Topological aspects of $\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}$ eigenfunctions for the Laplacian on $S^{2}$

To set the stage for what is to come fix an even, positive integer to be denoted by 2n and let $\operatorname{\mathfrak{p}}$ denote a chosen collection of $2n$ distinct points in the round 2-sphere. With $\operatorname{\mathfrak{p}}$ in hand, let $\operatorname{\mathcal{I}_{\mathfrak{p}}}$ denote the real line bundle over $S^{2}-\operatorname{\mathfrak{p}}$ with monodromy $-1$ on any embedded circle in $S^{2}-\operatorname{\mathfrak{p}}$ linking any given point from $\operatorname{\mathfrak{p}}$ and no others. This is viewed as an associated vector bundle to the principle $\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}$ bundle that is defined by the 2-1 branched cover of $S^{2}$ with $\operatorname{\mathfrak{p}}$ being the branch locus. (All real line bundles in this paper have implicit fiber metrics that associate them to principle $\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}$ bundles). Viewed in this way, the bundle $\operatorname{\mathcal{I}_{\mathfrak{p}}}$ has a canonical fiber metric which can and will be used to define the norm of sections and, with the round metric’s inner product, the norm of derivatives of sections. With regards to derivatives: There is a canonical, metric compatible ‘covariant derivative’ for sections of $\operatorname{\mathcal{I}_{\mathfrak{p}}}$ and $\operatorname{\mathcal{I}_{\mathfrak{p}}}$ valued tensors on $S^{2}$. This is because these are defined locally up to multiplication by the constant $-1$. If $f$ denotes a section, then $df$ is used to denote the corresponding covariant derivative which is a section of $\operatorname{\mathcal{I}_{\mathfrak{p}}}\otimes T^{*}S^{2}$ over $S^{2}-\operatorname{\mathfrak{p}}$. (The covariant derivative on $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-valued 1-forms and tensors is defined with the help of the round metric’s Levi-Civita connection.)

Let $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ denote the Hilbert space completion of the space of smooth, compactly supported sections of $\operatorname{\mathcal{I}_{\mathfrak{p}}}$ over $S^{2}-\operatorname{\mathfrak{p}}$ using the norm whose square is defined by the rule

$$f\longrightarrow\|f\|_{\mathbb{H}}^{2}=\int_{S^{2}-\operatorname{\mathfrak{p}}}|df|^{2}+\int_{S^{2}}|f|^{2}$$

(1.1)

As explained in [11], there is a largest positive number to be denoted by $\operatorname{\mathrm{E}_{\mathfrak{p}}}$ such that the inequality

$$\int_{S^{2}-\operatorname{\mathfrak{p}}}|df|^{2}\geq E_{\operatorname{\mathfrak{p}}}\int_{S^{2}}|f|^{2}$$

(1.2)

holds for every $f\in\operatorname{\mathbb{H}_{\mathfrak{p}}}$. (As a consequence, the right most integral on the right hand side of (1.1) is redundant except in the well understood case when $n=0$.)

Of interest in this article are those elements in $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ that are eigensections of the Laplacian: A section $f$ is an eigensection if there is a number $\lambda\in[\operatorname{\mathrm{E}_{\mathfrak{p}}},\infty)$ such that

$$\int_{S^{2}-\operatorname{\mathfrak{p}}}\langle df,dh\rangle=\lambda\int_{S^{2}-\operatorname{\mathfrak{p}}}\langle f,h\rangle$$

(1.3)

for all $h\in\operatorname{\mathbb{H}_{\mathfrak{p}}}$. The number $\lambda$ is called the eigenvalue. (The notation here and subsequently uses $\langle,\rangle$ to denote the round metric’s inner product on both $T^{*}S^{2}$ and on $T^{*}S^{2}\otimes\operatorname{\mathcal{I}_{\mathfrak{p}}}$.) Any given eigensection for the Laplacian is a smooth section of $\operatorname{\mathcal{I}_{\mathfrak{p}}}$. This is because the eigensection (denoted by $f$) can be canonically viewed (up to multiplication by $-1$) as an ordinary function on any disk that is disjoint from $\operatorname{\mathfrak{p}}$ where it obeys the spherical metric’s Laplace equation

$$d^{\dagger}df=\lambda f$$

(1.4)

with $\lambda$ denoting the corresponding eigenvalue. (What is denoted by $d^{\dagger}$ signifies the formal $L^{2}$ adjoint of $d$.)

An eigensection from $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ for the Laplacian is said in what follows to be an $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigensection and its corresponding eigenvalue is said to be an $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigenvalue.

Proposition 2.1 in [11] asserts that $\operatorname{\mathrm{E}_{\mathfrak{p}}}$ is an $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigenvalue. The technology used to prove that proposition can be employed in a straightforward way to prove the proposition that follows:

Let $\operatorname{\mathcal{C}_{2n}}$ denote the space of unordered $2n$-tuples of points in $S^{2}$. Each such $2n$-tuple of points has a corresponding set of $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigensections and the associated set of $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigenvalues: Our goal in this paper is explore how these $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigensections and their $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigenvalues change as $\operatorname{\mathfrak{p}}$ moves in $\operatorname{\mathcal{C}_{2n}}$.

We start by investigating the behavior of the eigenvalues as functions of the configurations. Our principle example is the lowest eigenvalue function, the function $\operatorname{\mathfrak{p}}\longrightarrow\operatorname{\mathrm{E}_{\mathfrak{p}}}$. This function on $\operatorname{\mathcal{C}_{2n}}$ has the following properties:

• •

  The function $\mathrm{E}_{(\cdot)}$ is continuous; it is differentiable only where the corresponding eigenspace is 1-dimensional.

• •

  The infimum of $\mathrm{E}_{(\cdot)}$ on $\operatorname{\mathcal{C}_{2n}}$ is zero, it is only achieved on $\operatorname{\mathcal{C}_{2n}}$ if $n=0$.

• •

  The supremum of $E_{\operatorname{\mathfrak{p}}}$ on $\operatorname{\mathcal{C}_{2n}}$ is greater than an $n$-independent multiple of $n$ and it is always achieved. But the function $\mathrm{E}_{(\cdot)}$ is not differentiable there (nor is it at any other putative critical point).

(1.5)

By way of an example for the first bullet: The eigenspace for the minimal eigenvalue $\operatorname{\mathrm{E}_{\mathfrak{p}}}$ on $\operatorname{\mathcal{C}_{2}}$ has dimension 2 when the two points in the configuration are antipodal (see Section 7). And, as we explain later, there are configurations in any $n>1$ version of $\operatorname{\mathcal{C}_{2n}}$ where the minimal eigenvalue eigenspace has multiplicity at least four. (This multiplicity business is surprising because mathematical physics/differential equation lore says that the lowest eigenvalue of a Laplacian must have multiplicity one.)

To elaborate on the second bullet: The function $\mathrm{E}_{(\cdot)}$ limits to zero along paths in $\operatorname{\mathcal{C}_{2n}}$ that bring all of the constituent points together. We study the behavior of the lowest eigenvalue function $\mathrm{E}_{(\cdot)}$ (and also higher $\mathcal{I}_{(\cdot)}$-eigenvalues) along sequences where subsets of configuration points coalesce by introducing a natural compactification of $\operatorname{\mathcal{C}_{2n}}$ as a stratified space with strata $\{\operatorname{\mathcal{C}_{2n}},\mathcal{C}_{2n-2},\mathcal{C}_{2n-4},\ldots,\mathcal{C}_{0}\}$. We then show that $\mathrm{E}_{(\cdot)}$ (in fact, all $\mathcal{I}_{(\cdot)}$-eigenvalues) extends continuously to this compactification. The compactification is denoted by $\operatorname{\overline{\mathcal{C}}_{2n}}$.

To elaborate on the third bullet: Although $\mathrm{E}_{(\cdot)}$ is differentiable where the corresponding eigenspace has multiplicity 1, it is not necessarily differentiable where the multiplicity is greater than 1. In this regard, we have an expression for the directional derivative of an eigenvalue function $\operatorname{\mathfrak{p}}\longrightarrow\lambda_{\operatorname{\mathfrak{p}}}$ where the eigenspace has multiplicity 1 which implies this:

The directional derivatives of $\lambda_{(\cdot)}$ at a given configuration p, are all zero if and only if the corresponding $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigensection is $\mathcal{O}\left(\mathrm{dist}(\operatorname{\mathfrak{p}},\cdot)^{3\over 2}\right)$ near $\operatorname{\mathfrak{p}}$.

(1.6)

By way of comparison, the arguments in Part 3 of Section 4 in [11] prove that the most that can be said about an $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigensection with no other a priori information is that its norm is $\mathcal{O}\left(\mathrm{dist}(p,\cdot)^{1\over 2}\right)$ near any given point $p\in\operatorname{\mathfrak{p}}$. As we explain below, the behavior in (1.6) implies that the eigenspace multiplicity of $\mathrm{E}_{(\cdot)}$ is greater than 1 at any configuration where all of the directional derivatives are zero. That implies, in turn, that $\mathrm{E}_{(\cdot)}$ does not have continuous derivatives where it is maximal.

What is said in (1.5) for the minimal eigenvalue function $\mathrm{E}_{(\cdot)}$ has an analog for the function that assigns to a configuration its $k$’th lowest eigenvalue (for any $k\geq 1$); see Section 3.4 for a precise definition of this function. The basic conclusion is that these eigenvalue functions are differentiable where they have multiplicity 1, and need only be continuous where their multiplicity is greater than 1. Moreover, any putative critical point must occur where the multiplicity is in fact greater than 1 and thus where the function might not be differentiable. (Ljusternick-Schnirrelman constructions – which is to say min-max – can be used to define the notion of a critical value and critical point for a function that is only continuous.) The preceding remarks about the lack of differentiability of the k’th eigenvalue functions are proved in Section 3.5.

Looking ahead: Sections 2 and 3 of this paper investigate the properties of these eigenvalue functions on $\operatorname{\mathcal{C}_{2n}}$ and $\operatorname{\overline{\mathcal{C}}_{2n}}$.

To understand the phenomena depicted in (1.5), we replace the eigenvalue functions on $\operatorname{\mathcal{C}_{2n}}$ with a ‘universal’ eigenvalue function on the space of pairs of the form $(\operatorname{\mathfrak{p}},f)$ where $\operatorname{\mathfrak{p}}$ is from $\operatorname{\mathcal{C}_{2n}}$ and $f$ is a section of $\operatorname{\mathcal{I}_{\mathfrak{p}}}$ with the $S^{2}$-integral of $f^{2}$ equal to 1. This function is denoted by $\mathcal{E}$ and it is defined by the rule

$$(\operatorname{\mathfrak{p}},f)\longrightarrow\int_{S^{2}-\operatorname{\mathfrak{p}}}|df|^{2}.$$

(1.7)

This function $\mathcal{E}$ is (formally) a smooth function of these sorts of pairs. Moreover, a perturbation theoretic argument says this: If $(\operatorname{\mathfrak{p}},f)$ is a critical point of $\mathcal{E}$, then $f$ is an $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigensection (this is Raleigh-Ritz) and $f$ as it nears any point in $\operatorname{\mathfrak{p}}$ vanishes as $\mathcal{O}\left(\mathrm{dist}(\operatorname{\mathfrak{p}},\cdot)^{3\over 2}\right)$ which is precisely the condition in (1.6).

Given $\operatorname{\mathfrak{p}}\in\operatorname{\mathcal{C}_{2n}}$, let $\operatorname{\mathbb{S}_{\mathfrak{p}}}$ denote the subset in the Hilbert space $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ (the norm is depicted in (1.1)) that consists of the elements whose square has $S^{2}$-integral equal to 1. The function $\mathcal{E}$ is formally a function on a ‘fiber bundle’ over $\operatorname{\mathcal{C}_{2n}}$ whose fiber of $\operatorname{\mathfrak{p}}$ is the subspace $\operatorname{\mathbb{S}_{\mathfrak{p}}}$. But now something interesting appears: There is no such fiber bundle. This is a very pretty example of what the physicists would call an anomaly, this one being a $\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}$ version of a integer characteristic class anomaly that was discussed at some length in an unpublished paper by G. Segal [9] in the context of certain observations by L. Fadeev [1]. (There are many papers on anomalies in physics; see the recent paper of L. Müller for other versions of anomalies and references [5].) In the context at hand, there exists an obstruction class in $H^{2}(\operatorname{\mathcal{C}_{2n}};\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}$) that is zero if and only if the assignments $\{\operatorname{\mathfrak{p}}\longrightarrow\operatorname{\mathbb{S}_{\mathfrak{p}}}\}_{\operatorname{\mathfrak{p}}\in\operatorname{\mathcal{C}_{2n}}}$ define a fiber bundle. This obstruction class is non-zero; it is denoted by $\omega$.

We can none-the-less define an $\operatorname{\mathbb{RP}}^{\infty}$ bundle over $\operatorname{\mathcal{C}_{2n}}$ whose fiber over any given configuration $\operatorname{\mathfrak{p}}$ is $\operatorname{\mathbb{S}_{\mathfrak{p}}}/\{\pm 1\}$ which is sufficient for the purposes at hand because $\mathcal{E}$ has the same value on $(\operatorname{\mathfrak{p}},f)$ as it has on $(\operatorname{\mathfrak{p}},-f)$. This bundle is denoted by $\operatorname{\mathbb{RP}}$ and $\mathcal{E}$ descends there as a smooth function. In particular, one can look for critical points of $\mathcal{E}$ on $\operatorname{\mathbb{RP}}$. This leads us to study the (co)homology of $\operatorname{\mathbb{RP}}$ because that (in principle) dictates critical points of $\mathcal{E}$. (This $\operatorname{\mathbb{RP}}$ bundle and its function $\mathcal{E}$ are described in Section 4 of this paper.)

By way of a parenthetical remark: In an alternate universe where the obstruction class is zero, one should still consider $\mathcal{E}$ on the quotient $\operatorname{\mathbb{RP}}^{\infty}$ to study its critical points.

The observation in the third bullet of (1.5) that the minimal eigenvalue function $\mathrm{E}_{(\cdot)}$ is not everywhere differentiable is an automatic consequence of the following observation about the function $\mathcal{E}$ on $\operatorname{\mathbb{RP}}$:

Suppose that $n$ ¿ 0 and that $(\operatorname{\mathfrak{p}},[f])$ is a critical point of the $\operatorname{\mathcal{C}_{2n}}$ version of $\mathcal{E}$. Then the corresponding critical value is not the lowest $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigenvalue (which is $\operatorname{\mathrm{E}_{\mathfrak{p}}}$); in fact, it is greater than the n’th $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigenvalue.

(1.8)

As noted in the first bullet of (1.5), the lack of differentiability of the minimal eigenvalue function $\mathrm{E}_{(\cdot)}$ is due to the existence of configurations where the minimal eigenvalue has an eigenspace with dimension greater than 1. The condition in (1.8) tells us slightly more; it tells us that there can’t be a differentiable section of $\operatorname{\mathbb{RP}}$ (as a bundle over $\operatorname{\mathcal{C}_{2n}}$) that assigns any given configuration to some $\mathcal{I}_{(\cdot)}$-eigensection (modulo $\pm 1$) with minimal eigenvalue. As explained in the next paragraph, the lack of a section of $\operatorname{\mathbb{RP}}$ is dictated by the existence of a non-zero anomaly class.

With regards to the anomaly and the existence of sections: The key fact about the anomaly class is that it’s pull-back to $\operatorname{\mathbb{RP}}$ via the projection map is zero; and this can’t be the case if there is a section because the composition of first the section and then the projection is the identity map on $\operatorname{\mathcal{C}_{2n}}$. (If the anomaly were zero, then there would be a section of the corresponding infinite dimensional sphere bundle; and thus a section of the $\operatorname{\mathbb{RP}}^{\infty}$ bundle. This is because a vector bundle over a finite dimensional manifold whose fiber dimension is greater than the base manifold dimension has a non-zero section.)

Morse theoretic arguments (min-max, for example) using the (co)homology of $\operatorname{\mathbb{RP}}$ would yield critical points of $\mathcal{E}$ were $\mathcal{E}$ a proper function on $\operatorname{\mathbb{RP}}$. Although $\mathcal{E}$ is proper along the fibers of the projection to $\operatorname{\mathcal{C}_{2n}}$, it is not globally proper because $\operatorname{\mathcal{C}_{2n}}$ isn’t compact.

As noted previously, the space $\operatorname{\mathcal{C}_{2n}}$ has a compactification as a stratified space (denoted by $\operatorname{\overline{\mathcal{C}}_{2n}}$) that accounts for the coalescence of subsets of configuration points. As it turns out, there is an extension of $\operatorname{\mathbb{RP}}$ over this compactification (denoted by $\operatorname{\overline{\mathbb{RP}}}$) that maps to the compactification $\operatorname{\overline{\mathcal{C}}_{2n}}$ and a corresponding, continuous extension of the function $\mathcal{E}$ as a function over $\operatorname{\overline{\mathbb{RP}}}$. This $\operatorname{\overline{\mathbb{RP}}}$ space has non-zero $\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}$ homology classes in degrees $4n+2m+1$ with $m$ any non-negative integer. It is possible that these classes yield critical points of $\mathcal{E}$ on large $n$ versions of $\operatorname{\overline{\mathcal{C}}_{2n}}$ because each such $(n,m)$ class is the image via a fiber preserving map to $\operatorname{\overline{\mathbb{RP}}}$ of the $\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}$ fundamental class of an $\operatorname{\mathbb{RP}}^{2m+1}$ bundle over $\operatorname{\overline{\mathcal{C}}_{2n}}$. (The class sees the whole of $\operatorname{\overline{\mathcal{C}}_{2n}}$ because of this; in particular, the class is not carried by any single strata in $\operatorname{\overline{\mathcal{C}}_{2n}}$.) We show that these classes do give critical points of $\mathcal{E}$ on $\operatorname{\overline{\mathbb{RP}}}$ and we show in Section 6 of this paper that the corresponding critical values (as functions of either $n$ or $m$) are unbounded. Even so, it remains to be seen whether these critical points all lie in some fixed $\mathcal{C}_{2j}$ strata with $j$ having an $n$ and $m$ independent upper bound. This is work for the future.

By way of a remark: The classes used here can be viewed as a homological work-around for the non-differentiability of the $k$’th eigenvalue function on $\operatorname{\mathcal{C}_{2n}}$ in the case when $k$ is odd. (There are also classes in the $\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}$ homology of $\operatorname{\overline{\mathcal{C}}_{2n}}$ with even degree – any even number – but their corresponding critical points are almost surely in the $\mathcal{C}_{0}$ or $\operatorname{\mathcal{C}_{2}}$ strata. These strata and the corresponding critical points are described in Section 7.)

Critical points of $\mathcal{E}$ on $\operatorname{\mathcal{C}_{2n}}$ (should they exist) can be used to construct homogeneous singularity models for $\operatorname{\mathbb{Z}}/2$ harmonic 1-forms and spinors on $\mathbb{R}^{3}$. To explain (see [11] and/or [10] for more), a homogeneous $\operatorname{\mathbb{Z}}/2$ harmonic 1-form on $\mathbb{R}^{3}$ consists of a data set $(Z,\mathcal{I},\nu)$ with $Z$ being a finite union of rays from the origin, with $\mathcal{I}$ being real line bundle over the complement of $Z$ and with $\nu$ being an $\mathcal{I}$-valued 1-form defined on $\mathbb{R}^{3}-Z$ that has the following properties:

• •

  $\nu$ is both closed and coclosed.

• •

  $|\nu|$ extends over $Z$ as a Hölder continuous function on $\mathbb{R}^{3}$.

• •

  $\nu$ is homogeneous with respect to pull-back by the rescaling diffeomorphism that multiplies the Euclidean coordinates by a non-zero real number.

(1.9)

As explained in [11], the data sets of this sort with $Z$ being the union of $2n$ rays are in 1-1 correspondence with pairs $(\operatorname{\mathfrak{p}},f)$ with $\operatorname{\mathfrak{p}}\in\operatorname{\mathcal{C}_{2n}}$ and with $f\in\operatorname{\mathbb{H}_{\mathfrak{p}}}$ being a non-zero eigensection of the Laplacian with $|f|\leq\mathcal{O}\left(\mathrm{dist}(\operatorname{\mathfrak{p}},\cdot)^{3\over 2}\right)$ near $\operatorname{\mathfrak{p}}$ (which is exactly the critical point criteria for the function $\mathcal{E}$). This same [11] explains how such a pair can also be used to define a homogeneous, $\operatorname{\mathbb{Z}}/2$ harmonic spinor on $\mathbb{R}^{3}$. To summarize the correspondence: The set $Z$ is the union of the rays from the origin through the points in the configuration $\operatorname{\mathfrak{p}}$, the line bundle $\mathcal{I}$ is the pull-back of $\operatorname{\mathcal{I}_{\mathfrak{p}}}$ via the map $\pi:x\longrightarrow{x\over|x|}$ from $\mathbb{R}^{3}\setminus\{0\}$ to $S^{2}$, and $\nu$ is given by the rule

$$\nu=d\left(|x|^{\mu}\pi^{*}f\right),$$

(1.10)

where $\mu$ is ${1\over 2}\left(1+(1+4\lambda)^{1\over 2}\right)$ with $\lambda$ denoting the corresponding $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigenvalue.

Before [11], the only example of a $\operatorname{\mathbb{Z}}/2$ harmonic 1-form on $\mathbb{R}^{3}$ had $Z$ being a straight line through the origin (which corresponds to a critical point of $\mathcal{E}$ on $\mathcal{C}_{2}$). The paper [11] used the symmetry groups of the Platonic solids to construct examples where $Z$ is the union of (respectively) four rays from the origin (pointing to the vertices of a regular tetrahedron), eight rays from the origin (pointing to the vertices of a cube) and 12 rays from the origin (pointing to the vertices of an iscosahedron.) Going in the reverse direction, the constructions in [11] give critical points of $\mathcal{E}$ on $\mathcal{C}_{4}$ and $\mathcal{C}_{8}$ and $\mathcal{C}_{12}$.

The first remark concerns the case of $n=1$: This case is illuminating because much can be seen explicitly in this case. The $n=1$ case is discussed in some detail in Section 7 and a reader is advised to refer to that section when new notions are introduced.

The second remark concerns two notational conventions that are used throughout this paper: The first convention has $c_{0}$ denoting a number greater than 1 whose value is independent of relevant background such as a configuration in $\operatorname{\mathcal{C}_{2n}}$ or the integer $n$. What precisely it is independent of will be clear from the context. The value of $c_{0}$ increases between consecutive incarnations. The second convention has $\chi$ denoting a chosen, smooth and non-increasing function on $\mathbb{R}$ that is equal to 1 on $\left(-\infty,{1\over 4}\right]$ and equal to 0 on $\left[{3\over 4},\infty\right)$. This fixed version of $\chi$ is implicitly used to construct ‘bump functions’ and ‘cut-off functions’.

What follows directly is the table of contents for this paper.

1. 1.

   Introduction

2. 2.

   Basic technology

3. 3.

   The lowest eigenvalue as a function on $\operatorname{\mathcal{C}_{2n}}$

4. 4.

   The $\operatorname{\mathbb{RP}}^{\infty}$ bundle

5. 5.

   The extension of $\operatorname{\mathbb{RP}}$ and $\mathcal{E}$ to $\operatorname{\overline{\mathcal{C}}_{2n}}$

6. 6.

   Min-max for $\mathcal{E}$ on $\operatorname{\overline{\mathbb{RP}}}$

7. 7.

   The case of $\mathcal{C}_{2}$

8. a.

   Appendix on the cohomology of a weak $\operatorname{\mathbb{RP}}^{\infty}$ bundle

Before starting, Y-Y. Wu wishes to acknowledge and thank the generous support of the Center of Mathematical Sciences and Applications (CMSA) at Harvard. Meanwhile, C.  H. Taubes wishes to acknowledge and thank the generous support of the National Science Foundation (DMS grant number 2002771).

The lemma that follows summarizes some basic a priori bounds on eigensections of the Laplacian for a given $\operatorname{\mathfrak{p}}\in\operatorname{\mathcal{C}_{2n}}$. These bounds are independent of $\operatorname{\mathfrak{p}}$.

This function is equal to 1 where the distance to $\operatorname{\mathfrak{p}}$ is greater than $(100\epsilon)^{1\over 2}$ and equal to zero where the distance to $\operatorname{\mathfrak{p}}$ is less than $100\epsilon$. If $\epsilon$ is much less than the distance from $q$ to $\operatorname{\mathfrak{p}}$, then $q$ will sit where $\chi_{\epsilon}$ is equal to 1. Note also that the derivative of $\chi_{\epsilon}$ has support only in the region where the distance to $\operatorname{\mathfrak{p}}$ is between $100\epsilon$ and $(100\epsilon)^{1\over 2}$ where it is bounded by $c_{0}{1\over|\ln\epsilon|}{1\over{\operatorname{dist}(\cdot,\operatorname{\mathfrak{p}})}}$ with $c_{0}$ being independent of $\operatorname{\mathfrak{p}}$. This implies in particular that the $S^{2}$ integral of $|d\chi_{\epsilon}|^{2}$ is bounded by $c_{0}{1\over|\ln\epsilon|}$ with $c_{0}$ again independent of $\operatorname{\mathfrak{p}}$. With these last points understood, multiply both sides of (2.1) by $\chi_{\epsilon}G_{q}$ with $\epsilon$ very small and then integrate by parts. The result of doing that leads to the inequality

$$|f|(q)\leq c_{0}\lambda\left(\int_{S^{2}}|f|^{2}\right)^{1\over 2}+c_{\operatorname{\mathfrak{p}},q}{1\over\sqrt{|\ln\epsilon|}}\left(\int_{S^{2}}|df|^{2}+\int_{S^{2}}|f|^{2}\right)^{1\over 2}$$

(2.3)

where $c_{0}$ is independent of $\operatorname{\mathfrak{p}}$ and $q$ and $f$ and $\epsilon$ whereas $c_{\operatorname{\mathfrak{p}},q}$ is only independent of $f$ and $\epsilon$ (it does depend on the distance from $q$ to $\operatorname{\mathfrak{p}}$). Take $\epsilon\to 0$ in (2.3) to obtain the second bullet’s bound for $|f|$.

As noted in the proof of Lemma 2.1, if $f$ is from $\operatorname{\mathbb{H}_{\mathfrak{p}}}$, then its norm is in the standard $L^{2}_{1}$ Sobolev space for the whole of $S^{2}$; and this implies via Sobolev inequalities that $|f|^{q}$ is integrable on $S^{2}$ for any finite, non-negative value of $q$. The following lemma concerns specifically $f$’s behavior near the points in $\operatorname{\mathfrak{p}}$; it says in effect that a function in $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ must vanish in a weak sense as a point in $\operatorname{\mathfrak{p}}$ is approached.

More can be said about behavior near the points in $\operatorname{\mathfrak{p}}$ when $f$ is an $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigensection. In this case, the necessary observations about the behavior of an $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigensection near a point $p\in\operatorname{\mathfrak{p}}$ invoke observations from Parts 2 and 3 of Section 4 in [11], specifically Equations (4.2)-(4.5) in [11] which hold for a Laplace eigensection whether or not $\operatorname{\mathfrak{p}}$ is invariant under a symmetry group action. These observations are summarized by the upcoming Lemma 2.3. This summary of what is said in [11] uses stereographic projection from $p$’s antipodal twin to define a complex coordinate to be denoted by $z$ near $\operatorname{\mathfrak{p}}$ with both $z=0$ and $|dz|=1$ at $p$. By way of notation: The lemma use $\operatorname{\mathfrak{Re}}(\cdot)$ to denote the real part of the indicated $\operatorname{\mathbb{C}}$-valued function or differential form.

When the point $p$ is germane to the discussion, then the corresponding versions of $\operatorname{\mathfrak{a}}$ and $n$ are written as $\operatorname{\mathfrak{a}}_{p}$ and $n_{p}$. When $f$ and $\operatorname{\mathfrak{p}}$ are germane, they are written as $\operatorname{\mathfrak{a}}_{p}(f)$ and $n_{p}(f)$.

By way of a definition: The subset $\operatorname{\mathfrak{p}}_{f}\in\operatorname{\mathfrak{p}}$ is defined to be the set of points with the integer $n$ being zero.

The identity that follows momentarily uses $\operatorname{\mathfrak{g}}$ to denote the round metric on $S^{2}$ when viewed as a section of $\operatorname{\mathrm{Sym}}^{2}(T^{*}S^{2})$. Supposing that $f$ denotes an $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigensection and $\lambda$ its eigenvalue, define the section $\mathbb{T}$ of $\operatorname{\mathrm{Sym}}^{2}(T^{*}S^{2})$ on $S^{2}-\operatorname{\mathfrak{p}}$ by the rule

$$\mathbb{T}=df\otimes df-{1\over 2}(|df|^{2}-\lambda f^{2})\operatorname{\mathfrak{g}}.$$

(2.5)

By virtue of (1.4), this section is divergence free, thus

$$\nabla^{\dagger}\mathbb{T}=0.$$

(2.6)

An integrated version of this identity will be presented momentarily in the upcoming Lemma 2.4. An observation is needed first which is this: Let $s$ denote a continuous section of $\operatorname{\mathrm{Sym}}^{2}(T^{*}S^{2})$ that is defined on the whole of $S^{2}$. Then the function $\langle s,\mathbb{T}\rangle$ has finite integral over $S^{2}$ and the assignment

$$s\longrightarrow\int_{S^{2}}\langle s,\mathbb{T}\rangle$$

(2.7)

defines a bounded linear functional on the Banach space of continuous sections of $\operatorname{\mathrm{Sym}}^{2}(T^{*}S^{2})$. This follows because $\mathbb{T}$’s norm is bounded by $c_{0}(|df|^{2}+\lambda f^{2})$ whose integral over $S^{2}$ is bounded by $c_{0}$ times the $\mathbb{H}$-norm of $f$. (It also follows from Lemma 2.3 that $\mathbb{T}$’s singularities at the points in $\operatorname{\mathfrak{p}}$ are integrable.)

The upcoming lemma also uses the convention whereby vector fields and 1-forms are identified using the metric $\operatorname{\mathfrak{g}}$.

The divergence identity in (2.6) for $\mathbb{T}$ as defined in (2.5) has a generalization that is also useful: To state the more general identity, suppose now that $f$ and $f^{\prime}$ are two $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigensections and let $\lambda$ and $\lambda^{\prime}$ denote their respective eigenvalues. Now define the symmetric tensor $\mathbb{S}$ by the rule

$$\mathbb{S}=df\otimes df^{\prime}+df^{\prime}\otimes df-\left(\langle df,df^{\prime}\rangle-{1\over 2}(\lambda+\lambda^{\prime})ff^{\prime}\right)\operatorname{\mathfrak{g}}.$$

(2.9)

The tensor $\mathbb{S}$ is divergence free if $\lambda=\lambda^{\prime}$, but otherwise

$$\nabla^{\dagger}\mathbb{S}={1\over 2}(\lambda^{\prime}-\lambda)(fdf^{\prime}-f^{\prime}df).$$

(2.10)

Much the same argument that led to Lemma 2.4 leads in this case to the following analog of (2.8):

$$\int_{S^{2}}\langle\nabla\nu,\mathbb{S}\rangle-{1\over 2}(\lambda^{\prime}-\lambda)\int_{S^{2}}\langle\nu,fdf^{\prime}-f^{\prime}df\rangle+\sum_{p\in\operatorname{\mathfrak{p}}_{f}}\lim_{\epsilon\to 0}\int_{\partial D_{\epsilon}(p)}\langle\hat{r}\otimes\nu,\mathbb{S}\rangle=0.$$

(2.11)

Now we can invoke the $f$ and $f^{\prime}$ versions of Lemma 2.3 to evaluate the limit in (2.11) and doing so leads to the following identity:

$$\int_{S^{2}}\langle\nabla\nu,\mathbb{S}\rangle-{1\over 2}(\lambda^{\prime}-\lambda)\int_{S^{2}}\langle\nu,fdf^{\prime}-f^{\prime}df\rangle=-{\pi\over 2}\sum_{p\in\operatorname{\mathfrak{p}}_{f}\cap\operatorname{\mathfrak{p}}_{f^{\prime}}}\langle\nu|_{p},\operatorname{\mathfrak{Re}}(\operatorname{\mathfrak{a}}_{p}(f)\operatorname{\mathfrak{a}}_{p}(f^{\prime})dz)\rangle.$$

(2.12)

This is the promised generalization of Lemma 2.4’s identity.

If $\operatorname{\mathfrak{p}}$ is a given $2n$-tuple of points in $S^{2}$ and $\operatorname{\mathfrak{p}}^{\prime}$ is some other $2n$-tuple, then there are diffeomorphism of $S^{2}$ that map the set $\operatorname{\mathfrak{p}}$ to $\operatorname{\mathfrak{p}}^{\prime}$. If $\phi$ is a diffeomorphism that does this, then $\phi^{*}\operatorname{\mathcal{I}_{\mathfrak{p}^{\prime}}}$ is isomorphic to $\operatorname{\mathcal{I}_{\mathfrak{p}}}$. As a consequence, the eigenvalues for the round metric’s Laplacian on sections of $\operatorname{\mathcal{I}_{\mathfrak{p}^{\prime}}}$ are precisely those of the Laplacian on sections of $\operatorname{\mathcal{I}_{\mathfrak{p}}}$ that is defined by the $\phi$-pull-back of the round metric (which won’t be the round metric unless $\phi$ is a rotation and/or reflection). By virtue of this pull-back correspondence, questions about the behavior of $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigenvalues and $\operatorname{\mathcal{I}_{\mathfrak{p}}}$-eigenfunctions of the round metric’s Laplacian on the $\operatorname{\mathcal{C}_{2n}}$-dependent family of domains are questions about the metric dependence of the eigenvalues and eigenfunctions for a $\operatorname{\mathcal{C}_{2n}}$-dependent family of Laplace operators on a fixed domain. The latter, alternate point of view is often taken in subsequent sections and it motivates the digression that follows directly about metric Laplacians acting on the sections of the fixed bundle $\operatorname{\mathcal{I}_{\mathfrak{p}}}$.

Any given smooth Riemannian metric on $S^{2}$ has a corresponding version of the Hilbert space $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ and a corresponding set of eigensections for its Laplacian. To say more about this, let $\operatorname{\mathfrak{m}}$ denote a given Riemannian metric. For convenience (this is not strictly necessary), we assume that the associated area 2-form is the same as that of the standard, round metric. This assumption is implicit in what follows. The metric $\operatorname{\mathfrak{m}}$’s inner product on $T^{*}S^{2}$ is denoted by $\langle,\rangle_{\operatorname{\mathfrak{m}}}$ and the associated norm by $|\cdot|_{\operatorname{\mathfrak{m}}}$. The corresponding Hilbert space is the completion of the space of smooth, compactly supported sections of $\operatorname{\mathcal{I}_{\mathfrak{p}}}$ using the version of (1.1) with $|df|^{2}$ replaced by $|df|_{\operatorname{\mathfrak{m}}}^{2}$. An element in this Hilbert space (denoted by $f$) is an eigensection of the metric m’s Laplacian with eigenvalue $\lambda$ when

$$\int_{S^{2}-\operatorname{\mathfrak{p}}}\langle df,dh\rangle_{\operatorname{\mathfrak{m}}}=\lambda\int_{S^{2}-\operatorname{\mathfrak{p}}^{2}}fh$$

(2.13)

for all elements $h$ from the Hilbert space. Eigensections are the elements in the Hilbert space that obey the Laplace equation for $\operatorname{\mathfrak{m}}$’s version of the Laplace operator.

With regards to this metric $\operatorname{\mathfrak{m}}$ Hilbert space: It is the original round metric’s version of $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ with a different but equivalent inner product. This is to say that any section in one is in the other and vice versa. This understood, the metric $\operatorname{\mathfrak{m}}$’s version of the eigenvalue/eigensection question will be viewed henceforth as a question concerning a bilinear form on the original $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ Hilbert space. As was the case with the round metric $\operatorname{\mathfrak{g}}$, there is a complete, orthonormal basis for $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ of eigensections for the bilinear form in (2.13) with the corresponding eigenvalue set being discrete with no accumulation points and with each eigenvalue having finite multiplicity.

Now suppose that $\mathcal{M}$ is a real analytic, finite dimensional manifold that parameterizes in a real analytic fashion a family of metrics on $S^{2}$ subject to the constraint that their area forms are the same as that of $\operatorname{\mathfrak{g}}$. There is a corresponding family of bilinear forms on $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ and one can ask how the sets of eigenvalues and eigenfunctions change as the domain $\mathcal{M}$ is changed. Exactly this sort of question is addressed in Chapters 6 and 7 of Tosio Kato’s classic book Perturbation theory for linear operators [4]. The rest of this section summarizes the implications of Kato’s discussion.

To start the story, fix a metric $\operatorname{\mathfrak{m}}_{0}\in\mathcal{M}$ and then fix a smooth coordinate chart for a neighborhood of $\operatorname{\mathfrak{m}}_{0}$ that identifies the chosen neighborhood with a ball about the origin in $\operatorname{\mathbb{R}}^{N}$. (The integer $N$ is the dimension of $\mathcal{M}$.) The identification is chosen so that $\operatorname{\mathfrak{m}}_{0}$ corresponds to the origin in $\operatorname{\mathbb{R}}^{N}$. A point in this ball is denoted by $y$ and the corresponding metric by $\operatorname{\mathfrak{m}}_{y}$. If $y$ is close to the origin in $\operatorname{\mathbb{R}}^{N}$, then $\operatorname{\mathfrak{m}}_{y}^{-1}$ can be written as

$$\operatorname{\mathfrak{m}}_{y}^{-1}=\operatorname{\mathfrak{m}}_{0}^{-1}(1+\operatorname{\mathfrak{k}}y+\operatorname{\mathfrak{e}}_{y})$$

(2.14)

with $\operatorname{\mathfrak{k}}$ denoting a linear map from $\operatorname{\mathbb{R}}^{N}$ to $C^{\infty}(S^{2};\operatorname{\mathrm{Sym}}^{2}(TS^{2}))$ and with $\operatorname{\mathfrak{e}}_{y}$ being a map from the coordinate ball in $\operatorname{\mathbb{R}}^{N}$ to $C^{\infty}(S^{2};\operatorname{\mathrm{Sym}}^{2}(TS^{2}))$ with $C^{k}$ norm (for any given non-negative integer $k$) bounded by a $k$-dependent multiple of $|y|^{2}$.  Now suppose that $\lambda_{0}$ is an eigenvalue for the $\operatorname{\mathfrak{m}}_{0}$ Laplacian on $\operatorname{\mathbb{H}_{\mathfrak{p}}}$. As noted in [4], there will be an eigenvalue of the $\operatorname{\mathfrak{m}}_{y}$ Laplacian near to $\lambda_{0}$ if $y$ is sufficiently close to the origin. Moreover, if $\lambda_{0}$ has multiplicity $n$, then there will be precisely $n$ linearly independent eigensections for the $\operatorname{\mathfrak{m}}_{y}$-Laplacian with eigenvalue very close to $\lambda_{0}$ if $y$ is small (much closer to $\lambda_{0}$ than to any other $\operatorname{\mathfrak{m}}_{0}$ eigenvalue.) In particular, the following is true: Given $\epsilon>0$, there exists $\delta>0$ such that the eigenvalues of these $n$ eigenvectors of the $\operatorname{\mathfrak{m}}_{y}$ Laplacian will differ from $\lambda_{0}$ by at most $\epsilon$ if $|y|<\delta$. In addition, if $|y|<\delta$, then the $n$-dimensional vector space spanned by these $n$ eigensections for the $\operatorname{\mathfrak{m}}_{y}$-Laplacian will have $\mathbb{H}_{p}$-distance at most $\epsilon$ from the $\lambda_{0}$ eigenspace of the $\operatorname{\mathfrak{m}}_{0}$ Laplacian. (This distance is measured by the $\mathbb{H}_{p}$-orthogonal projection.) The subsequent paragraphs and the upcoming Proposition 2.5 say more about the relationship between respective $\operatorname{\mathfrak{m}}_{y}$ and $\operatorname{\mathfrak{m}}_{0}$ Laplacian eigenvalues and eigenspaces.

The simplest case to consider is that where $\lambda_{0}$ has multiplicity 1 in which case there is one nearby eigenvalue when $y$ is near 0 and that eigenvalue varies smoothly with $y$ on a neighborhood of 0 with value $\lambda_{0}$ at $y=0$. This $y$-dependent eigenvalue can be written as $\lambda_{0}+y\cdot\lambda^{\prime}+\mathcal{O}(|y|^{2})$ with $\lambda^{\prime}$ given by the rule

$$\lambda^{\prime}=\int_{S^{2}}\langle\operatorname{\mathfrak{k}},df_{0}\otimes df_{0}\rangle$$

(2.15)

In this case, there is one corresponding eigensection of the $\operatorname{\mathfrak{m}}_{y}$ Laplacian that is very near to a normalized $\lambda_{0}$ eigensection when $y$ has small norm and it also varies smoothly with $y$ near the origin. (This $\lambda_{0}$ eigensection is denoted by $f_{0}$; it is normalized so that the $S^{2}$ integral of its square is equal to 1.) In particular, this $y$-dependent element in $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ can written as a Taylor’s expansion as $f_{0}+y\cdot\operatorname{\mathfrak{f}}+r_{y}$ with $\|r_{y}\|_{\mathbb{H}}$ bounded by $c_{0}|y|^{2}$ and with the linear map $\operatorname{\mathfrak{f}}$ from $\operatorname{\mathbb{R}}^{N}$ to $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ determined via the following two rules:

1. (1)

   $\int_{S^{2}}\operatorname{\mathfrak{f}}f_{0}=0,$

2. (2)

   $\int_{S^{2}}\langle d\operatorname{\mathfrak{f}},dh\rangle_{\operatorname{\mathfrak{m}}_{0}}-\lambda_{0}\int_{S^{2}}\operatorname{\mathfrak{f}}h+\int_{S^{2}}\langle\operatorname{\mathfrak{k}},df_{0}\otimes dh\rangle=0$ for all $h\in\mathbb{H}$ that obey $\int_{S^{2}}hf_{0}=0.$

(2.16)

With this simple case understood, consider now the case when $\lambda_{0}$ has multiplicity $n\geq 1$. To set the stage for this case, introduce by way of notation $\mathbb{V}$ to denote the corresponding $n$-dimensional vector space of eigensections of the Laplacian in $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ with eigenvalue $\lambda_{0}$. Given a pair $f$ and $f^{\prime}$ from $V$, define a vector in $\operatorname{\mathbb{R}}^{N}$ which is denoted by $\mathbf{k}(f,f^{\prime})$ by the rule

$$\mathbf{k}(f,f^{\prime})\equiv\int_{S^{2}}\langle\operatorname{\mathfrak{k}},df\otimes df^{\prime}\rangle.$$

(2.17)

Supposing that $y\in\operatorname{\mathbb{R}}^{N}$, then the assignment of any given pair of elements in $\mathbb{V}$ to the value of $y\cdot\mathbf{k}$ on that pair defines a symmetric bilinear form on $\mathbb{V}$. An element $f\in\mathbb{V}$ is said to be an eigensection for this bilinear form when there exists a number $e(y)$ such that

$$y\cdot\mathbf{k}(f,f^{\prime})=e(y)\int_{S^{2}}ff^{\prime}\text{ for all }f^{\prime}\in\mathbb{V}.$$

(2.18)

With regards to the $y$-dependence: If $f$ is an eigensection for $y\cdot\mathbf{k}$, then it is likewise for $ry\cdot\mathbf{k}$ for any given real number $r$, the corresponding eigenvalue $e(ry)$ being $re(y)$.

As noted, this proposition summarizes the consequences of various observations from Chapters 6 and 7 of [4]. An important point to keep in mind with regards to this proposition: The correspondence between the eigensections of the $\operatorname{\mathfrak{m}}_{y}$ Laplacian with eigenvalue near $\lambda_{0}$ and those of the bilinear form $y\cdot\mathbf{k}$ is one-to-one when $|y|$ is small if the latter has distinct eigenvalues. If it doesn’t, then there might only be a one-to-one correspondence with a particular basis of its eigensections.

Fix a positive integer to be denoted by $N$ and then fix a linear map from $\operatorname{\mathbb{R}}^{N}$ into the subspace of divergence zero elements in $C^{\infty}(S^{2};TS^{2})$; the map is denoted by $\nu$ in what follows. Thus, if $y\in\operatorname{\mathbb{R}}^{N}$, then $y\cdot\nu$ is a divergence zero vector field on $S^{2}$. Taking the time 1 point on the integral curves of these vector fields defines a smooth map from a ball about the origin in $\operatorname{\mathbb{R}}^{N}$ to the subspace of area preserving diffeomorphisms of $S^{2}$ which is equal to the identity at $y=0$ and whose differential at $y=0$ is given by $\nu$. (The differential at $y=0$ is a linear map from $\operatorname{\mathbb{R}}^{N}$ into the tangent space to the space of area preserving diffeomorphisms of $S^{2}$ at the identity element which is the space of divergence zero vector fields on $S^{2}$.) Let $\mathbf{B}\subset\operatorname{\mathbb{R}}^{N}$ denote this ball and, supposing that $y\in\mathbf{B}$, let $\phi_{y}$ denote the diffeomorphism that is defined by $y$. Also: Let $\operatorname{\mathfrak{m}}_{y}$ denote the pull-back of the round metric on $S^{2}$ by $\phi_{y}$. Because $\phi_{y}$ is area preserving, the area form for the metric $\operatorname{\mathfrak{m}}_{y}$ is the round metric’s area form.

By taking the radius of $\mathbf{B}$ to be smaller if necessary, each $y\in\mathbf{B}$ version of the inverse metric to $\operatorname{\mathfrak{m}}_{y}$ can be written as in (2.14) with $\operatorname{\mathfrak{k}}$ as follows:

$$\operatorname{\mathfrak{k}}=-(\nabla v+(\nabla v)^{T})$$

(2.19)

with it understood that the round metric has been used to identify $TS^{2}$ with $T^{*}S^{2}$ and to define the transpose that is used in (2.19) to make a symmetric tensor from $\nabla\nu$.

Now suppose that $\lambda_{0}$ is an eigenvalue of the round metric’s Laplacian on $\operatorname{\mathbb{H}_{\mathfrak{p}}}$ and let $\mathbb{V}$ denote the corresponding eigenspace. Then the $\operatorname{\mathbb{R}}^{N}$-dependent bilinear form on $\mathbb{V}$ that is depicted in (2.17) can be written as

$$\operatorname{\mathbf{k}}(f,f^{\prime})=-\int_{S^{2}}\langle\nabla\nu,\mathbb{S}\rangle$$

(2.20)

with $\mathbb{S}$ given by (2.9). (The contributions to $\langle\nabla v,\mathbb{S}\rangle$ from the terms proportional to $\operatorname{\mathfrak{g}}$ are zero because $\nu$ is divergence free.) This being the case, $\mathbf{k}$ can be written using (2.12):

$$\mathbf{k}(f,f^{\prime})={\pi\over 2}\sum_{p\in\operatorname{\mathfrak{p}}_{f}\cap\operatorname{\mathfrak{p}}_{f^{\prime}}}\langle\nu|_{p},\mathfrak{Re}(\operatorname{\mathfrak{a}}_{p}(f)\operatorname{\mathfrak{a}}_{p}(f^{\prime})dz)\rangle.$$

(2.21)

This implies in particular that the eigenvalues of any $y\in\mathbf{B}$ version of $y\cdot\mathbf{k}$ are determined by the asymptotics of the eigensections in $\mathbb{V}$ at the points in $\operatorname{\mathfrak{p}}$.

The applications of the preceding formulas will use divergence zero vector fields of the sort that are described in this subsection. Each vector field of interest is specified by a data set consisting of a point in $S^{2}$ to be denoted by $q$, a number from $(-1,1)$ to be denoted by $r$, a positive number which is denoted by $a$, and a real number to be denoted by $s$. To describe the corresponding vector field, consider first the case where $q$ is the north pole. Introduce spherical coordinates $(\theta,\phi)$ on $S^{2}$ by viewing $S^{2}$ as the set of length 1 vectors in $\mathbb{R}^{3}$ and then writing the Cartesian coordinates $(x_{1},x_{2},x_{3})$ of any given point on $S^{2}$ as $(x_{1}=\sin\theta\cos\varphi,x_{2}=\sin\theta\sin\varphi,x_{3}=\cos\theta$). Granted this definition, set

$$v_{(q,r,\mathfrak{a},s)}\equiv s\chi_{r,a}(\theta){\partial\over\partial\varphi}\text{ where }\chi_{r,a}\text{ is }\chi\left({1\over a}(\cos\theta-r)\right)\chi\left({1\over a}(r-\cos\theta)\right)$$

(2.22)

Note in particular that $v_{(q,r,\mathfrak{a},s)}$ is equal to $s{\partial\over\partial\varphi}$ where $\cos\theta=r$ and that it has compact support where $|\cos\theta-r|<a$. The definition of $v_{(q,r,\mathfrak{a},s)}$ for any other choice of $q$ is the push-forward of the north polar version of $v_{(q,r,\mathfrak{a},s)}$ depicted above by the rotation of $S^{2}$ that moves the north pole to $q$ along the short great circle between them. (In the case of the south pole, the formula is given by the analog of (2.22) with $\theta$ replaced by $\pi-\theta$ and with the $s$ changed to $-s$.) Let $\mathcal{V}$ denote the set of all such $v_{(q,r,a,s)}$. The next paragraph and the subsequent ones point out some of the salient properties of the vector fields in the set $\mathcal{V}$.

The first point of note is this: Given a point $p\in\operatorname{\mathfrak{p}}$, and a non-zero vector $\nu\in TS^{2}|_{p}$, there are vectors fields in $\mathcal{V}$ that are zero on a neighborhood in $S^{2}$ of $\operatorname{\mathfrak{p}}-\{p\}$ and a positive multiple of $v$ at $p$. To find such a vector field, fix a Gaussian coordinate chart centered at $p$. Use this chart to identify a small radius disk around $p$ with the corresponding radius disk about the origin in $\operatorname{\mathbb{R}}^{2}$. Take the radius of this disk to be much less than the distance to any point in $\operatorname{\mathfrak{p}}-p$. The Gaussian coordinate chart is uniquely specified by requiring that it identify $v$ with a vector at the origin pointing in the positive direction along the $x$ axis. Now fix a point (which will be $q$) in the disk in $\operatorname{\mathbb{R}}^{2}$ on the positive $y$ axis and take $r$ to be the distance from the origin to this point. Take the number $a$ to be any positive number that is much less than $r$. The corresponding $v_{(q,r,\mathfrak{a},+)}$ is a non-zero, positive multiple of $\nu$ at the point $p$ and it is zero at all other points in $\operatorname{\mathfrak{p}}$.