the asymptotic behavior of eigenvalues of the Laplace operator in $X$ (with Dirichlet boundary conditions). Marc Kac’s famously asked “can you hear the shape of a drum”, or if a Riemannian manifold with boundary is determined up to isometry by the spectrum of its Dirichlet Laplacian. The spectrum of the Dirichlet Laplacian of a drum would be the frequencies heard when playing said drum, thus motivating the formulation.

We may paraphrase Kac’s question for magnitudes: “can you magnitude the shape of a drum”?

Kac’s question was answered negatively for closed manifolds by Milnor, who gave examples of two non-isometric 4-dimensional tori with the same spectrum of the associated Laplacians. For domains in Euclidean space, Gordon, Webb and Wolpert gave an example of two non-isometric polygonal domains in the plane with the same spectrum of the associated Dirichlet Laplacian. We note that the case of domains with smooth boundary is open, in all dimensions. Returning to magnitudes, we formulate our problem.

Problem D. Are there two non-isometric domains in odd-dimensional Euclidean space with the same magnitude function? Are there infinitely many? What about the case for closed manifolds?

A simple example for non-isometric four-point metric spaces with the same magnitude function may be found in Example 2.3.5 of this paper by Leinster.

A second intriguing problem is to compute the geometric meaning of all expansion coefficients $c_j(X)$ in the expansion $\mathcal{M}_X(R)\sim \sum_{j=0}^\infty c_j(X)R^{n-j}$, when $X$ is a smooth domain. The results in our paper give a recipe and some structural results about $c_j(X)$. For $j=0,1,2$ we compute it and recent computations have been done for $j=3$ (the result is announced in our third blog post). At first we wanted to pose the computation of $c_j(X)$ as a problem, but already computing $c_{4}(X)$ by brute force looks like a complicated task that we would not give even to the worst of our enemies. So we refrained. New ideas seem to be needed. One idea from geometric analysis is that complicated invariants sometimes have simple variational formulas. For instance, analytic torsion is a complicated spectral invariant which is hard to compute even in examples, but variations in the input data can be computed by local expressions through the so called anomaly formula due to Brüning-Ma.

Problem E. Let $B$ denote the unit ball and $X$ a domain with smooth boundary. Compute the values of

$c_{j,k}(X):=\frac{\mathrm{d}^k}{\mathrm{d}\epsilon^k}\bigg|_{\epsilon=0} c_j(X+\epsilon B).$

Here $+$ denotes the Minkowski sum.

Compare this to Weyl’s tube formula, which gives a variational formula for the volume in terms of intrinsic volumes. Using that $c_0(X)=\mathrm{vol}(X)$, it follows that $c_{0,k}(X)=V_k(X)$ (the intrinsic volumes) if $X$ is convex. More generaly, for $j=0,1,2$, the invariants $c_{j,k}$ can be computed in terms of mixed volumes.

In the special case of a ball, Problem E relates to an observation by Simon Willerton, Conjecture 17 of his preprint.

Some analytic problems

The magnitude function extends meromorphically to $\mathbb{C}$ for compact metric spaces of a geometric origin, such as compact domains in $\mathbb{R}^n$ ($n$ odd) or closed Riemannian manifolds. The pole structure of the magnitude function for odd-dimensional balls is intriguingly regular, as is seen in the plots of poles for the magnitudes of the $n$-dimensional ball here, for $n=13,17,21$. The picture for the zeroes of the magnitude of these odd-dimensional balls is similar. Computations of Willerton show that the magnitude function of such an $n$-dimensional ball is rational, with at most $\frac{n^2-3n+3}{8}$ poles which are empirically observed to lie in the sector $\{R \in \mathbb{C} : |\mathrm{arg}(R)|\geq \frac{3\pi}{4}\}$. Is there a deeper explanation? What is the role of the poles? For instance, why is $R=-3$ a pole for the magnitude function of the $5$-ball?

Quite little is known in the case of closed Riemannian manifolds. It is currently not even clear if there could be poles on the positive half-axis $R\gt 0$. The example of $S^2$ computed by Willerton (and discussed in our third blog post) shows that $\mathcal{M}_{S^2}(R)=\frac{2R+2}{1-\mathrm{e}^{-\pi R}}$ so the set of poles of magnitude of $S^2$ is $2i\mathbb{Z}$ (all simple poles).

Problem F. Is there a closed manifold with finitely many poles of magnitude?

More is known in the case of compact domains (with smooth boundary) in $\mathbb{R}^n$, $n$ odd. The poles are situated outside the sector where $|\mathrm{arg}(R)|\leq \pi/(n+1)$, and for any $C\gt 0$ there are finitely many poles in the sector $\mathrm{Re}(R)\gt C |\mathrm{Im}(R)|$. In scattering theory, convex domains have scattering poles that are contained even further into the left half-plane (even as far as to the left of some curve $|\mathrm{Im}(R)|^3=|\mathrm{Re}(R)|+C$ in the half-plane $\mathrm{Re}(R)\lt 0$ by results in the PhD thesis of Long Jin). This suggests that the convex situation is better behaved.

Problem G. What can be said about the number and location of poles of the magnitude function for a convex domain? Is the number of poles finite?

An interesting answer to this problem could be an example of a convex domain with infinitely many poles of magnitude.

Let us end this blog post by mentioning a formula that holds for spheres and flat two-dimensional torii. We have no reason to believe that it holds nor fails in general, but it is simply too beautiful to not mention. If $X$ is a sphere in its round metric or a flat two-dimensional torus, it holds that

$\chi(X)=\mathrm{Res}_{R=0} \mathcal{M}_X(R).$

For spheres, this formula follows from Willerton’s computations for the magnitude function of spheres and for torii it follows from a short computation. In geometric analysis, residues of zeta functions (and eta functions) play an important role. They can for instance be used to equate global invariants (like a Fredholm index) to a local invariant, and it would be most delightful if the above equation was not just a funny coincidence. Seeing that the asymptotics of the magnitude might not capture the Euler characteristic (as seen in dimension $3$ from the computation of $c_3$ mentioned in our third blog post), it would be interesting if the Euler characteristics could be found in the pole structure of the magnitude function instead.

Posted at January 27, 2019 11:38 AM UTC

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Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Problem D: If one allows disconnected domains, then one cannot magnitude the shape of a one-dimensional drum. Suppose that $X \subseteq \mathbb{R}$ is obtained by taking a closed interval of length $L$, and removing (finitely or countably many) disjoint open subintervals of lengths $\ell_1, \ell_2, \ldots$. Then Corollary 4.3 of my survey paper with Tom shows that $\mathcal{M}_X (R) = 1 + \frac{1}{2}\left(L - \sum_i \ell_i\right) R + \sum_{i} \tanh \left(\frac{\ell_i}{2} R\right).$ In particular, this is independent of where the removed subintervals are, and if infinitely many are removed then we get infinitely many distinct domains with the same magnitude functions.

This of course leaves wide open the problem for higher dimensional connected or convex domains.

Posted by: Mark Meckes on January 27, 2019 6:58 PM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Actually we get infinitely many distinct domains even if only one subinterval is removed, from different locations. (Before I was only thinking about permuting them.)

Posted by: Mark Meckes on January 27, 2019 10:02 PM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Thanks for this comment Mark! Your example is nice since it shows that already in one dimension, there is enough elbow room to get the same magnitude function when allowing disconnected domains. The point seems to be that when moving these intervals around, the Dirichlet-Neumann operator for the exterior problem moves along with it (since the exterior problem studied by you and Barcelo-Carbery only takes place on these open intervals). The problem in higher dimensions would analogously be to deform higher-dimensional domains while still keeping control of the Dirichlet-Neumann operator along the deformation. That seems harder.

Posted by: Magnus Goffeng on January 28, 2019 1:45 PM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

So what about $X = B(0,1) \setminus B(x, \varepsilon)$, where $B(x,r)$ denotes the ball of radius $r$ about $x$ and $\| x \| < 1-\varepsilon$?

Posted by: Mark Meckes on January 28, 2019 6:58 PM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

This example will not provide a counterexample as the computation from our third post shows. For instance, if the magnitude function at two values $\epsilon$ and $\epsilon'$ coincide then $\begin{cases} 2\epsilon^2+6\epsilon=2\epsilon'^2+6\epsilon',\\ 2\epsilon^2+2=2\epsilon'^2+2,\\ 4\epsilon=4\epsilon' \end{cases}.$ And these are only the coefficients for the asymptotic piece. So in this case, if we have two identical magnitude asymptotics for $\epsilon$ and $\epsilon'$, then $\epsilon=\epsilon'$.

In dimension $3$, the deformation we look for must preserve volume, boundary area, average mean curvature and Willmore energy (which is $c_3(X)-\chi(X)$ modulo normalizing constants).

After altering your suggestion slighly, and look at $B(0,r_1)\setminus \overline{B(0,r_2)}$, it is possible to obtain two non-isometric domains with the same magnitude asymptotics. The equations for equal magnitude asymptotics look like $\begin{cases} (r_1-r_2)^3=(r_1'-r_2')^3,\\ r_1^2+r_2^2=r_1'^2+r_2'^2,\\ r_1-r_2=r_1'-r_2'\end{cases}.$ The constant term is scale invariant and is constant in the family. The first equation follows from the third. So this equation is equivalent to $\begin{cases} 2r_1r_2=r_2'(r_2'-2r_2+2r_1),\\ r_1-r_2=r_1'-r_2'\end{cases}.$ We get a whole family of examples satisfying this by setting $r_1=\frac{12}{5}r_2, \; r_2'=\frac{1}{2}r_1,\; r_1'=r_1-r_2+r_2'=\frac{19}{10}r_2.$ By similar computations as in the third post, the remainders are $O(R^{-\infty})$. Just looking at them briefly indicates that it will be harder to equate them for two different values of the parameters.

Posted by: Magnus Goffeng on January 29, 2019 7:56 AM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Crap. I computed this 4 minutes before my teaching started. The correct equation is $\begin{cases} r_1r_2=r_2'(r_2'-r_2+r_1)\\ r_1-r_2=r_1'-r_2'\end{cases}.$ The only solution to this with $r_1 \gt r_2 \gt 0$ and $r_1' \gt r_2' \gt 0$ is $r_1=r_1'$ and $r_2=r_2'$. Sorry for this numerical error. More is needed even for the asymptotics to coincide.

Posted by: Magnus Goffeng on January 29, 2019 11:39 AM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Sorry, I should have been clearer. I meant to consider a single value of $\varepsilon$, but different locations $x$ of the center of the removed inner ball.

Posted by: Mark Meckes on January 29, 2019 1:18 PM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

This is a brilliant suggestion! It’s so remarkably simple that I feel a bit silly. Here I was sitting with some dreadful integrals over flat torii thinking that it would be easier than looking for non-isometric domains. Thanks!

By the formula in Proposition 8 of our paper , translation invariance implies that the magnitude function of $\overline{B(0,1)}\setminus B(x,\epsilon)$ is independent of $x$ with $|x| \lt 1-\epsilon$ if the function $h_{R,x}$ solving the exterior problem for $\overline{B(0,1)}\setminus B(x,\epsilon)$ is given by $h_{R,x}(y)= \begin{cases} h_{R,0}(y), \; &|y| \gt 1,\\ h_{R,0}(y-x), \; &|y-x| \lt \epsilon.\end{cases}$ And it is again clear from translation invariance that the right hand side of this equation solves the exterior problem. Uniqueness of solutions imply the equality. If we vary $x$, we obtain non-isometric domains but the magnitude function remains the same. This solves Problem D.

Posted by: Magnus Goffeng on January 29, 2019 2:01 PM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Great! So next we can pose Problem D’: Are there two non-isometric convex domains in odd-dimensional Euclidean space with the same magnitude function? What about star-shaped domains? What about closed manifolds?

(Also, a tip: to use > and < here, you need to use HTML codes for those characters: &gt; and &lt;. It took me a long time to figure that one out!)

Posted by: Mark Meckes on January 29, 2019 4:08 PM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Re the $\lt$ and $\gt$ signs: thanks, Mark, for mentioning that. I’ve fixed Magnus’s comments.

You can also use \lt and \gt for less than and greater than.

Posted by: Tom Leinster on January 29, 2019 11:27 PM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Let me see if I understand the outcome of this discussion correctly.

As I understand it, you’ve shown:

Theorem   Let $n \geq 1$ be an odd integer and let $0 \lt \varepsilon \lt 1$. Then for $x \in B(0, 1 - \varepsilon)$, the magnitude function of the space $\overline{B}(0, 1) \setminus B(x, \varepsilon)$ is independent of $x$, where $B$ means ball in $\mathbb{R}^n$.

Is that correct?

If so, it’s an interesting class of examples for a bunch of reasons, one of which is that it would be true if magnitude satisfied the inclusion-exclusion principle. Of course, the inclusion-exclusion principle isn’t satisfied — but if it was, the magnitude of this complement would be $|\overline{B}(0, 1)| - |\overline{B}(0, \varepsilon)| + |S(0, \varepsilon)|$ (where $S$ means sphere), which is evidently independent of $x$.

Posted by: Tom Leinster on January 30, 2019 12:03 AM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Thanks for taking care of the inequalities Tom, and thanks Mark for learning me how to do it properly.

Your theorem Tom is a correct interpretation of the statement I made above. What clinches the deal seem to be that the exterior problem can be solved independently on the components of the complement. So more generally, whenever you have a domain, the magnitude function is unaltered by deformations obtained from translating components of the complement (as long as no boundaries cross/touch each other).

Posted by: Magnus Goffeng on January 30, 2019 8:23 AM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Let me add a trivial observation to Tom’s comment about the inclusion-exclusion principle: If $X, Y \subset \mathbb{R}^n$ are compact domains and $Y \subset \mathrm{int} X$, then the inclusion-exclusion principle holds for the magnitude function by Mark’s observation.

Posted by: Heiko Gimperlein on January 30, 2019 11:24 AM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

After talking with a couple convex geometer colleagues, I have the following candidates for Problem D’: consider a ball with two disjoint caps removed. Concretely, say $X_{y,z, \varepsilon} = \{ x \in \mathbb{R}^n : \|x \| \le 1, \langle x, y \rangle \le 1 - \varepsilon, \langle x, z \rangle \le 1 - \varepsilon \},$ where $0 < \varepsilon < 1$ and $y$ and $z$ are two unit vectors, sufficiently far apart. If we vary the distance between $y$ and $z$, then we get many non-isometric convex bodies which have all the same intrinsic volumes. We could also smooth these $X_{y,z,\varepsilon}$ so that all the integrals of curvatures on the boundaries are equal. Are their magnitude functions also equal?

Posted by: Mark Meckes on January 29, 2019 9:15 PM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

Another remark about this candidate: the two cases ($y = e_1 = -z$) and ($y=e_1$, $z = e_2$) yield the same magnitude functions with respect to the $\ell_1$ metric (as follows from Tom’s formula for the $\ell_1$-magnitude of a convex body in our survey paper).

Posted by: Mark Meckes on January 30, 2019 4:00 PM | Permalink | Reply to this

Re: On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b)

The paper on which these blog posts were based is now accepted in American Journal of Math. The final version can be found on arXiv: https://arxiv.org/abs/1706.06839

Posted by: Magnus Goffeng on March 18, 2020 9:13 AM | Permalink | Reply to this

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