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Comptes Rendus
Mathématique
Charles Collot, Thomas Duyckaerts, Carlos Kenig and Frank Merle
Remark on the energy channel property for the radial linear wave equation
Volume 363 (2025), p. 1457-1460
Online since: 28 November 2025
https://doi.org/10.5802/crmath.803
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 1457-1460
https://doi.org/10.5802/crmath.803
Research article
Partial differential equations
Remark on the energy channel property for
the radial linear wave equation
Charles Collot a, Thomas Duyckaerts b,c, Carlos Kenig d and FrankMerle a,e
a AGM (UMR 8088), CY Cergy Paris Université, France
b LAGA (UMR 7539), Université Sorbonne Paris Nord, France
c Département de mathématiques et applications (UMR 8553), École Normale
Supérieure, France
d University of Chicago, USA
e Institut des Hautes Études Scientifiques, France
E-mails: charles.collot@cyu.fr, duyckaer@math.univ-paris13.fr,
cek@math.uchicago.edu, frankemerle@gmail.com
In the memory of Haïm Brezis
Abstract. This note concerns an exterior energy bound (channel of energy property) for the linear radial wave
equations, which is crucial in the proof of the soliton resolution for the energy-critical wave equations. We
give a short and synthetic proof that this property in general odd space dimension follows from the case of
space dimension 3, which is elementary. This gives a simple proof of the channel of energy property in all
odd space dimensions. We also show by the same proof that the analogous bound in even space dimensions
follows from the case of space dimension 4.
2020 Mathematics Subject Classification. 35L05, 35B40.
Funding. C.C. is supported by the CY Initiative of Excellence Grant “Investissements d’Avenir” ANR-16-IDEX-
0008 via Labex MME-DII, the Grant “Chaire Professeur Junior” ANR-22-CPJ2-0018-01 of the French National
Research Agency and the Grant BOURGEONS ANR-23-CE40-0014-01 of the French National Research Agency.
C.K. is partially supported by NSF Grants DMS-2153794 and DMS-2052710.
Note. Article submitted by invitation.
Manuscript received 16 June 2025, revised 25 July 2025, accepted 16 October 2025.
1. Introduction and statement of the result
Let N ≥ 3, and denote by Ḣ 1N (R) and L2N (R) the vector space of radial functions in
{
x ∈ RN ,
|x| > R}, with the norms:
∥ f ∥2
Ḣ 1N
=
∫ ∞
R
∣∣ f ′(r )∣∣2r N−1 dr, ∥g∥2
L2N
=
∫ ∞
R
∣∣g (r )∣∣2r N−1 dr.
When there is no ambiguity on the dimension, we will drop the index N . We will consider the
radial wave equation in space dimension N :
∂2t u −∂2r u −
N −1
r
∂r u = 0, r > R +|t |, (1)
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1458 Charles Collot et al.
with initial data
(u,∂t u)↾t=0 = (u0,u1) ∈ Ḣ 1(R)×L2(R). (2)
Extending the initial data (u0,u1) to an element of Ḣ 1×L2(RN ), we see that the solution u of (1), (2)
is defined without ambiguity on
{
r > R+|t |}. Indeed, by finite speed of propagation the restriction
of u to this set is independent of the choice of the extension of (u0,u1).
We define the following subspace of Ḣ 1N (R) (respectively L
2
N (R)):
PN (R) = span
{
1
r N−2ℓ
∣∣∣∣ ℓ= 1, . . . ,⌊ N +14
⌋}
,
QN (R) = span
{
1
r N−2ℓ
∣∣∣∣ ℓ= 1, . . . ,⌊ N −14
⌋}
,
and note that PN (R) (respectively QN (R)) is exactly the intersection of the generalized kernel of∆
with Ḣ 1N (R) (respectively L
2
N (R)).
Theorem 1. There exists a constant C > 0, independent of N , with the following property. Assume
that N ≥ 3 is an integer, and R > 0. Let u be a solution of (1), (2).
• Assume that N is odd, and that (u0,u1) is in the orthogonal of PN (R)×QN (R) in Ḣ 1(R)×
L2(R). Then:∑
±
lim
t→±∞
∫ ∞
R+|t |
(
∂t ,r u(t ,r )
)2r N−1 dr ≥ ∥∥(u0,u1)∥∥2Ḣ 1(R)×L2(R). (3)
• Assume that N = 4k, k ∈N, and that u0 is orthogonal to PN (R) in Ḣ 1(R). Then:∑
±
lim
t→±∞
∫ ∞
R+|t |
(
∂t ,r u(t ,r )
)2r N−1 dr ≥ 1
C
∥u0∥2Ḣ 1(R). (4)
• Assume that N = 4k +2, k ∈N\ {0}, and that u1 is orthogonal to QN (R) in L2(R). Then:∑
±
lim
t→±∞
∫ ∞
R+|t |
(
∂t ,r u(t ,r )
)2r N−1 dr ≥ 1
C
∥u1∥2L2(R). (5)
The estimates of Theorem 1 are crucial for the proof of the soliton resolution for the energy
critical semi-linear wave equation: see [1,5,7,8].
Theorem 1 was proved in space dimension 3 in [6] and generalized to higher odd space
dimensions in [9]. The difficulty of the proof in [9] increases with the dimensions.
Channels of energy in even space dimensions were first studied in [2], where the limit of
the outer energy (the left-hand side of (4) and (5)) is computed in the case R = 0. Theorem 1
was proved for N = 4 and N = 6 in [5], using the formula in [2] and explicit computation. The
general case of Theorem 1 in even space dimensions was obtained in [10]. The proof in [10] relies
on explicit formulas for the solution of the wave equation whose complexity increases with the
dimension. We refer to [3,4] for other recent works on channels of energy.
In this note, we remark, using a short argument, that it is sufficient to prove Theorem 1 in space
dimensions 3 and 4. More precisely, the estimate (3) in all odd space dimensions follows from the
same estimate in space dimension 3, and the estimates (4) and (5) in all even space dimensions
follow from the estimate (4) in space dimension 4.
Theorem 1 in the case N = 3 (proved in [6]) is elementary: it follows immediately from the
explicit formula for radial solutions of (1), (2) for N = 3:
u(t ,r ) = 1
r
(
ϕ(r + t )−ϕ(t − r )), (6)
where
ϕ(η) = 1
2
ηu0(|η|)+ 1
2
∫ η
0
σu1(|σ|)dσ. (7)
We thus have a very simple proof of the odd dimension channels of energy estimate (3) in every
odd dimension.
Charles Collot et al. 1459
The fourth author is particularly pleased to honour Haïm with a short and elegant proof, which
was one of the pleasures he has cherished in mathematics.
2. Proof
The proof, by induction, is based on a transformation of the radial linear wave equation, that
was introduced in [9], and also used in [5] to deduce the case N = 6 from the case N = 4. More
precisely, we will introduce a transformation that maps a solution u of (1) to a solution v of:
∂2t v −∂2r v −
N −3
r
∂r v = 0, r > R +|t |. (8)
We prove Theorem 1 when N is even. The proof when N is odd is quasi identical.
We thus fix N ≥ 6 even and assume that the conclusion of the theorem holds with N−2 instead
of N . We consider a solution u of (1). We let
v(t ,r ) =
∫ ∞
r
ρ∂t u(t ,ρ)dρ, v0(r ) = v(0,r ), v1(r ) = ∂t v(0,r ). (9)
Then by (1),
∂r v =−r∂t u, ∂t v =−(N −2)u(t ,r )− r∂r u(t ,r ), (10)
so that v satisfies (8). Also, ∫ ∞
R+|t |
(∂r v)
2r N−3 dr =
∫ ∞
R+|t |
(∂t u)
2r N−1 dr (11)
and (by a straightforward integration by parts)∫ ∞
R+|t |
(∂t v)
2r N−3 dr =
∫ ∞
R+|t |
(∂r u)
2r N−1 dr − (N −2)(R +|t |)N−2u2(t ,R +|t |), (12)
and thus
lim
t→±∞
∫ ∞
R+|t |
(∂t v)
2r N−3 dr ≤ lim
t→±∞
∫ ∞
R+|t |
(∂r u)
2r N−1 dr (13)
(indeed it easy to show the equality between these two limits but the inequality (13) is sufficient
for our purpose).
We will denote by P⊥N (R) the orthogonal of PN (R) in Ḣ
1(R), and Q⊥N (R) the orthogonal of QN (R)
in L2(R).
Case 1. We assume N = 4k, k ∈N, u1 = 0 and u0 ∈ P⊥N (R). By (10), v0 = 0. We claim
v1 ∈Q⊥N−2(R), (14)
i.e. that v1 is orthogonal to 1/r 2 j , j ∈
{
k, . . . ,2k−2} in L2N−2(R). For such j , we compute, using (10),∫ ∞
R
v1(r )
1
r 2 j
r N−3 dr =−
∫ ∞
R
(
(N −2)u0(r )+ r∂r u0(r )
) 1
r 2 j
r N−3 dr.
We have ∫ ∞
R
r
du0
dr
1
r 2 j
r N−3 dr =
∫ ∞
R
du0
dr
1
r 2 j+1
r N−1 dr = 0,
since u0 ∈ P⊥N (R). On the other hand, integrating by parts, we obtain∫ ∞
R
u0(r )r
N−3−2 j dr =− 1
N −2−2 j
∫ ∞
R
du0
dr
1
r 1+2 j
r N−1 dr − 1
N −2−2 j R
N−2−2 j u0(R).
Furthermore
u0(R) =−
∫ ∞
R
du0
dr
dr =−
∫ ∞
R
du0
dr
1
r 4k−1
r N−1 dr = 0,
where the last equality follows again from the fact that u0 ∈ P⊥N (R), and thus that u0 is orthogonal
to 1/r 4k−2 in Ḣ 1N (R). This concludes the proof of (14).
1460 Charles Collot et al.
By (13), (14) and the induction hypothesis,∑
±
lim
t→±∞
∫ ∞
R
∣∣∂t ,r u(t ,r )∣∣2r N−1 dr ≥ 1
C
∥v1∥2L2N−2(R). (15)
By (12)
∥v1∥2L2N−2(R) = ∥u0∥
2
Ḣ 1N (R)
− (N −2)RN−2u20(R).
Since we have proved that u0(R) = 0, this implies the desired inequality (4).
Case 2. We assume N = 4k +2, u0 = 0 and u1 ∈Q⊥N (R). We claim
v0 ∈ P⊥N−2(R), (16)
that is v0 is orthogonal to
1
r 2 j
, k ≤ j ≤ 2k −1. Indeed, by (10), if k ≤ j ≤ 2k −1,∫ ∞
R
dv0
dr
1
r 2 j+1
r N−3 dr =−
∫ ∞
R
u1
1
r 2 j+2
r N−1 dr = 0,
by the assumption that u0 ∈Q⊥N (R). This proves (16). By (13), (16), and the induction hypothesis,∑
±
lim
t→±∞
∫ ∞
R+|t |
(
∂t ,r u(t ,r )
)2r N−1 dr ≥ 1
C
∥v0∥2Ḣ 1N−2(R). (17)
By (11), ∥v0∥2Ḣ 1N−2(R) = ∥u1∥
2
L2N (R)
, which yields (5) and concludes the proof.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] C. Collot, T. Duyckaerts, C. E. Kenig and F. Merle, “Soliton resolution for the radial quadratic
wave equation in space dimension 6”, Vietnam J. Math. 52 (2024), no. 3, pp. 735–773.
[2] R. Côte, C. E. Kenig and W. Schlag, “Energy partition for the linear radial wave equation”,
Math. Ann. 358 (2014), no. 3–4, pp. 573–607.
[3] R. Côte and C. Laurent, “Concentration close to the cone for linear waves”, Rev. Mat.
Iberoam. 40 (2024), no. 1, pp. 201–250.
[4] J.-M. Delort, “Microlocal partition of energy for linear wave or Schrödinger equations”,
Tunis. J. Math. 4 (2022), no. 2, pp. 329–385.
[5] T. Duyckaerts, C. E. Kenig, Y. Martel and F. Merle, “Soliton resolution for critical co-
rotational wave maps and radial cubic wave equation”, Commun. Math. Phys. 391 (2022),
no. 2, pp. 779–871.
[6] T. Duyckaerts, C. E. Kenig and F. Merle, “Universality of blow-up profile for small radial
type II blow-up solutions of the energy-critical wave equation”, J. Eur. Math. Soc. 13 (2011),
no. 3, pp. 533–599.
[7] T. Duyckaerts, C. E. Kenig and F. Merle, “Classification of radial solutions of the focusing,
energy-critical wave equation”, Camb. J. Math. 1 (2013), no. 1, pp. 75–144.
[8] T. Duyckaerts, C. E. Kenig and F. Merle, “Soliton resolution for the radial critical wave
equation in all odd space dimensions”, Acta Math. 230 (2023), no. 1, pp. 1–92.
[9] C. E. Kenig, A. Lawrie, B. Liu and W. Schlag, “Channels of energy for the linear radial wave
equation”, Adv. Math. 285 (2015), pp. 877–936.
[10] L. Li, R. Shen and L. Wei, “Explicit formula of radiation fields of free waves with applications
on channel of energy”, Anal. PDE 17 (2024), no. 2, pp. 723–748.


Remark on the energy channel property for the radial linear wave equation

Mumtaz Hussain

Rebecca Smith

Zhenliang Zhang

Fractional dimension of some exceptional sets in continued fractions

Jared Marx-Kuo

Comptes Rendus
Mathématique
Jared Marx-Kuo
The isospectral problem for p-widths: an application of Zoll metrics
Volume 363 (2025), p. 565-570
Online since: 5 June 2025
https://doi.org/10.5802/crmath.708
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2025, Vol. 363, p. 565-570
https://doi.org/10.5802/crmath.708
Research article / Article de recherche
Geometry and Topology / Géométrie et Topologie
The isospectral problem for p-widths: an
application of Zoll metrics
Le problème isospectral pour les p-largeurs : une
application des métriques de Zoll
JaredMarx-Kuo a
a 450 Jane Stanford Way, Stanford, United States of America
E-mail: jmarxkuo@gmail.com
Abstract. We pose the isospectral problem for the p-widths: is a Riemannian manifold (Mn+1, g ) uniquely
determined by its p-widths,
{
ωp (M , g )
}∞
p=1? We construct many counterexamples on S2 using Zoll metrics
and the fact that geodesic p-widths are given by unions of immersed geodesics.
Résumé. Nous posons le problème isospectral pour les p-largeurs : une variété Riemannienne (Mn+1, g ) est-
elle uniquement déterminée par ses p-largeurs,
{
ωp (M , g )
}∞
p=1 ? Nous construisons de nombreux contre-
exemples sur S2 en utilisant les métriques de Zoll et le fait que les p-largeurs géodésiques sont données par
des unions de géodésiques immergées.
Keywords. Min-max, p-widths, Zoll, geodesics.
Mots-clés. Min-max, p-largeurs, Zoll, géodesiques.
2020 Mathematics Subject Classification. 53C22.
Funding. The author was supported by a Stanford SGF fellowship and an NSF grant number DGE-1656518.
Manuscript received 13 June 2024, revised 19 November 2024 and 3 December 2024, accepted 3 December 2024.
1. Introduction
In [15], Gromov introduced the p-widths, {ωp }, of a Riemannian manifold, (M n+1, g ), as an
analogue of the spectrum of the Laplacian, {λk }, the discrete collection of eigenvalues. Intuitively,
one replaces the Rayleigh quotient in the definition of eigenvalues with the area functional in
the definition of ωp – see Section 2, as well as [10,16,17,19] for more background. The p-widths
have proven to be extremely useful due to the Almgren–Pitts/Marques–Neves Morse theory
program for the area functional (see [2,3,25–27,30,34,38]). In [24], Liokumovich–Marques–Neves
demonstrated that {ωp } satisfy a Weyl law, which later led to a resolution of Yau’s conjecture [37]:
any closed three-dimensional manifold must contain an infinite number of immersed minimal
surfaces. In some sense, the resolution was stronger than expected:
Theorem 1 ([8,21,28,31,36]). On any closed manifold (M n+1, g ) with 3 ≤ n + 1 ≤ 7, there exist
infinitely many embedded minimal hypersurfaces.
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
566 Jared Marx-Kuo
We emphasize the result of embedded surfaces over immersed ones, and also note that em-
bedded minimal hypersurfaces are automatically smooth when 3 ≤ n +1 ≤ 7. We recall the fol-
lowing timeline: Marques–Neves [28] first proved the result in the Ricg > 0 setting, Irie–Marques–
Neves [21] proved the result for generic metrics, Chodosh–Mantoulidis [8] proved the result us-
ing the Allen–Cahn equation for a generic metric when n +1 = 3, and Song [36] proved the result
for non-generic metrics. We also note the work of Li [23] who showed the existence of infinitely
many immersed minimal surfaces with optimal regularity when n + 1 ≥ 8. We refer the reader
to the introduction of [9] for a more thorough overview of the history and applications of the
p-widths.
Recall the isospectral problem for the Laplacian:
Problem 2. Is a closed manifold, (M n+1, g ), uniquely determined (up to isometry) by its discrete
spectrum, {λk }?
Informally, this is known as “can you hear the shape of a drum?” and was made famous
by Kac [22]. In [33], Milnor constructed two 16-dimensional tori which are isospectral but not
isometric, providing the first resolution of Problem 2. Gordon–Webb–Wolpert [12] constructed
simpler examples within the class of polygons in R2. By contrast, Berger [6] showed that S2 is
spectrally rigid (see also Hersch [20] who deduced the same from only the first 3 eigenvalues).
Given the analogy between {ωp } and {λp }, it is natural to pose the p-width isospectral problem:
Problem 3. Is a closed manifold, (M n+1, g ), uniquely determined (up to isometry) by the values
ωp (g )?
We show that the answer is “no”:
Theorem 4. For any p > 0, ωp is constant on any fixed connected component of the space of Zoll
metrics on S2.
As a corollary, we can compute the p-widths explicitly for new metrics.
Corollary 5. Let g be any Zoll metric on S2 which lies in the connected component of ground in the
space of Zoll metrics on S2. Then ωp (g ) =ωp (ground) = 2π⌊pp⌋ for all p.
The above gives rise to a continuous deformation of p-width isospectral metrics (com-
pare [13]). In particular, any odd function f : S2 → R gives rise to a 1-parameter family of geo-
metrically distinct Zoll metrics near the round metric (see [18]). The p-widths of (S2, ground) were
computed explicitly in [9], and this was the first example of the p-width spectrum being known
for all p. Corollary 5 now gives uncountably more manifolds where the full p-width spectrum is
known.
The proof of Theorem 4 uses the defining properties of Zoll metrics on S2, which are metrics
for which all geodesics (with multiplicity one) are simple and of length 2π (note this choice of
length, as opposed to any constant c, agrees with [18]). These were first constructed by Zoll [39],
and we refer the reader to [7,11,14,18] for a by no means complete list of background sources.
The proof utilizes a recent result of Chodosh–Mantoulidis [9], which states that eachωp (M 2, g ) is
achieved by a union of closed, potentially immersed geodesics.
This paper is organized as follows: in Section 2, we briefly define the p-widths, and recall the
results of Chodosh–Mantoulidis [9]. We also define Zoll metrics and list some relevant properties.
In Section 3 we prove the main theorem, and in Section 4 we pose some open questions.
2. Background
We recall the essential notation from the min-max theory of Almgren–Pitts [2] and Marques–
Neves [29]. Let M be an (n+1)-dimensional manifold, let X ⊆ [0,1]k denote a cubical subcomplex
Jared Marx-Kuo 567
of the k-dimensional unit cube (cf. [29, Section 2.2]), and let Zn(M ;Z2) denote the set of n-cycles
mod Z2 – informally, a cycle is the boundary of a nice (Caccioppoli) open set. In [2], Almgren
showed that Zn(M ;Z2) is weakly homotopic to RP∞. We let λ ∈ H 1
(
Zn(M ;Z2),Z2
) ∼= Z2 denote
the generator of this group, so that λ
p
generates H p
(
Zn(M ;Z2),Z2
) ∼= Z2. For T ∈ Zn(M ;Z2),
there is a natural associated varifold to T and we let ∥T ∥ denote the corresponding Radon
measure, with ∥T ∥(U ) denoting the integral of the measure over U . We further let M(T ) = ∥T ∥(M).
Definition 6 ([28, Definition 4.1]). A map Φ : X →Zn(M ;Z2) is a p-sweepout if it is continuous
(with the standard flat norm topology on Zn(M ;Z2)) andΦ∗(λ
p
) ̸= 0 ∈ H p (X ).
Definition 7 ([28, Section 3.3]). A map Φ : X → Zn(M ;Z2) is said to have no concentration of
mass if
lim
r→0 sup
{
∥Φ(x)∥(Br (p)) : x ∈ X , p ∈ M}= 0.
Definition 8 ([16,19]). We define Pp =Pp (M) to be the set of all p-sweepouts, out of any cubical
subcomplex X , with no concentration of mass. The p-width of (M , g ) is
ωp (M , g ) := inf
Φ∈Pp
sup
{
M
(
Φ(x)
)
: x ∈ domain(Φ)
}
.
To see the analogy between ωp and λp , consider the following characterization of λp (M) as
λp (M) := inf
V ⊆C∞(M)
dimV =p+1
sup
f ∈V \{0}
∫
M |∇ f |2∫
M f
2
.
Note the parallels between the min-max definitions of ωp ,λp , with Φ∗(λ
p
) ̸= 0 being analogous
to dim(V ) = p +1.
Having defined the p-widths, one expects from the min-max theory of Almgren–Pitts [3,34]
that ωp = ∑i M(Vi ) where Vi are stationary varifolds for M. When 3 ≤ n + 1 ≤ 7, the regularity
theory of Schoen–Simon [35] shows that such varifolds are induced by smooth minimal hyper-
surfaces, potentially with multiplicity (see also [27, Section 3]). However, when n + 1 = 2, the
Vi may correspond to geodesic networks, as opposed to unions of closed geodesics (cf. [27, Re-
mark 1.1] and [9, Section 5], as well as [32, Figure 1] for a visual). In [9], Chodosh–Mantoulidis
used the Allen–Cahn equation to prove that the p-widths can always be achieved by a union of
closed geodesics:
Theorem 9 ([9, Theorem 1.2]). Let (M 2, g ) be a closed Riemannian manifold. For every p ∈ Z+,
there exists a collection of primitive, closed geodesics {σp, j } such that
ωp (M , g ) =
N (p)∑
j=1
m j ·Lg (σp, j ), m j ∈Z+.
Here, “primitive” means that the geodesic is connected and transversed with multiplicity one.
They also showed:
Theorem 10 ([9, Theorem 1.4]). For g0 the unit round metric on S2
∀ p ∈N, ωp (S2, g0) = 2π⌊pp⌋.
We now collect relevant facts about Zoll metrics on S2.
Definition 11. A metric, g , on S2is called a Zoll metric if all of the primitive geodesics are closed
and of length 2π.
Zoll [39], Funk [11], Guillemin [18], and Gromoll–Grove [14] each contributed significantly
to the theory of Zoll metrics. In particular, Gromoll–Grove showed that all of the primitive
geodesics on a Zoll surface are simple. Furthermore, the combined work of Guillemin and Funk
568 Jared Marx-Kuo
demonstrated that the tangent space of Zoll metrics at the round metric is given by odd functions
f : S2 → R. We also mention a higher dimensional analogue of these metrics constructed by
Ambrozio–Marques–Neves [4].
Let Z (S2) denote the space of Zoll metrics on S2, Zα, an arbitrary connected component of
Z (S2), and Z0, the connected component containing g0, the unit round metric.
3. Proof of main theorem
Let M∞(S2) denote the set of all smooth metrics on S2, and let g0 denote the round metric. For
fixed p, ωp : M∞(S2) →R is continuous due to the following lemma of Marques–Neves–Song:
Lemma 12 ([31, Lemma 1]). Let C1 <C2. There exists a K (C1,C2), such that
|ωp (g )−ωp (g ′)| ≤ K (C1,C2)p1/2∥g − g ′∥C 0(g0)
for any g , g ′ such that C1g0 ≤ g ≤C2g0 and C1g0 ≤ g ′ ≤C2g0.
Take any Zoll metric, g , on S2. Applying Theorem 9, we have
ωp (S
2, g ) =
N (p)∑
j=1
m j ·Lg (σp, j )
= 2π
N (p)∑
j=1
m j ∈ 2πZ+
since each σp, j is primitive and Lg (σp, j ) = 2π by definition of Zoll metric. Let Zα denote a fixed,
connected component of Z . Consider the restriction of ωp : Zα → R. Its image is discrete, lying
inside 2πN, butωp is continuous in each component due to Lemma 12, thus the map is constant
on connected components, proving Theorem 4.
4. Open questions
Having shown a counterexample to p-width spectral rigidity, it would be interesting to know any
of the following:
• Do there exist counterexamples to the isospectral problem in dimensions 3 ≤ n +1 ≤ 7?
The minimal hypersurfaces which achieve the p-widths behave differently (e.g. are em-
bedded and disjoint) in these dimensions compared to n +1 = 2, where self-intersecting
geodesics, or two geodesics intersecting may occur. See [1,32] which highlight the dif-
ferences between multiplicity, immersion vs. embeddedness, and index bounds of the
underlying minimal objects in dimensions n +1 = 2 vs. 3 ≤ n +1 ≤ 7.
• Are there other surfaces for which (Σ2, g ) is p-width spectrally rigid up to isometry? We
note that shortly after the release of this work, Ambrozio–Marques–Neves [5] demon-
strated that RP2 with the round metric is p-width spectrally rigid.
• Do there exist other reasonable geometric conditions that one can impose on a manifold
so that isospectral rigidity holds?
Acknowledgments
The author thanks Otis Chodosh for his conversations on p-widths and Lucas Ambrozio for his
conversations on Zoll metrics. The author is also thankful to the referee for their in-depth and
thoughtful feedback.
Jared Marx-Kuo 569
Declaration of interests
The author does not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and has declared no affiliations other than their research
organizations.
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