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Without loss of generality, assume that [EQUATION] Then from the construction of the graph and from the assumption of the lemma [EQUATION] |
are edges, hence [MATH] and [MATH] are connected. Likewise, [MATH] and [MATH] are connected. Therefore any connected component of [MATH] must contain either [MATH] or [MATH] |
8.2. The number of nodal domains of generic eigenfunctions Let [MATH] be a principal [MATH] bundle over a connected smooth compact Riemannian surface [MATH] with the covering map [MATH] . Let [MATH] be a fixed integer, and assume that [MATH] satisfies the following conditions: |
Condition 8.1 For any small open [MATH] such that [MATH] , there exists a local coordinate [MATH] of [MATH] such that (i) [MATH] |
(ii) the zero set of [MATH] (resp. [MATH] ) gives rise to a partition [MATH] (resp. [MATH] ) of [MATH] , and (iii) [MATH] is a generic pair of partitions of [MATH] |
In this section, we prove the following theorem. {theo} Fix any point [MATH] such that [MATH] . Then any nodal domain of [MATH] has a nonempty intersection with [MATH] . In particular, the number of nodal domains of [MATH] is [MATH] . Assume further that [MATH] has a regular zero. Then the number of nodal domains of [M... |
We begin with few observations in terms of fixed [MATH] and a local coordinate [MATH] of [MATH] Proposition 13 If [MATH] is positive on two open sets [MATH] and [MATH] for some integer [MATH] , and if [MATH] , then [MATH] and [MATH] are contained in the same nodal domain of [MATH] |
Proof 8.3 Let [MATH] be a point in the intersection [MATH] . Then from the equation [EQUATION] we see that [MATH] is positive along the curve |
[EQUATION] which connects [MATH] and [MATH] . Therefore [MATH] and [MATH] are contained in the same nodal domain. Proposition 14 |
Any nodal domain of [MATH] must intersect [MATH] nontrivially for some integer [MATH] Proof 8.4 Assume for contradiction that [MATH] is a nodal domain of [MATH] that is contained in |
[EQUATION] From the equation [EQUATION] we see that for each fixed [MATH] [MATH] either vanishes identically or has at most one sign change along the curve |
[EQUATION] This implies that if [MATH] , then [EQUATION] which contradicts the assumption that the zero set of [MATH] gives rise to a partition of [MATH] |
From these two propositions, we see that the nodal domains of [MATH] can be understood from the nodal domains of the restrictions of [MATH] to the [MATH] -hypersurfaces |
[EQUATION] In particular, if we define [MATH] and [MATH] in terms of the sign of [MATH] and [MATH] , then the number of connected components of [MATH] is equal to the number of nodal domains of [MATH] |
Proof 8.5 (Proof of Theorem 8.2 Let [MATH] be a point where [MATH] , and let [MATH] be a sufficiently small neighborhood of [MATH] . We may assume without loss of generality that the vertices |
[EQUATION] of [MATH] correspond to the nodal domains of the restrictions of [MATH] to the hypersurfaces [EQUATION] that intersect the fiber [MATH] . Then Lemma 8.1 implies that any nodal domain of [MATH] must intersect [MATH] |
Now assume that [MATH] is another point in [MATH] . Then we may restate this as “any nodal domain of [MATH] that intersect [MATH] must intersect [MATH] ”, and equivalently, “any nodal domain of [MATH] that intersect [MATH] must intersect [MATH] ”. Because we assumed that [MATH] is connected, by the freedom of choice of... |
For the latter part of the theorem, let [MATH] be a regular zero of [MATH] , i.e., [EQUATION] is a surjection. Choose a sufficiently small neighborhood [MATH] of [MATH] , and let [MATH] be the function that satisfies |
[EQUATION] If [MATH] and [MATH] are linearly dependent, then a straightforward computation implies that [MATH] has rank [MATH] , so [MATH] and [MATH] are linearly independent. |
This implies that [MATH] is a regular zero of both [MATH] and [MATH] . Also, linear independency implies that locally around [MATH] [MATH] and [MATH] define two curves intersecting transversally at [MATH] . From this, we may find four open sets near [MATH] that are required for Lemma 8.1 , and we infer that the number ... |
Now because any nodal domain of [MATH] must intersect with [MATH] for some [MATH] , any nodal domain of [MATH] must contain one of the nodal domains of [MATH] , from which we conclude that [MATH] has only two nodal domains. |
We are ready to prove our main theorem, Theorem 1.3 Proof 8.6 It is sufficient to verify the assumptions in Theorem 8.2 is satisfied. The first condition is trivial to verify. For the other conditions, note from the assumption that [MATH] is non-trivial, [MATH] is non-empty, and Theorem implies that it is discrete and ... |
{rema} If [MATH] contains a closed curve that divides [MATH] into two connected components, then the number of nodal domain can be large. For instance, if [MATH] vanishes on the boundary of small open disc [MATH] , and if it does not vanish on [MATH] , then [MATH] vanish identically on [MATH] , and therein, [MATH] has ... |
9. Surfaces of constant curvature In this section, we illustrate the geometry of Kaluza-Klein metrics and the Kaluza-Klein eigenvalue problem on unit tangent bundles of surfaces of constant curvature. |
9.1. Flat tori Let [MATH] . We use coordinates [MATH] . Its unit tangent bundle is [MATH] . The connection is flat and [MATH] is simply the Laplacian of [MATH] . The Kaluza-Klein Laplacian is that [MATH] on [MATH] . The Kaluza-Klein eigenfunctions are linear combinations of the product eigenfunctions, |
[EQUATION] The multiplicity of the eigenvalue with fixed [MATH] is the number of ways of representing [MATH] as a sum of two squares. They correspond to eigendifferentials |
[EQUATION] In the notation ( 1.2 ), [EQUATION] The nodal sets of the imaginary part are given by, [EQUATION] [MATH] contains the set |
[EQUATION] Note that [MATH] has no zeros on [MATH] and [MATH] has no zeros as an [MATH] -differential on [MATH] If we change the lattice to a general lattice [MATH] , the eigenfunctions of [MATH] change to [MATH] where [MATH] , the dual lattice. For generic [MATH] , the eigenvalues have multiplicity [MATH] and the eige... |
[MATH] or equivalently by [MATH] and its complex conjugate [MATH] . The same is true of the Kaluza-Klein eigenfunctions [MATH] . Again, [MATH] has no zeros. Using the bifurcation of nodal sets of eigenfunctions under generic paths of metrics of |
, one can show that Conjecture 9.1 for generic Kaluza-Klein metrics on [MATH] the joint eigenfunctions [MATH] have no zeros. We now give an explicit orthonormal eigenbasis of [MATH] such that all of them have exactly two nodal domains, hence proving Theorem 1.4.1 |
To begin with, let [MATH] and [MATH] . Then [EQUATION] is an orthogonal eigenbasis of [MATH] . We consider four cases. Case 1: [MATH] |
We first have [EQUATION] Assume without loss of generality that [MATH] . Then [EQUATION] has two nodal domains by Theorem 8.2 , because |
[EQUATION] has a regular zero. Case 2: exactly one [MATH] is zero, and the other two are different From the same reasoning, each eigenfunction in the new basis in the following has two nodal domains: |
[EQUATION] [EQUATION] and [EQUATION] Case 3: exactly one [MATH] is zero, and the other two are equal Again by the same reasoning, each of the following |
[EQUATION] has two nodal domains, and these are the basis of [EQUATION] Case 4: exactly one [MATH] is nonzero In this case, we consider orthogonal eigenfunctions |
[EQUATION] which span [EQUATION] Each of these has only two nodal domains from the following lemma. {lemm} Let [MATH] be a positive integer. Then |
[EQUATION] has only two nodal domains. Proof 9.2 Let [MATH] [MATH] , and [MATH] . Then [EQUATION] and from Theorem 8.2 , it is sufficient to prove that |
[EQUATION] has a regular zero. Let [MATH] and [MATH] , then this is equivalent to [EQUATION] having a regular zero. Since [MATH] and [MATH] do not have singular points, it is sufficient to check if these two functions have a common zero, in other words, if |
[EQUATION] has a solution. Note that this is equivalent to [EQUATION] Because [EQUATION] for [MATH] such that [MATH] , there is [MATH] satisfying ( 9.1 ), and this completes the proof. |
9.2. Kaluza-Klein metrics on [MATH] Let [MATH] be the [MATH] -sphere with its standard metric of curvature [MATH] Then its unit tangent [MATH] and the Kaluza-Klein metric is the standard metric of constant sectional curvature [MATH] on [MATH] (divided by the antipodal group [MATH] ). The Kaluza-Klein Laplacian is there... |
Since [MATH] is a group, [MATH] where [MATH] is an irreducible representation of [MATH] of dimension [MATH] . Alternatively, the eigenfunctions of |
[MATH] are harmonic homogeneous polynomials on [MATH] Moreover, [MATH] . The eigenfunctions of [MATH] are those where [MATH] is even. |
We need explicit separation of variables expressions for equivariant spherical harmonics, and therefore need to introduce coordinate systems. We use ‘axis - angle’ Hopf coordinates [MATH] on [MATH] defined by |
[EQUATION] Here [MATH] . This corresponds to writing [EQUATION] There exist two commuting isometric [MATH] actions [EQUATION] The metric is |
[MATH] In these coordinates one has an orthogonal basis of eigenfunctions given by [EQUATION] where [MATH] is a Jacobi polynomial and where |
[EQUATION] Here, weight [MATH] in our sense means that the eigenfunctions transform by [MATH] [MATH] are also known as “Wigner D-functions” on [MATH] . Another expression is |
[EQUATION] Here [MATH] in our notation. These are manifestly joint eigenfunctions of [MATH] and of [MATH] {lemm} The nodal sets of the equivariant eigenfunctions [MATH] (or equivalently [MATH] ) have real dimension [MATH] |
Proof 9.3 The only factors with zeros are the [MATH] -functions. These have roughly [MATH] discrete zeros in [MATH] . Hence, the complex nodal set is a union |
[EQUATION] and thus has real dimension [MATH] As a result, these eigenfunctions do not satisfy the conditions of the generic Kaluza-Klein metrics to which our results apply, and their nodal sets are quite different. |
As mentioned in the introduction, the numerical experiments of A. Barnett et all show that random spherical harmonics of degree [MATH] on [MATH] also have different types of nodal sets than our generic eigenfunctions. Namely, the expected number of nodal domains has the asymptotics [MATH] for a certain [MATH] . As prov... |
, the nodal sets of real/imaginary parts of random equivariant eigenfunctions with fixed [MATH] (a subspace isomorphic to [MATH] ) have connected nodal sets. The difference is due to the fact that our random equivariant spherical harmonics are a thin subset of the random spherical harmonics of degree [MATH] on [MATH] |
{rema} In , we compute the expected genus of the single component of the nodal sets of real/imaginary parts of random equivariant spherical harmonics of degree [MATH] where [MATH] . The expected Euler characteristic is of the form [MATH] modulo lower order terms. |
9.3. Hyperbolic surfaces [MATH] Although it differs from our prior discussion in the compact case, let us consider a finite area hyperbolic real Riemann surface of constant negative curvature [MATH] . Then [MATH] where [MATH] . The total space [MATH] carries a Lorentz Cartan-Killing metric with indefinite Laplacian the... |
[MATH] with fibers given by [MATH] -orbits. They are necessarily totally geodesic. It follows that the horizontal Laplacian [MATH] commutes with the vertical Laplacian [MATH] . This is obvious because [MATH] |
The joint eigenfunctions of [MATH] are denoted by [MATH] When [MATH] they are pullbacks of eigenfunctions of [MATH] In particular the number of nodal domains of [MATH] on [MATH] is the same as the number of nodal domains of [MATH] on [MATH] . The former nodal sets are [MATH] -invariant and in the case of regular nodal ... |
The lift of weight [MATH] of an [MATH] -differential [MATH] is given by [EQUATION] Here, the Kähler potential is [MATH] [MATH] [MATH] . Also, |
[MATH] and [MATH] . The Maass operator is [EQUATION] and [EQUATION] Breaking up into real and imaginary parts gives the system, [EQUATION] |
The raising/lowering operators are the Maass operators defined by [EQUATION] Then, [EQUATION] and [EQUATION] and [EQUATION] 9.3.1. Automorphic forms on the full modular group |
Now we consider the case [MATH] . Note that the quotient [MATH] is non-compact in this case. Nevertheless, it is known that [MATH] has infinitely many discrete spectrum, where corresponding [MATH] integrable eigenfunctions can be chosen so that they are in one-to-one correspondence with Maass–Hecke cusp forms or holomo... |
for detailed background. {theo} Let [MATH] , and let [MATH] be a weight [MATH] Maass–Hecke cusp form on [EQUATION] Assume that the zeros of [MATH] are isolated. Then [MATH] has only two nodal domains. |
Proof 9.4 The first statement of Condition 8.1 follows from the definition of Maass–Hecke cusp form, and the second statement follow from the fact that [MATH] can not be scaled to a real-valued function, and that [MATH] is analytic. The third statement follows from the assumption. |
Now, because the first Hecke eigenvalue is [MATH] , the first Fourier coefficient of [MATH] at the cusp does not vanish, meaning that [MATH] is a regular zero of [MATH] . We conclude the proof by applying Theorem 8.2 |
{rema} It is not hard to see that in the constant curvature case, the nodal set of [MATH] consists of the fibers over the critical point set [MATH] of [MATH] At this time, it does not seem to be known whether [MATH] is necessarily a discrete set of points in the case of hyperbolic surfaces. This cannot be proved by a p... |
critical point sets. One can put any negatively curved [MATH] invariant metric on [MATH] and obtain the same result, so it is not an effect of constant curvature. We conjecture that for compact hyperbolic surfaces without boundary, |
[MATH] is a finite set for every eigenfunction. When we have holomorphicity of [MATH] , we may remove the assumption that the zeros of [MATH] being isolated. For instance, we have: |
{theo} Let [MATH] , and let [MATH] be a Laplacian eigenfunction on [MATH] corresponding to a holomorphic Hecke cusp form [MATH] of weight [MATH] . Then [MATH] has only two nodal domains. |
Proof 9.5 We first note that [MATH] , and [MATH] is holomorphic. Therefore Condition 8.1 is satisfied. Because we assumed that [MATH] is a Hecke cusp form, the first Hecke eigenvalue is [MATH] . Therefore [MATH] is a regular zero of [MATH] , and now the theorem follows from Theorem 8.2 |
{coro} There exist eigenfunctions on [MATH] that have only two nodal domains but with arbitrarily large eigenvalues. We remark here that Theorem 9.3.1 is false, without the assumption that [MATH] is a Hecke cusp form. To construct a counter example, let [MATH] be the discriminant modular form given by |
[EQUATION] where [MATH] . This is a weight [MATH] modular form on [MATH] . Thus [MATH] is a modular form of weight [MATH] on [MATH] , and |
[EQUATION] is a Laplacian eigenfunction on [MATH] of weight [MATH] . To count the number of nodal domains of this eigenfunction, we let |
[EQUATION] be the fundamental domain of [MATH] , and let [MATH] We then consider the restrictions of [MATH] to the top [MATH] , side [MATH] , and front [MATH] of the solid [MATH] |
It can be shown that the nodal set of [MATH] on the side is that of [MATH] , and on the front is that of [MATH] , where we define [MATH] . We compute the nodal set of the restriction to the top numerically using Mathematica. |
The nodal set of [MATH] on the front, the side, and the top of the solid [MATH] Note that we may obtain [MATH] from [MATH] by gluing the sides via [MATH] (corresponding to [MATH] ), the top and the bottom via [MATH] (corresponding to [MATH] ), and then the front with itself via [MATH] and [MATH] (corresponding to [MATH... |
From these, one can verify that [MATH] has exactly four nodal domains, where in the pictures above, two positive nodal domains are colored differently with red and orange. |
# Source: arxiv 1806.04721 # Title: Convergence of Cauchy Sequences for the covariant Gromov-Hausdorff propinquity # Sections: all # Downloaded: 2026-03-03T05:18:04.178596+00:00 |
url] Convergence of Cauchy Sequences for the Covariant Gromov-Hausdorff Propinquity Abstract The covariant Gromov-Hausdorff propinquity is a distance on Lipschitz dynamical systems over quantum compact metric spaces, up to equivariant full quantum isometry. It is built from the dual Gromov-Hausdorff propinquity which, ... |
Introduction The covariant Gromov-Hausdorff propinquity is a distance, up to equivariant full quantum isometry, on the class of Lipschitz dynamical systems, defined as the class of quantum compact metric space endowed with a strongly continuous action of a proper monoid by Lipschitz morphisms or even Lipschitz linear m... |
that the covariant propinquity is a metric up to equivariant full quantum isometry — namely, distance zero implies the existence of a full quantum isometry between the quantum compact metric spaces as well as an isometric isomorphism between the acting monoids, which intertwine the actions in a natural fashion. We illu... |
our metric by showing that fuzzy tori with their dual actions converge to quantum tori with their own dual actions for the covariant propinquity. |
The covariant propinquity is built from the dual Gromov-Hausdorff propinquity , which actually enjoys some natural covariance properties |
, though it is only defined on quantum compact metric spaces and thus does not fully capture the structure of a Lipschitz dynamical system. Nonetheless, our work in |
suggests that a covariant propinquity, as introduced in , is a natural object to construct. The covariant propinquity between two Lipschitz dynamical systems dominate the propinquity between the underlying quantum compact metric spaces and the pointed Gromov-Hausdorff distance |
between the underlying proper monoids, and both these last two distances are in particular complete. We are thus left with a very natural question: what classes of Lipschitz dynamical systems are complete when endowed with the covariant propinquity? |
This question is the subject of the present paper. As the covariant propinquity is built using a covariant version of the pointed Gromov-Hausdorff distance between proper monoids, we begin with finding natural classes of proper monoids complete for the monoid-adapted Gromov-Hausdorff distance. We then discover that com... |
We then turn to the matter of convergence for Cauchy sequences for the covariant propinquity. We provide a sufficient condition which includes the condition exhibited for the monoid-Gromov-Hausdorff distance, and a similar condition on the actions themselves. The reason for this double condition is simply that we actua... |
We begin our paper with a background section on noncommutative metric geometry and the covariant propinquity to set up the framework of this paper. |
Acknowledgment: This work is part of the project supported by the grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS and grant #3542/H2020/2016/2 of the Polish Ministry of Science and Higher Education. |
The covariant Gromov-Hausdorff Propinquity The covariant propinquity is defined on Lipschitz dynamical systems, which are proper monoid actions on quantum compact metric spaces by Lipschitz maps. A quantum compact metric space is a noncommutative analogue of the algebra of Lipschitz functions over a compact metric spac... |
Notation 2.1 Throughout this paper, for any unital C*-algebra [MATH] , the norm of [MATH] is denoted by [MATH] , the space of self-adjoint elements in [MATH] is denoted by [MATH] , the unit of [MATH] is denoted by [MATH] and the state space of [MATH] is denoted by [MATH] . We also adopt the convention that if a seminor... |
Definition 2.2 quantum compact metric space [MATH] is an ordered pair of a unital C*-algebra [MATH] and a seminorm [MATH] , called an L-seminorm , defined on a dense Jordan-Lie subalgebra [MATH] of [MATH] , such that: |
1. [MATH] 2. the Monge-Kantorovich metric [MATH] defined for any two states [MATH] by: [EQUATION] metrizes the weak* topology restricted to [MATH] |
3. [MATH] satisfies the [MATH] -quasi-Leibniz inequality, i.e. for all [MATH] [EQUATION] for some permissible function [MATH] , i.e. a function [MATH] , increasing when [MATH] is endowed with the product order, and such that for all [MATH] we have [MATH] |
4. [MATH] is lower semi-continuous with respect to [MATH] We say that [MATH] is Leibniz when [MATH] can be chosen to be [MATH] . More generally, if [MATH] satisfies the [MATH] -quasi-Leibniz inequality for some [MATH] then [MATH] is called a [MATH] -quantum compact metric space. |
We refer to for various examples of quantum compact metric spaces, including quantum tori, certain group C*-algebras, AF algebras, noncommutative solenoids, Podles spheres, and more. |
Quantum compact metric spaces form a category for the appropriate choices of morphisms. We refer to for some observations on the definition of Lipschitz morphisms and some of their applications. The definition of quantum isometry relies on a key observation of Rieffel in |
Definition 2.3 Let [MATH] and [MATH] be quantum compact metric spaces. 1. A positive unital linear map [MATH] is Lipschitz when there exists [MATH] such that [MATH] |
2. Lipschitz morphism [MATH] is a unital *-morphism from [MATH] to [MATH] when there exists [MATH] such that [MATH] 3. quantum isometry |
[MATH] is a *-epimorphism from [MATH] onto [MATH] such that for all [MATH] [EQUATION] 4. full quantum isometry [MATH] is a *-isomorphism from [MATH] onto [MATH] such that [MATH] |
We now equip quantum compact metric spaces with actions of proper monoids. As a matter of definition, we recall: Definition 2.4 metric monoid |
[MATH] (resp. group) is a monoid (resp. a group) [MATH] and a left invariant metric [MATH] on [MATH] for which the multiplication is continuous (resp. the multiplication and the inverse function are continuous). |
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