Comptes Rendus Mathématique

Rodolphe Richard

Andrei Yafaev

Comptes RendusMathématiqueRodolphe Richard and Andrei YafaevGeneralised André–Pink–Zannier conjecture for Shimura varieties of abelian typeVolume 363 (2025), p. 873-878Online since: 21 July 2025https://doi.org/10.5802/crmath.751This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2025, Vol. 363, p. 873-878https://doi.org/10.5802/crmath.751Research article / Article de rechercheAlgebraic geometry, Number theory / Géométrie algébrique, Théorie des nombresGeneralised André–Pink–Zannier conjecturefor Shimura varieties of abelian typeConjecture d’André–Pink–Zannier généralisée pour lesvariétés de Shimura de type abélienRodolphe Richard a and Andrei Yafaev ∗,aa UCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UKE-mails: r.richard@ucl.ac.uk, yafaev@ucl.ac.ukAbstract. This note describes the results of [6]. The main result is the proof of the Generalised André–Pink–Zannier conjecture in Shimura varieties of abelian type. The core result is a lower bound, in termsof height functions defined in [7], for the sizes of Galois orbits of points in generalised Hecke orbits, which isunconditional for Shimura varieties of abelian type.Résumé. Dans cette note nous décrivons les résultats de [6]. Le résultat principal est la preuve de la Conjec-ture d’André–Pink–Zannier genéralisée pour les variétés de Shimura de type abélien. Le résultat centralénonce des bornes inférieures, en termes des fontions hauteurs de [7], pour la taille des orbites galoisiennesdans une orbite de Hecke généralisée, qui sont inconditionnelles pour les variétés de Shimura de type abelien.Keywords. Shimura varieties, Hecke orbits, Zilber–Pink, heights, Siegel sets, Mumford–Tate conjecture, adeliclinear groups.Mots-clés. Variétés de Shimura, orbites de Hecke, Zilber–Pink, hauteurs, ensembles de Siegel, conjecture deMumford–Tate, groupes linéaires adéliques.2020 Mathematics Subject Classification. 03C64, 11G18, 11G50, 11F80, 14L30, 20G35, 15A16, 14G35.Funding. This project was funded by Leverhulme Trust grant RPG-2019-180.Manuscript received 21 December 2022, revised 17 November 2023 and 1 May 2025, accepted 16 April 2025.1. Statements of resultsIn this note we describe the results of [6], which is the sequel to [7]. A previous note [8] describedthe results of [7]. We refer to [2] for the notions related to Shimura varieties, and to [9, Section 3]for the notion of weakly special subvarieties. We study the following conjecture, introduced inthe following form in [7]. We refer to the previous note [8] for the conjecture and for the notionsinvolved.Conjecture 1 (Generalised André–Pink–Zannier Conjecture [7, Conjecture 1.1]). Let S be aShimura variety and Σ a subset of a generalised Hecke orbit in S. Then, irreducible componentsof the Zariski closure of Σ are weakly special subvarieties.∗Corresponding authorISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/874 Rodolphe Richard and Andrei YafaevOur most notable result is the following. We refer to [2] for the notion of Shimura of abeliantype. This is the most important class of Shimura varieties, and it contains in particular themoduli spaces of abelian varieties Ag ,d .Theorem 2. If S is of abelian type, and Σ is a subset of a generalised Hecke orbit in S, then theZariski closure of Σ is a finite union of weakly special subvarieties.The main strategy underlying the proof of Theorem 2 is the Pila–Zannier strategy in the formlaid out in [7]. This relies on a variant [5] of Pila–Wilkie theorem, some functional transcendenceresults [4] and the so-called “geometric part of André–Oort conjecture” [9]. The article [7]introduces some height functions with good properties which are required for this strategy towork. We refer to [8] for more details.A main difficulty in this approach is to obtain lower bounds for the sizes of some Galois orbits.This is Theorem 7. The use of Theorem 5, which is deduced from Faltings’ theorems, allows usto reduce Theorem 2 to Theorem 6. The proof of Theorem 6 follows from [7] and Theorem 7.Theorem 7 is reduced to Theorem 8, whose proof is discussed in Section 2. This proof relies onthe tool presented in Section 3.1.1. Main resultsTheorem 2 follows from two statements. The first one, Theorem 5, is a refined version ofFaltings theorems’ on Tate conjecture for abelian varieties. We encode this refined form intothe Definition 3 of “uniform integral Tate property”. The second statement, Theorem 6, provesConjecture 1 assuming that a point s ∈Σ satisfies this uniform integral Tate property.In order to state Definition 3 we let (G , X ) be the Shimura datum (G , X ) and we let K be theopen compact subgroup of G(A f ) such that S = ShK (G , X ). We write s = [x,1] ∈ S with x ∈ X . Wedenote by M ≤G the Mumford–Tate group of x, choose a field E ≤ C such that s ∈ S(E) and suchthat we have a Galois representation, as in [7, Section 3],ρx : Gal(E/E) → M(A f )∩K . (1)The following definition makes sense for two Q-algebraic groups M ≤ G and a faithful repre-sentation ρG : G → GL(d). It does not depend on the choice of ρG .Definition 3 (Uniform integral Tate property). Let U ≤ M(A f ) be a compact subgroup. Let Up :=U ∩M(Qp ) and, for p ≫ 0, let U (p) ≤ GL(d ,Fp ) be the image of Up .We say that U satisfies the uniform integral Tate property with respect to M , G and ρG if thefollowing is satisfied.(1) For every p,ZGQp (Up ) = ZGQp (Up 0) = ZG (M)Qp (2a)andthe action of Up onQdp is semisimple.1 (2b)(2) For every D, there exists an integer M(D) such that, for every p ≥ M(D) and every U ′ ≤U (p)of index[U (p) : U ′]≤ D, we haveZGFp (U′) = ZGFp (MFp ) (3a)andthe action of U ′ on Fpdis semisimple. (3b)Definition 4. We say that x0 satisfies the uniform integral Tate conjecture if U = ρx0(Gal(E/E))satisfies the uniform integral Tate property with respect to M, G in the sense of Definition 3.1Also known as completely reducible.Rodolphe Richard and Andrei Yafaev 875The following is deduced (see [6, Section 6.2]) from Faltings’ theorems, a specialisation argu-ment of Noot, and a result of Serre.Theorem 5. In a Shimura variety of abelian type, every point satisfies the uniform integral Tateconjecture in the sense of Definition 4.Our main result towards Conjecture 1 is the following. We recall that a generalised Hecke orbitis a finite union of “geometric Hecke orbits” ([7, Definition 2.3 and Theorem 2.4]).Theorem 6. Let s0 = [x0,1] be a point in a Shimura variety ShK (G , X ), and assume the uniformintegral Tate conjecture for x0 in X in the sense of Definition 4. Let Z be a subvariety whoseintersection with the geometric Hecke orbit of s0 is Zariski dense in Z . Then, Z is a finite unionof weakly special subvarieties of S.We observe that the “uniform integral Tate property” follows from the “weakly adelicMumford–Tate property” of [7]. As a result, Theorem 6 supersedes the main result of [7].2. Polynomial Galois boundsThe proof of Theorem 6 is found in [6, Section 3]. It follows [7], which uses Pila–Zannier strategy,as summarised in [8]. The main new difficulty which is dealt with in [6] is to prove lower boundson Galois orbits assuming the uniform integral Tate property.Theorem 7 (Galois bounds in Shimura varieties). Let S = ShK (G , X ) be a Shimura variety,let M ≤ G be the Mumford–Tate group of an element x ∈ X , and denote by W the conjugacyclass G ·φ0 ≃ G/ZG (M) of the inclusion φ0 : M → G (viewed as a Q-algebraic variety). We chooseintegral structures mZ of m and gZ of g, and consider the “finite height function” H f : W (Q) →Z≥1defined byH f (φ) = min{k ∈Z≥1∣∣ dφ(mZ) ≤ 1k ·gZ},where dφ : m→ g denotes the differential of φ : M →G.We define W + :=G(R)+ ·φ0 ⊆W (R) and W (Q)+ :=W +∩W (Q).Let E ≤C be a subfield of finite type overQ and such that x0 satisfies Definition 4.Then, as functions of φ ∈W (Q)+,#Gal(E/E) · [φ◦x0,1] ≈ H f (φ).We recall that{[φ◦x0,1]∣∣φ ∈W (Q)+} is the geometric Hecke orbit of x0.Using [7, Section 3.3], we deduce Theorem 7 from Theorem 8 applied to U as in Definition 4.The assumption (1) in Theorem 8 follows from the properties of canonical models of Shimuravarieties. The assumption (2) in Theorem 8 follows from the assumption that x0 satisfies Defini-tion 4.We believe Theorem 8 to be of independent interest.Theorem 8 ([6, Theorem 5.1]). Let M ≤G be connected reductive Q-groups. Let U ≤ M(A f ) be asubgroup satisfying the following.(1) The image of U in M ab is MT in M ab in the sense of [7, Defintion 6.1].(2) The group U satisfies the uniform integral Tate property as in Definition 3.Denote by φ0 : M → G the identity homomorphism and W = G ·φ0 its conjugacy class. Then as φvaries in W (A f ), for any compact open subgroup K ≤G(A f ), where KM = K ∩M(A f ), we have[φ(U ) :φ(U )∩K ]≈ [φ(KM ) :φ(KM )∩K ]≽ H f (φ).as functions W (A f ) →Z≥1.876 Rodolphe Richard and Andrei YafaevTheorem 8 can be studied prime by prime. Without loss of generality, we assume K = ∏p Kpwhere Kp := GL(d ,Zp )∩G(Qp ). We write M(Zp ) := GL(d ,Zp )∩M(Qp ). It is enough to prove thefollowing.• For every prime p, we have∃ a(p), c(p) ∈R>0, ∀φ ∈W (Qp ),[φ(U ) :φ(U )∩K ]≥ a(p) ·[φ(M(Zp )) :φ(M(Zp ))∩Kp]c(p). (4)• There exists a,c ∈R>0 such that, for all but finitely many primes,∀φ ∈W (Qp ),[φ(Up ) :φ(Up )∩Kp]≥ a · [φ(M(Zp )) :φ(M(Zp ))∩Kp]c . (5)At a given prime, property (2a) ensures that the conjugacy classes of M(Zp ) → G(Qp ) and Up ,→G(Qp ) are the same variety W . Then (4) follows from the functoriality of height.The main difficulty is to ensure uniform and sharper estimates (5) for p ≫ 0, and properties (2)in Definition 3 are crucial. This is the most technical part of [6].The approach is as follows (see [6, Section 5] for details).We construct from Up a vector, say v , in a representation of G . This representation is a productof copies of the adjoint representation G → GL(g) and of a fixed representation G → GL(d). Inparticular the weights are bounded. Moreover, the orbit (G ·v)(Qp ) can be identified with W (Qp ).Using the tools of [7], we can bound from below[φ(Up ) : φ(Up )∩Kp]in terms of the p-adicheight Hp (g · v) = max{1;∥g · v∥}, where φ ∈W (Qp ) is identified with g · v ∈ (G · v)(Qp ).Let v ′ be the vector constructed from the case Up = M(Zp ). The local height Hp (g · v ′) is thelocal factor of a global height function W (A f ) →Z≥1. Theorem 10 is used, for p ≫ 0, to comparethe local height Hp (g · v) = max{1;∥g · v∥} with the local height Hp (g · v ′) = max{1;∥g · v ′∥}.Because the constant C in (13) does not depend on p, but only on the weights of the represen-tation, we obtain the uniformity in (5).3. p-adic Kempf–Ness theoremTheorem 9 is key in order to make use of part (2) of Definition 3. Theorem 9 can be seen as ananalogue of Kempf–Ness theorem [3, Theorem 0.1(b)]. Kempf–Ness theorem deals with closedorbits in linear representations equipped with an archimedean metric. Theorem 9 is a p-adicanalogue. While Kempf–Ness theorem is easy to prove, our p-adic analogue requires a fine studyof the “good reduction” of the orbit under consideration and the use of group schemes.We believe this is a result of independent interest.Theorem 9 (p-adic Kempf–Ness Theorem). Let FZp ≤GZp ≤ GL(n)Zp be smooth reductive groupschemes such that FZp →GZp → GL(n)Zp are closed immersions and GZp is connected.Let v ∈Znp , denote by v ∈ Fnp its reduction and assume thatStabGQp (v) = FQp and dim(StabGFp (v))= dim(FFp ), (6)(using Krull dimensions) and assume that the orbitsGQp · v ⊆AnQp and GFp · v ⊆AnFp(7)are closed.Then, denoting by Zp [G/F ] := Zp [G] ∩Qp [G]F the algebra of F -invariant functions G → A1defined over Zp , for every g ∈G(Qp ), we have,g · v ∈Zp n if and only if ∀ f ∈Zp [G/F ], f (g ) ∈Zp . (8)Moreover, Spec(Zp [G/F ])is smooth over Zp , and we have(G(Qp ) · v)∩Zp n =G(Zp ) · v. (9)Rodolphe Richard and Andrei Yafaev 877Remarks. Some of the hypotheses can be rephrased as follows.The Qp -algebraic groups F and G are reductive, the compact subgroups FZp (Zp ) ≤ F (Qp )and GZp (Zp ) ≤ G(Qp ) are hyperspecial subgroups, and we have FZp (Zp ) = F (Qp ) ∩ GL(n,Zp )and GZp (Zp ) =G(Qp )∩GL(n,Zp ).The property (7) is related to semi-stability and residual semi-stability of the vector v in thesense of [1].In (6), the hypothesis on dimensions means that StabGFp (v)0,red (the reduced subgroup of theneutral component) is equal to (FFp )0. Equivalently, StabGFp (v)0(Fp ) = F 0(Fp ). This is implied bythe stronger conditionStabG(Fp )(v) = F (Fp ) (10)and the stronger oneStabGFp (v) = FFp as group schemes. (11)3.1. Comparison of local height functionsTheorem 9 is a key result in proving the following, which can be seen as a refined, more precise,version of the functoriality of local heights.Theorem 10 (Local relative stability estimates). Let F ≤G ρ→ GL(d) be as in Theorem 9. Let v, v ′ ∈Zdp be non zero vectors, denote by v , v′ ∈ Fdp their reduction, and assume that(1) the orbits GQp · v and GQp · v ′ ⊆AdQp are closed subvarieties;(2) the stabiliser groups Fv := StabG (v),Fv ′ := StabG (v ′) satisfyFv = Fv ′ = F ;and that(3) the orbits GFp · v and GFp · v ′ ⊆AdFp are closed subvarieties;(4) the stabiliser groups Fv := StabG (v), Fv ′ := StabG (v ′) satisfy, as group schemes2,Fv = Fv ′ = FFp . (12)We denote by ∥·∥ : Cp d →R the standard ultrametric norm, and we define two functions G(Cp ) →Rgiven byHv : g 7→ max{1;∥g · v∥} and Hv ′ : g 7→ max{1;∥g · v ′∥}.Then, the functions hv = log Hv and hv ′ = log Hv ′ satisfyhv ≤C ·hv ′ and hv ′ ≤C ·hv , (13)in which C =C(Σ(ρ)) depends only on the set of weights of ρ (cf. [6, 6.1.1.2]).Declaration of interestsThe authors do not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and have declared no affiliations other than their researchorganizations.2It amounts to the property that Fv and Fv are smooth.878 Rodolphe Richard and Andrei YafaevReferences[1] J.-F. Burnol, “Remarques sur la stabilité en arithmétique”, Int. Math. Res. Not. (1992), no. 6,pp. 117–127.[2] P. Deligne, “Variétés de Shimura: interprétation modulaire, et techniques de construction demodèles canoniques”, in Automorphic forms, representations and L-functions (Proc. Sym-pos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proceedings of Symposiain Pure Mathematics, vol. XXXIII, American Mathematical Society, 1979, pp. 247–289.[3] G. Kempf and L. Ness, “The length of vectors in representation spaces”, in Algebraic geome-try (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Mathe-matics, vol. 732, Springer, 1979, pp. 233–243.[4] B. Klingler, E. Ullmo and A. Yafaev, “The hyperbolic Ax–Lindemann–Weierstrass conjecture”,Publ. Math., Inst. Hautes Étud. Sci. 123 (2016), pp. 333–360.[5] J. Pila, “On the algebraic points of a definable set”, Sel. Math., New Ser. 15 (2009), no. 1,pp. 151–170.[6] R. Richard and A. Yafaev, “Generalised André–Pink–Zannier Conjecture for Shimura vari-eties of abelian type”, 2021. Online at https://arxiv.org/abs/2111.11216v2.[7] R. Richard and A. Yafaev, “Height functions on Hecke orbits and the generalised André–Pink–Zannier conjecture”, 2021. Online at https://arxiv.org/abs/2109.13718v1.[8] R. Richard and A. Yafaev, “On the Generalised André–Pink–Zannier conjecture”. To appearin C. R. Math.[9] E. Ullmo, “Applications du théorème d’Ax–Lindemann hyperbolique”, Compos. Math. 150(2014), no. 2, pp. 175–190.

Generalised André–Pink–Zannier conjecture for Shimura varieties of abelian type

Pham Viet Hai

Waleed Noor

Osmar Reis Severiano

Cyclicity of composition operators on the Paley–Wiener spaces

Alain Bretto

Alain Faisant

François Hennecart

Comptes RendusMathématiqueAlain Bretto, Alain Faisant and François HennecartAbout Berge–Füredi’s conjecture on the chromatic index of hypergraphsVolume 363 (2025), p. 323-328Online since: 8 April 2025https://doi.org/10.5802/crmath.739This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2025, Vol. 363, p. 323-328https://doi.org/10.5802/crmath.739Research article / Article de rechercheCombinatorics / CombinatoireAbout Berge–Füredi’s conjecture on thechromatic index of hypergraphsSur la conjecture de Berge–Füredi sur l’indicechromatique des hypergraphesAlain Bretto a, Alain Faisant b and François Hennecart ∗,ba Université Caen Normandie, GREYC CNRS-UMR 6072, Caen, Franceb Université Jean Monnet, ICJ UMR5208, CNRS, École Centrale de Lyon, INSA Lyon, Université ClaudeBernard Lyon 1, 42023 Saint-Étienne, FranceE-mails: alain.bretto@info.unicaen.fr, faisant@univ-st-etienne.fr, francois.hennecart@univ-st-etienne.frAbstract. We prove that the chromatic index of a hypergraph H satisfies the Berge–Füredi conjectured boundq(H ≤ ∆([H2])+ 1 under certain hypotheses on the antirank ar(H ) or on the maximum degree ∆(H ).This provides sharp information in connection with the Erdős–Faber–Lovász conjecture which deals withthe coloring of a family of cliques that intersect pairwise in at most one vertex.Résumé. Nous montrons que l’indice chromatique d’un hypergraphe satisfait la borne conjecturée indépen-damment par Berge et Füredi, q(H ≤∆([H2])+1 sous certaines conditions portant sur son antirang ar(H )ou son degré maximum ∆(H ). Ces résultats fournissent des informations sur la conjecture de Erdős, Faberand Lovász qui traite de la coloration d’une famille de cliques se coupant deux à deux en au plus un sommet.Keywords. Hypergraphs, hyperedge coloring, chromatic index.Mots-clés. Hypergraphes, coloration des hyperarêtes, indice chromatique.2020 Mathematics Subject Classification. 05C15.Manuscript received 12 March 2024, revised 10 March 2025, accepted 11 March 2025.1. IntroductionBerge–Füredi’s conjecture (cf. [1,2,6] and Meyniel, unpublished communication) extends tolinear hypergraphs Vizing’s theorem on the chromatic index of graphs (see [3, Theorem 7.2.2]). Itasserts that the chromatic index of any linear hypergraph H = (V ,E) without loop satisfiesq(H ) ≤∆([H ]2)+1. (1)This bound was first conjectured around 1986. Since then, to our knowledge, only few advanceshave been made. It has been recently proved [8] that (1) holds whenever ∆([H ]2) ≤ 5. In thepresent paper we obtain new conditional results where the antirank ar(H ) and the maximumdegree ∆(H ) are involved together with the maximum degree ∆([H ]2)of the 2-section of H .If H is linear then∆([H ]2)≤ |V |−1. Hence bound (1) implies Erdős–Faber–Lovász conjectureasserting that q(H ) ≤ |V | for any linear hypergraph H .∗Corresponding authorISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/324 Alain Bretto, Alain Faisant and François Hennecart2. Notation and statementsLet H = (V ,E) be a hypergraph. For x ∈V and e ∈ E , we letdegH (x) =∣∣∣{a ∈ E : x ∈ a}∣∣∣, dH (e) = ∣∣∣{a ∈ E ∖ {e} : a ∩e ̸=∅}∣∣∣,be the degree of vertex x and the degree of hyperedge e in H , respectively. Then, we definerespectively the rank, the antirank, the maximum degree and the minimum degree of H byr(H ) = maxe∈E|e|, ar(H ) = mine∈E|e|, ∆(H ) = maxx∈VdegH (x), δ(H ) = minx∈V degH (x).A loop in a hypergraph H is a hyperedge e such that |e| = 1.The 2-section of H = (V ,E) is the loopless multigraph [H ]2 = (V , A) with k many edgesbetween x, y ∈V , x ̸= y , if there are k hyperedges in H containing both x and y . We have∆([H ]2)≥ (ar(H )−1)∆(H ). (2)The chromatic index of H , denoted by q(H ), is the least integer q such that there exists a mapc : E → {1,2, . . . , q} satisfying∀ e,e ′ ∈ E , (c(e) = c(e ′) =⇒ e ∩e ′ =∅ or e = e ′).The map c is called a hyperedge-coloring of H and the image of E through c forms the set ofcolors.A hypergraph H = (V ,E) is said linear if for any pair of distinct hyperedges e,e ′ ∈ E , we have|e ∩e ′| ≤ 1. In this case [H ]2 reduces to a simple graph.Recently it has been established that if H is a linear hypergraph satisfyingar(H ) ≥√|V |, (3)then q(H ) ≤ |V | (cf. [7]). This upper bound has been improved in [4] where it is shown that underthe same hypotheses on H , then (1) actually holds. In this note we show that condition (3) canbe weakened by proving the following sharper result.Theorem 1. Let H = (V ,E) be a loopless hypergraph such that ar(H ) ≥√∆([H ]2)+1. Thenq(H ) ≤∆([H ]2)+1.Note that we do not assume in the above statement that H is linear. We will propose anapproach supported by the notion of critical hypergraph (see also Section 6.1 for additionalcomments).We will prove the following key result which has its own interest. H is said k-uniform (resp.d-regular) if ∀ e ∈ E , |e| = k (resp. ∀ x ∈V , degH (x) = d).Theorem 2. Let k ≥ 2 and H = (V ,E) be a k-uniform and (k+1)-regular linear hypergraph. Thenq(H ) ≤∆([H ]2)+1.The following corollary will be derived from Theorem 1.Corollary 3. Let H = (V ,E) be a loopless hypergraph such that∆(H ) ≤√∆([H ]2)+1+1. (4)Then q(H ) ≤∆([H ]2)+1.Alain Bretto, Alain Faisant and François Hennecart 3253. Proof of Theorem 1In [4] we exploited the notion of critical hyperedge. A hyperedge e is said critical if q(H ∖ e) =q(H ) − 1. Hypergraph H is a critical hypergraph if all its hyperedges are critical. Since theaddition of e to H ∖ e needs a new color, e must intersect every color of H ∖ e. So we havethe following.Lemma 4 ([4, Lemma 3.2]). Let H = (V ,E) be a hypergraph without loop and e ∈ E be a criticalhyperedge. Then q(H )−1 ≤ dH (e).We may assume that H is connected and critical since ar(H ′) ≥ ar(H ) and ∆([H ′]2) ≤∆([H ]2)for any partial hypergraph H ′ of H . Let e ∈ E be of minimum cardinality k := ar(H ).We havedH (e) =∑a∈E∖{e}a∩e ̸=∅1 ≤ ∑a∈E∖{e}|a ∩e| = ∑x∈V∑a∈E∖{e}a∋x1 = ∑x∈e(degH (x)−1), (5)and equality occurs when H is linear. Then by Lemma 4 we getq(H )−1 = q(H ∖e) ≤ dH (e) ≤ k(∆(H )−1)= (k −1)∆(H )− (k −∆(H )),soq(H ) ≤∆([H ]2)+1− (k −∆(H )). (6)Our hypothesis implies ∆([H ]2)≤ k2 −1. We distinguish two cases.• If ∆([H ]2) < k2 −1 we get by (2) that ∆(H ) < k +1 thus ∆(H ) ≤ k, yielding the requiredupper bound for q(H ) by (6).• Assume that ∆([H ]2)= k2 −1. We thus have to prove that q(H ) ≤ k2. If there exists e ∈ Esuch that |e| = k and dH (e) ≤ k2 −1 then q(H ) ≤ dH (e)+1 ≤ k2 by Lemma 4 and we aredone. OtherwisedH (e) ≥ k2 for any hyperedge e of cardinality k. (7)We shall see that H is a very special hypergraph.– Let e ∈ E be of cardinality k. By (2) we have ∆(H ) ≤ k +1, thus by (7) and (5) we getk2 ≤ dH (e) ≤∑x∈e(degH (x)−1)≤ k(∆(H )−1)≤ k(k +1−1) = k2.We inferdH (e) = k2 and degH (x)−1 = k for any x ∈ e. (8)– Let a ∈ E such that a ∩e ̸=∅. Let x ∈ a ∩e. We havek2 −1 =∆([H ]2)≥ deg[H ]2 (x) = ∑b∈Eb∋x(|b|−1)= |a|−1+ ∑b∈E∖{a}b∋x(|b|−1)≥ |a|−1+ (degH (x)−1)(k −1)= |a|−1+k(k −1)≥ k2 −1,giving |a| = k.– Since H is connected we infer that |a| = k for any a ∈ E . Hence H is k-uniform and(k + 1)-regular. By (5) and (8), we have |a ∩ e| = 1 whenever a ∩ e ̸= ∅. Thus H islinear. The proof will be completed when Theorem 2 will be established.326 Alain Bretto, Alain Faisant and François Hennecart4. Proof of Theorem 2By assumption we havek|e| = (k +1)|V |, ∆([H ]2)= k2 −1 and ∀ e ∈ E , dH (e) = k2. (9)Since H is linear, its line-graph L(H ) is the simple graph whose vertices are hyperedges of Hand such that there is an edge between e,e ′ ∈ E , e ̸= e ′, if e ∩ e ′ ̸=∅. We have degL(H )(e) = dH (e)for any e ∈ E .By (9), each vertex e of L(H ) has degree degL(H )(e) = k2. Hence L(H ) is k2-regular. Sinceq(H ) coincides with χ(L(H )), namely the chromatic number of the simple graph L(H ), we inferfrom Brooks’ theorem (see [3, Theorem 7.1.9, p. 238]) that eitherq(H ) ≤∆(L(H ))= k2 =∆([H ]2)+1or L(H ) is complete or an odd-length cycle. If L(H ) were a complete graph, we would have|e| = k2 +1 implying by (9) that (k +1) divides k(k2 +1), which gives a contradiction since k ≥ 2.If L(H ) were a cycle, we would have dH (e) = degL(H )(e) = 2 for any e ∈ E , a contradiction. Thisends the proof of Theorem 2.5. Proof of Corollary 3Let H ′ = (V ,E ′) be a critical partial hypergraph of H such that q(H ′) = q(H ).• If ar(H ′) ≥√∆([H ′]2)+1 we infer from Theorem 1 thatq(H ) = q(H ′) ≤∆([H ′]2)+1 ≤∆([H ]2)+1.• If ar(H ′) <√∆([H ′]2)+1 then we take e ∈ E ′ such that |e| = ar(H ′). By Lemma 4 andusing (5) with H ′ in place of H , we getq(H )−1 = q(H ′)−1 ≤ dH ′ (e)≤ ∑x∈e(degH ′ (x)−1)≤ |e|(∆(H ′)−1)= ar(H ′)(∆(H ′)−1)<√∆([H ′]2)+1(∆(H ′)−1)≤√∆([H ]2)+1(∆(H )−1)≤∆([H ]2)+1by (4). Hence q(H ) ≤∆([H ]2)+1 as wanted.6. Remarks and questions6.1.We should notice that when ar(H ) >√∆([H ]2)+1 then the desired bound q(H ) ≤∆([H ]2)+1can be equally deduced from the greedy algorithm for coloring hyperedges (see for instance [5,p. 586]) which yieldsq(H ) ≤ maxr(H )≥k≥ar(H )k⌊∆([H ]2)k −1 −1⌋+1.Here we use a different approach which allows us to solve in addition the equality case ar(H ) =√∆([H ]2)+1.Alain Bretto, Alain Faisant and François Hennecart 3276.2.When the maximum degree ∆(H ) satisfies ∆(H ) >√∆([H ]2)+1+ 1, Corollary 3 is no longerusable. However at the cost of an additional condition on the rank r(H ) = maxe∈E |e|, we cannotice that (1) remains valid for linear hypergraphs H . Let δ2 :=√∆([H ]2)and assume thatr(H ) = δ2 −u, u > 0,∆(H ) = δ2 +1+ v, v > 0.Thenq(H ) =χ(L(H ))≤∆(L(H ))+1 ≤ r(H )(∆(H )−1)+1 = (δ2 −u)(δ2 + v)+1.Thus (1) holds whenever (v −u)δ2 ≤ uv , that is 1u ≤ 1v + 1δ2 .6.3.If H is supposed linear, in the extremal case ∆([H ]2) = |V | − 1, ar(H ) = |V |1/2 in Theorem 1the authors obtained in [4] a much better upper bound for q(H ), namely q(H ) ≤ 12 (∆([H ]2)+1+√∆([H ]2)+1)+ 1. How can the upper bound for q(H ) be improved under the sole condi-tion ar(H ) =√∆([H ]2)+1?6.4.Theorem 2 deals with k-uniform and d-regular linear hypergraphs when d = k +1 and providesthe required bound (1). For general k,d satisfying k ≤ d − 2, the line-graph L(H ) is k(d − 1)-regular and Brooks’ theorem provides the (non sufficiently sharp) inequality q(H ) = χ(L(H )) ≤∆(L(H ))= k(d −1) while ∆([H ]2)= d(k −1).6.5.More generally let H be a k-uniform linear hypergraph with k ≥ 2 and set δ2 :=√∆([H ]2).Then (1) holds if at least one of the following conditions is satisfied:(1) k = 2: Vizing’s theorem (see [3, Theorem 7.2.2, p. 249]);(2) k ≥√∆([H ]2)+1: Theorem 1;(3) ∆([H ]2)= k2 which implies ∆(H ) ≤√∆([H ]2)+1: Corollary 3;(4) 3 ≤ k < δ2 and 1δ2−k ≤1∆(H )−δ2−1 +1δ2: see Section 6.2.In order to solve Berge–Füredi’s conjecture for all linear uniform hypergraphs, it thus remains toconsider the case where3 ≤ k < δ2 <∆(H )−1 and 1δ2 −k> 1∆(H )−δ2 −1+ 1δ2.We may likewise ask the following question: is Berge–Füredi’s conjecture in all its generalityimplied by its validity for linear uniform hypergraphs?Declaration of interestsThe authors do not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and have declared no affiliations other than their researchorganizations.328 Alain Bretto, Alain Faisant and François HennecartReferences[1] C. Berge, “On the chromatic index of a linear hypergraph and the Chvátal Conjecture”, Ann.N.Y. Acad. Sci. 555 (1989), pp. 40–44.[2] C. Berge, “Motivations and history of some of my conjectures”, Discrete Math. 165-166(1997), pp. 61–70.[3] A. Bretto, A. Faisant and F. Hennecart, Elements of Graph Theory: From Basic Concepts toModern Developments, EMS Textbooks in Mathematics, European Mathematical Society,2022, pp. xv+486.[4] A. Bretto, A. Faisant and F. Hennecart, “The Berge–Füredi Conjecture on the chromatic indexof hypergraphs with large hyperedges”, 2024. Online at https://arxiv.org/abs/2403.06850.[5] T. Dvořák, “Chromatic index of hypergraphs and Shannon’s Theorem”, Eur. J. Comb. 21(2000), pp. 585–591.[6] Z. Füredi, “The chromatic index of simple hypergraphs”, Graphs Comb. 2 (1986), no. 1,pp. 89–92.[7] Z. Wang, “The Erdős–Faber–Lovász Conjecture for gap-restricted hypergraphs”, Engineering16 (2024), pp. 47–59.[8] G.-H. Zhang and B. Skinner, “Chromatic index of simple hypergraphs”, Discrete Math. 343(2020), no. 12, article no. 112087 (7 pages).

About Berge–Füredi’s conjecture on the chromatic index of hypergraphs

Alonso Beaumont

Comptes RendusMathématiqueAlonso BeaumontOn the nilpotency of locally pro-p contraction groupsVolume 363 (2025), p. 267-270Online since: 8 April 2025https://doi.org/10.5802/crmath.728This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2025, Vol. 363, p. 267-270https://doi.org/10.5802/crmath.728Research article / Article de rechercheGroup theory / Théorie des groupesOn the nilpotency of locally pro-p contractiongroupsSur la nilpotence des groupes de contraction localementpro-pAlonso Beaumont aa IRMAR, Université de Rennes, 263 Avenue du Général Leclerc, 35042 Rennes Cedex, FranceE-mail: alonso.beaumont@univ-rennes.frAbstract. H. Glöckner and G. A. Willis have recently shown [2] that locally pro-p contraction groups arenilpotent. The proof hinges on a fixed point result [2, Theorem B]: if the local field Fp ((t )) acts on its d-thpower Fp ((t ))d additively, continuously, and in an appropriately equivariant manner, then the action has anon-zero fixed point. We provide a short proof of this theorem.Résumé. H. Glöckner et G. A. Willis ont récemment démontré [2] que les groupes de contraction localementpro-p sont nilpotents. Leur démonstration repose sur un résultat de point fixe [2, Theorem B]: si le corpslocal Fp ((t )) agit sur sa d-ième puissance Fp ((t ))d additivement, continûment et de manière convenablementéquivariante, alors l’action a un point fixe non nul. Nous présentons une démonstration courte de cethéorème.2020 Mathematics Subject Classification. 22D05, 20E36, 20F18.Manuscript received 8 January 2025, revised 30 January 2025, accepted 1 February 2025.1. IntroductionA locally compact topological group G is said to be a contraction group if it has a continuousautomorphism α whose forward orbits converge to the identity element e:∀ g ∈G , αn(g ) −→n→∞ e.This dynamical property has interesting connections with the purely group-theoretic notion ofnilpotency. This relationship is best illustrated in the case where G is a Lie group: a contractingautomorphism on G provides a positive gradation for its Lie algebra g, proving in particular that gis nilpotent (see [5]). A thorough study of totally disconnected locally compact (tdlc) contractiongroups is carried out in [1]. One of its main structural results asserts that any tdlc contractiongroup is a direct product of a torsion-free contraction group, which is always nilpotent, and atorsion contraction group [1, Theorem B]. The remaining questions concerning the torsion partwere treated in [2]. In order to accommodate for torsion, an additional hypothesis is needed: atdlc group is called locally pro-p if it contains an open pro-p subgroup. For example, analyticgroups over the field Qp of p-adic numbers, or the field Fp ((t )) of Laurent series with coefficientsISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/268 Alonso Beaumontin Fp , are locally pro-p. In [2] it is shown that locally pro-p contraction groups are nilpotent. Themajor part of the proof consists in establishing the following fixed point result [2, Theorem B]:Theorem 1. Let d ≥ 1 and consider a continuous action ψ : (Fp ((t )),+) × (Fp ((t ))d ,+) →(Fp ((t ))d ,+)by group automorphisms. If∀ x ∈ Fp ((t )), ∀ u ∈ Fp ((t ))d , ψ(t x, tu) = tψ(x,u)then the action has a non-zero fixed point.Glöckner and Willis’s proof of this result involves representing endomorphisms of the abeliangroup Fp ((t ))d as infinite block matrices satisfying certain conditions. We refer the reader toSection 3 of [2] for details. The purpose of this note is to provide an alternative, shorter proofof this result. It has two main ingredients: a study of Fp -representations of elementary abelianp-groups, and an application of Ascoli’s theorem to the automorphism group Aut(Fp ((t ))d).2. Proof of the theoremWe begin with a lemma. A linear representation V of a group G is said to be locally finite if itsfinitely generated subrepresentations are finite-dimensional. We denote by V G the set of G-fixedpoints in V .Lemma 2. Let G be an elementary abelian p-group, and V a locally finite representation of G overFp . If V is infinite-dimensional, then eitherV G or(V /V G)Gis infinite-dimensional.Proof. First, suppose that G is finite. We will show that V G is infinite-dimensional. In order togive a self-contained presentation, we will provide a direct proof of the lemma in this case1. WriteG = 〈g1, . . . , gr 〉. By the local finiteness of V , there is a strictly increasing family (Vn)n≥0 of finitesubrepresentations. For all n ≥ 0, write dn = dimVn and consider the sequence∀ i ∈ {0, . . . , p}, dn(i ) = dim(ker(g1 − id)i ∩Vn).By a standard application of the rank-nullity theorem, (dn(i + 1) − dn(i ))0≤i<p is a decreasingsequence. Moreover, dn(p) = dn since (g1− id)p = g p1 −id = 0. We deduce that dn(1) = dimV〈g1〉n ≥dn/p. The elements g1 and g2 commute, so we can consider the action of 〈g2〉 on V 〈g1〉. Byrepeating the argument above we obtain dimV 〈g1,g2〉n ≥ dn/p2 and by a finite induction,dimV G ≥ dimV Gn ≥ dn/pr −→n→+∞+∞.Now suppose that G is infinite, and that both V G and (V /V G )G are finite-dimensional. Thenthe preimage V ′ of (V /V G )G in V is finite-dimensional. Therefore, the G-action on V ′ factorsthrough a finite quotient of G . Since G is elementary abelian, this quotient is a direct summand.We can therefore write G =G0 ×G1, where G0 is finite and G1 acts trivially on V ′. In particular, G1acts trivially on V ′∩V G0 so V ′∩V G0 ⊂ (V G0 )G1 =V G . Hence,(V ′∩V G0)/V G = (V G0 /V G )G = 0.Suppose V G0 /V G is non-trivial. Since V G0 /V G is locally finite, it would have a non-trivial finite-dimensional subrepresentation, and hence a non-zero G-fixed point, because G acts as anabelian unipotent group. We deduce that V G0 /V G = 0. This is absurd since G0 is finite and thusV G0 is infinite-dimensional. □1An alternative line of argument is as follows. If G is finite then Fp [G] is a finite-dimensional Fp -algebra. If the Fp [G]-module V G were finite then its essential extension V would also be finite (see for instance [4]), so V G must be infinite-dimensional.Alonso Beaumont 269Let d ≥ 1. We consider Fp ((t ))d as an additive tdlc group, equipped with a continuousautomorphism given by scalar multiplication by t : u = (u1, . . . ,ud ) 7→ tu = (tu1, . . . , tud ). Thegroup Aut(Fp ((t ))d)of continuous automorphisms is endowed with the compact-open topology.In particular, a continuous action G × Fp ((t ))d → Fp ((t ))d by group automorphisms induces acontinuous morphism ϕ : G → Aut(Fp ((t ))d ). We restate Theorem 1 in the following manner:Theorem 3. A continuous morphism ϕ : Fp ((t )) → Aut(Fp ((t ))d)satisfying∀ x ∈ Fp ((t )), ∀ u ∈ Fp ((t ))d , ϕ(t x)(u) = tϕ(x)(t−1u) (1)has a non-zero fixed point: there exists u ∈ Fp ((t ))d \{0} such that ϕ(x)(u) = u for all x ∈ Fp ((t )).Proof. The proof will be done in two steps. We will first construct a nested family of invariantsubgroups, and then we will use it to locate increasingly smaller orbits that will converge to afixed point. We will use the following notation: G = ϕ(Fp ((t ))), Gℓ = ϕ(t−ℓFpt) for all ℓ ≥ 0,B = Fptd , and S = B\tB .Step 1. Let M be the largest G-invariant subgroup contained in B . It can be defined as the groupgenerated by all G-invariant subgroups in B . We will prove the following: M is non-trivial andclosed, and A :=⋃n≥0 t−n M is a G-invariant Fp ((t ))-subspace of Fp ((t ))d in which M is open.Since ϕ is continuous, Gℓ is compact, and so by Ascoli’s theorem [3, Theorem 47.1] it is anequicontinuous family of automorphisms of Fp ((t ))d . In particular, there exists a neighbourhoodof 0 whose Gℓ-orbit is contained in B . The subgroup generated by this orbit is therefore open, Gℓ-invariant, and contained in B . Let Mℓ be the largest Gℓ-invariant subgroup in B . By the previousargument Mℓ contains an open subgroup and thus it is itself open. By property (1), we have∀ x ∈ t−ℓFpt, ϕ(x)(t−1Mℓ) = t−1ϕ(t x)(Mℓ) = t−1Mℓand so t−1Mℓ is also Gℓ-invariant. This means that Mℓ and S have a non-empty intersection, forotherwise the subgroup Mℓ+ t−1Mℓ, which is Gℓ-invariant and strictly larger that Mℓ, would becontained in B . Therefore, the decreasing sequence (Mℓ∩S)ℓ≥0 consists of non-empty compactsets, and hence its intersection is non-empty. We have proven thatM := ⋂ℓ≥0Mℓis a non-trivial subgroup of Fp ((t ))d . Every G-invariant subgroup of B is Gℓ-invariant, hencecontained in Mℓ for each ℓ. Therefore M is the largest G-invariant subgroup of B . Again using (1),we see that t M is G-invariant and so t M ⊂ M . As an intersection of closed subgroups, M is closed.We deduce that M is an Fpt-module and thusA := ⋃n≥0t−n Mis a G-invariant Fp ((t ))-subspace of Fp ((t ))d . In particular, A is closed in Fp ((t ))d and therefore it isa Baire space. As such, some t−n M , or equivalently all of them, are open in A.Step 2. For all n ≥ 0, let Vn := t−n M/M . This quotient is an Fp -vector space of finite dimensionendowed with a linear G-action. This means that V := A/M =⋃n≥0 Vn satisfies the hypotheses ofLemma 2. Let us show that V has infinitely many G-fixed points. Suppose otherwise: V G wouldthen be contained in Vn for some n ≥ 0. On the other hand, by Lemma 2 (V /V G )G would beinfinite: since V /V G surjects G-equivariantly onto V /Vn with finite kernel, (V /Vn)G would alsobe infinite. We claim this is absurd because V G and (V /Vn)G are in bijection with one another.Indeed, we may choose a section V /Vn = A/t−n M → A and compose it with t n · id: A → Afollowed by A → V . This defines an isomorphism V /Vn → V which by property (1) preservesG-fixed points.270 Alonso BeaumontWe have shown that V G is infinite. In particular, for all n ≥ 0, V G is not contained in Vn : wecan find some vn ∈ (A/M)G that lifts to vn ∈ A\t−n M . Let kn > n be the unique integer such thatwn := t kn vn ∈ M\t M . Applying (1), we have∀ x ∈ Fp ((t )), ϕ(x)(wn)−wn = t kn(ϕ(t−kn x)(vn)− vn) ∈ t kn M .Since M\t M is compact, the family (wn)n≥0 has an accumulation point w . For all x ∈ Fp ((t )) andinfinitely many n, we have ϕ(x)(w)−w ∈ t kn M , so ϕ(x)(w) = w : we have constructed a non-zeroG-fixed point in A. □Remark 4. Glöckner and Willis state Theorem 3 slightly differently: they endow Aut(Fp ((t ))d)with the finer Braconnier topology, and assume that ϕ is continuous with respect to it. Thisassumption is in fact no stronger than ours (see [6, Lemma 9.13]). In light of this, Step 1 of theproof above may be started as follows. The subgroup{g ∈ Gℓ∣∣g (B) = B} is open in Gℓ for theBraconnier topology, and therefore it is of finite index. Thus, the intersection⋂g∈Gℓ g (B) is finite,yielding an open Gℓ-invariant subgroup in B .Remark 5. Let (G1,α1) and (G2,α2) be two tdlc contraction groups, equipped with their respec-tive contracting automorphisms. A continuous actionψ : G1×G2 →G2 may be called equivariantif ψ(α1(x),α2(y)) = α2(ψ(x, y)) for all x ∈ G1, y ∈ G2. Then the argument of Step 1 can be ap-plied in this more general setting: there exists a compact, non-trivial subgroup H of G2 such thatα2(H) ⊂ H and ψ(G1 ×H) = H .AcknowledgmentsI’d like to thank David J. Benson for his contribution to Lemma 2, as well as Adrien Le Boudec andHelge Glöckner for their helpful comments.Declaration of interestsThe author does not work for, consult, own shares in or receive funding from any companyor organization that would benefit from this article, and has disclosed no relevant affiliationsbeyond his research organization.References[1] H. Glöckner and G. A. Willis, “Classification of the simple factors appearing in composi-tion series of totally disconnected contraction groups”, J. Reine Angew. Math. 2010 (2010),no. 643, pp. 141–169.[2] H. Glöckner and G. A. Willis, “Locally pro-p contraction groups are nilpotent”, J. ReineAngew. Math. 781 (2021), pp. 85–103.[3] J. Munkres, Topology, Second edition, Pearson Education, 2014, pp. i+503.[4] A. Rosenberg and D. Zelinsky, “Finiteness of the injective hull”, Math. Z. 70 (1958), pp. 372–380.[5] E. Siebert, “Contractive automorphisms on locally compact groups”, Math. Z. 191 (1986),pp. 73–90.[6] M. Stroppel, Locally compact groups, EMS Textbooks in Mathematics, European Mathemat-ical Society, 2006, pp. x+302.

On the nilpotency of locally pro-$p$ contraction groups

Claudemir Alcantara

Edgard Pimentel

José Miguel Urbano

Comptes RendusMathématiqueClaudemir Alcantara, Edgard Pimentel and José Miguel UrbanoHessian regularity in Hölder spaces for a semi-linear bi-Laplacian equationVolume 363 (2025), p. 533-539Online since: 26 May 2025https://doi.org/10.5802/crmath.737This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2025, Vol. 363, p. 533-539https://doi.org/10.5802/crmath.737Research article / Article de recherchePartial differential equations / Équations aux dérivées partiellesHessian regularity in Hölder spaces for asemi-linear bi-Laplacian equationRégularité du hessien dans les espaces de Hölder pourune équation bi-laplacienne semi-linéaireClaudemir Alcantara a, Edgard Pimentel b and José Miguel Urbano b,ca Department of Mathematics, Pontifical Catholic University of Rio de Janeiro (PUC-Rio),22451-900 Rio de Janeiro, Brazilb CMUC, Department of Mathematics, University of Coimbra, 3000-143 Coimbra, Portugalc Applied Mathematics and Computational Sciences Program (AMCS), Computer, Electricaland Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University ofScience and Technology (KAUST), Thuwal, 23955 -6900, Kingdom of Saudi ArabiaE-mails: alcantara@mat.puc-rio.br, edgard.pimentel@mat.uc.pt, miguel.urbano@kaust.edu.saAbstract. We examine a semi-linear variant of the bi-Laplacian equation in the superlinear, subquadraticsetting and obtain C 2,σ-regularity estimates, depending on the growth regime of the nonlinearity. Ourstrategy is to render this fourth-order problem as a system of two Poisson equations and explore the interplaybetween the integrability and smoothness available for each equation taken isolated.Résumé. Nous examinons une variante semi-linéaire de l’équation bi-laplacienne dans un cadre superli-néaire et subquadratique, et obtenons des estimations de régularité C 2,σ dépendant du régime de croissancede la non-linéarité. Notre stratégie consiste à transformer le problème du quatrième ordre en un système dedeux équations de Poisson, puis à explorer l’interaction entre l’intégrabilité et la régularité disponibles pourchaque équation prise isolément.Keywords. Bi-Laplacian operator, semi-linear equations, Hessian regularity, Hölder spaces.Mots-clés. Opérateur bi-laplacien, équations semi-linéaires, régularité hessienne, espaces de Hölder.2020 Mathematics Subject Classification. 35B65, 35J91, 35G20.Funding. EP and JMU are supported by the Centre for Mathematics of the University of Coimbra (CMUC,funded by the Portuguese Government through FCT/MCTES, UIDB/00324/2020). JMU is partially supportedby the King Abdullah University of Science and Technology (KAUST). CA was partially supported by theCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) – Brasil and by Stone Instituição dePagamento S.A., under the Project StoneLab. This study was financed in part by the CAPES-Brazil – FinanceCode 001.Manuscript received 29 January 2025, accepted 27 February 2025.ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/534 Claudemir Alcantara, Edgard Pimentel and José Miguel Urbano1. IntroductionWe examine the regularity of weak solutions to the semi-linear bi-Laplacian equation∆2u = f (x,u,Du) inΩ, (1)where Ω ⊂ Rd is a bounded smooth domain, ∆2u := ∆(∆u) denotes the bi-Laplacian operator,and the nonlinearity f : Ω×R×Rd → R satisfies a polynomial growth condition. Our findingsreport on C 2,σ-regularity estimates for the solutions to (1), depending on the growth regime ofthe nonlinearity.Elliptic equations driven by operators of higher order play an important role across disciplinesin pure mathematics and find relevant applications in various realms of life and social sciences.We mention differential geometry, calculus of variations, free boundary problems, the mechanicsof deformable media (mainly in the mathematical theory of elasticity), and the dynamics of slowviscous fluids; see the monograph [10] and the references therein.From the perspective of partial differential equations (PDE), the study of bi-Laplacian equa-tions has covered a plethora of topics. These include the existence of solutions, fundamentalproperties (such as the validity of the maximum principle and the positivity of the Green’s func-tion), and regularity estimates. We notice the existence of relevant literature examining thoseproperties in connection with the geometry of the domain. Of particular interest is the fact thatmerely Lipschitz-regular domains entail further difficulties for the analysis. In this connection,we refer the reader to [9] and the extensive list of references therein.Regarding regularity estimates, the study of the bi-Laplacian operator has been pursued inseveral contexts. In the realm of obstacle problems, it appears as the operator governing avariational inequality. More precisely, given a function ϕ : Ω→R, let u ∈W 2,2(Ω) be such that∆2u ≥ 0 inΩ, (2)with u ≥ϕ, and∆2u · (u −ϕ) = 0 inΩ. (3)In [7], the author proves that weak solutions to this problem are in W 2,∞(Ω). The analysis of thefree boundary associated with (2)–(3) is the subject of [1]. In that paper, the authors establish thelocal boundedness of the Hessian and verify that∆2u is a non-negative measure with finite mass.In addition, they examine the planar case, showing that solutions are C 2-regular and that the freeboundary is contained in a continuously differentiable curve.The bi-Laplacian operator has also been studied in [2] in the context of the two-phase freeboundary obstacle problem∆2u = 0 in B+1 ,u = g on (∂B1)+,∂xd+1 u = 0 on B ′1,∂xd+1∆u =λ−(u−)p−1 −λ+(u+)p−1 on B ′1,(4)where p > 1, λ− and λ+ are positive constants, g ∈ W 2,q (B+1 ) for q := max(2, p), and the sets B+1and B ′1 are defined asB+1 :={(x, xd+1) ∈ B1 ⊂Rd ×R∣∣ xd+1 > 0}andB ′1 := B1 ∩ {xd+1 = 0}.The authors prove that both u and ∆u are locally bounded. They also establish regularityestimates in Hölder spaces of the type C p+1,γ, for every γ ∈ (0,1), and show their findings areoptimal in the case of p ∈ N. They also consider an Almgren’s type frequency formula and aMonneau monotonicity formula and perform a thorough analysis of the singular set associatedClaudemir Alcantara, Edgard Pimentel and José Miguel Urbano 535with (4). If B1 is replaced with Rd , it is worth noticing that the formulation in (4) can be regardedas a Dirichlet-to-Neumann extension, in the spirit of Caffarelli and Silvestre, for the operator(−∆)3/2u =λ−(u−)p−1 −λ+(u+)p−1 in Rd ,with u → 0 as |x|→∞.Interior regularity estimates for the pure equation∆2u = f prescribed in a domainΩ have alsobeen pursued in the literature. Of particular interest is the analysis of polyharmonic equations ofthe form(−∆)mu = f inΩ,where 2 ≤ d ≤ 2m + 1 and f ∈ C∞c (Ω). In [9], the authors prove that solutions to this problemsatisfyDm−d2 + 12 u ∈ L∞(Ω) if d = 2k +1,andDm−d2 u ∈ L∞(Ω) if d = 2k,with k ∈N. In the concrete case of (1), were f = f (x) is a smooth function, the analysis would leadto u ∈ L∞(Ω) for dimensions d = 4,5, and Du ∈ L∞(Ω) for dimensions d = 2,3.Concerning the semi-linear formulation of the bi-Laplacian equation, we mention the devel-opments reported in [3]. In that paper, the authors produce regularity estimates for the weaksolutions to∆2u +a(x)u = g (x,u) in Rd . (5)They work under natural assumptions on the functions a = a(x) and g = g (x,u), includingpolynomial growth conditions on g , and prove that solutions to (5) are in W 4,2(Rd )∩W 2,s (Rd )for every 1 ≤ s ≤∞. Their arguments rely on asymptotic properties of the fundamental solutionassociated with the operator∆2+k2, for k ∈N. Apparently, these methods fall short in addressingthe dependence on the gradient Du, not covering the case of (1). To the best of our knowledge,the analysis of semi-linear bi-Laplacian equations with explicit dependence on Du has hithertonot been addressed in the literature.Our reasoning in this note relies on a reduction argument. Namely, we write the fourth-orderPDE as a system of two second-order equations. This strategy is inspired by ideas introduced inthe works [5,6] of Evans. The gist of those papers is to design new PDE methods to address theMather minimization principle in dynamics, as well as the weak KAM theory.Our approach stems from this class of ideas as we write the single equation in (1) as a coupling.Indeed, the integrability available for D2u in the class of weak solutions allows us to define afunction m ∈ Lq (Ω) as m :=∆u. The definition of weak solution to (1) then implies that m is a veryweak solution to the Poisson equation ∆m = f . As a consequence, we render the bi-Laplacianequation as the system {∆u = m inΩ,∆m = f (x,u,Du) inΩ. (6)The unknown for (6) is a pair (u,m) in a suitable functional space, where u is a strong solutionfor the first equation in the system, while m satisfies the second one in the very weak sense (wemake these matters precise further on in the note).In this context, the regularity of the solutions to (1) benefits from the interplay between theregularity estimates available at the level of the equations taken on their own. An Lq -regularitytheory for very weak solutions builds upon standard Sobolev embeddings to produce improvedintegrability for m in Lebesgue spaces. In turn, the integrability of m affects the regularity of u.Indeed, we prove that very weak solutions to∆m = f (x,u,Du) inΩ536 Claudemir Alcantara, Edgard Pimentel and José Miguel Urbanoare in W 2,sloc (Ω), for some s ∈ (d/2,d ]. Hence, u solves a Poisson equation with right-hand side inC 0,2−d/sloc (Ω). This fact unlocks a Schauder regularity theory for the solutions to (1).The remainder of this note is split into two sections, one detailing preliminary definitions anduseful facts and the other encompassing our main result.2. Notions of solution and auxiliary resultsBecause our strategy is to render the bi-Laplacian equation as a system, we rely on three differentnotions of solutions. Namely, weak distributional solutions, very weak solutions, and Ld -strongsolutions. Requiring u to be in W 2,q (Ω), for values of q ∈ (d/2,d ] depending on the growth regimesatisfied by f , allows us, in particular, to switch between these different notions of solutionseamlessly. We start by defining a weak solution for the bi-Laplacian equation in a distributionalsense.Definition 1 (Local weak solution of the bi-Laplacian equation). A function u ∈ W 2,2loc (Ω) is alocal weak solution to (1) if ∫Ω∆u∆ϕdx =∫Ωf (x,u,Du)ϕdx,for every ϕ ∈C∞c (Ω).Concerning Poisson’s equation∆w = g inΩ, (7)we will explore two notions of solution, those of very weak and Lq -strong solution. For complete-ness, we recall these notions in what follows.Definition 2 (Very weak solution of the Poisson equation). Let g ∈ L1loc(Ω). A function w ∈L1loc(Ω) is a very weak solution to (7) if∫Ωw∆ϕdx =∫Ωg ϕdx,for every ϕ ∈C∞c (Ω).Definition 3 (Lq -strong solution of the Poisson equation). Let g ∈ Lq (Ω) for q > 1. We say thatw ∈W 2,q (Ω) is an Lq -strong solution to (7) if∆w(x) = g (x), a.e. x ∈Ω.We proceed by defining a notion of solution to the system in (6), relating (1) with the latter.Definition 4. The pair (u,m) ∈W 2,q (Ω)×Lq (Ω) is a solution to (6) if u is an Lq -strong solution tothe first equation in (6) whereas m is a very weak solution to the second equation in (6).Next, we recall a result on the regularity of very weak solutions to the Poisson equation.Proposition 5 (Sobolev regularity for very weak solutions). Fix 1 < s <∞. Let w ∈ L1loc(Ω) be avery weak solution to (7), with g ∈ Lsloc(Ω). Then Dw ∈ W 1,sloc (Ω). If w ∈ Lsloc(Ω), then w ∈ W2,sloc (Ω).Moreover, forΩ′′ ⋐Ω′ ⋐Ω, there exists C > 0 such that∥w∥W 2,s (Ω′′) ≤C(∥w∥Ls (Ω) +∥g∥Ls (Ω)). (8)Proof. To verify that w ∈ W 2,sloc (Ω), we resort to [4, Theorem 3]. Once we have w ∈ W2,sloc (Ω), wbecomes a strong solution to ∆w = g in Ω′. Standard results in elliptic regularity theory (see, forinstance, [8, Theorem 9.11]) yield (8). □We conclude this section by verifying that a weak solution to (1) yields a pair (u,m) in a suitablefunctional space, solving (6).Claudemir Alcantara, Edgard Pimentel and José Miguel Urbano 537Lemma 6. Let u ∈W 2,q (Ω) be a local weak solution to (1), with q ≥ 2. Then, there exists m ∈ Lq (Ω)such that (u,m) is a solution to (6) according to Definition 4.Proof. It is clear that if u ∈W 2,q (Ω),∆u ∈ Lq (Ω). Set m :=∆u and notice that m ∈ Lq (Ω) is definedalmost everywhere inΩ. In addition, the weak formulation of (1) implies∫Ωm∆ϕdx =∫Ω∆u∆ϕdx =∫Ωf (x,u,Du)ϕdx,for every ϕ ∈C∞c (Ω). Hence, m is a very weak solution to∆m = f (x,u,Du) inΩ,and the proof is complete. □3. Improved regularity in Hölder spacesIn this section, we state and prove the main result in this note. For ease of presentation, weset Ω ≡ B1, where B1 stands for the unit ball in Rd ; standard covering arguments ensure thisreduction entails no further restrictions on the problem.Theorem 7. Let 2 ≤ q ≤ d and u ∈W 2,q (B1) be a local weak solution to∆2u = f (x,u,Du) in B1.Assume the nonlinearity satisfies the growth condition∣∣ f (x,r, p)∣∣≤ h(x)+C(|r |α+|p|β), (9)for h ∈ Ld (B1) and fixed constants C > 0 andα,β ∈ [1,2). (10)Suppose further thatmax(α,β)d2< q ≤ d . (11)Then u ∈C 2,σloc (B1) forσ := 2− d max(α,β)q∈ (0,1).Moreover, there exists C > 0 such that∥u∥C 2,σ(B7/8) ≤C(∥h∥Ld (B1) +∥u∥max(α,β)W 2,q (B1)).Proof. We start by choosings := qmax(α,β),noting that, due to (10) and (11), we have s ∈ (d/2, q]. Moreover, for B9/10 ⋐ B1 it follows from (9)that ∥∥ f ( · ,u,Du)∥∥Ls (B9/10) ≤C(∥h∥Ls (B9/10) +∥u∥αLαs (B9/10) +∥Du∥βLβs (B9/10))≤C(∥h∥Ld (B1) +∥u∥max(α,β)W 2,q (B1)),which is finite since u ∈ W 2,q (B1). The constant C here depends on q and the ingredients in (9).By Lemma 6, there exists m ∈ Lq (B1) such that m is a very weak solution to∆m = f (x,u,Du) in B1.538 Claudemir Alcantara, Edgard Pimentel and José Miguel UrbanoDue to Proposition 5, we conclude Dm ∈ W 1,sloc (B99/100). But, since d/2 < s ≤ q , we also havem ∈ Ls (B1) and therefore m ∈W 2,sloc (B99/100). In addition, there exists C > 0 such that∥m∥W 2,s (B9/10) ≤C(∥h∥Ld (B1) +∥m∥Ls (B1) +∥u∥max(α,β)W 2,q (B1))≤C(∥h∥Ld (B1) +∥u∥W 2,q (B1) +∥u∥max(α,β)W 2,q (B1)),where the second inequality follows from the fact that∥m∥Ls (B9/10) = ∥∆u∥Ls (B9/10) ≤ ∥u∥W 2,q (B1).Because of Gagliardo–Nirenberg–Sobolev’s embedding theorem, we obtain m ∈C 0,σ(B8/9), withσ := 2− d max(α,β)q.Since u is an Lq -strong solution to ∆u = m, we have u ∈C 2,σ(B8/9) (see [8, Theorem 9.19]). Also,by Schauder’s theory, there exists a positive constant C =C (d ,α,β, q), such that∥u∥C 2,σ(B7/8) ≤C(∥u∥L∞(B8/9) +∥m∥C 0,σ(B8/9)).To complete the proof, we notice that∥m∥C 0,σ(B8/9) ≤C∥m∥W 2,s (B8/9)≤C∥∥ f ( · ,u,Du)∥∥Ls (B9/10)≤C(∥h∥Ld (B1) +∥u∥max(α,β)W 2,q (B1)),and the estimate in the theorem follows. □Remark 8. A fundamental question arises in the context of assumption (11): one must ensurethatmax(α,β)d2< d ,so the range for q is nonempty. The above inequality is indeed satisfied due precisely to (10).The explicit description of the modulus of continuity is appealing, as it provides asymptoticinformation. As q → d and the growth conditions for f approach the linear regime, the exponentσ→ 1, yielding asymptotic estimates for (1) in C 2,1. We also notice the explicit gains of regularitystemming from (1). Indeed, we start with a function u ∈ W 2,q (Ω) and the equation yieldsu ∈C 2,σloc (Ω), with estimates.We conclude this note with a corollary to Theorem 7 yielding smoothness of the solutions to (1)in the case α=β= 1, under the assumption that h ∈C∞(Ω).Corollary 9 (C∞-regularity estimates). Let u ∈ W 2,q (B1) be a weak solution to (1), with q ≥ 2satisfying (11). Suppose (9) is in force, withf (x,r, p) := h(x)+a(x)r + c(x) ·p,where h, a ∈C∞(B1) and c ∈C∞(B1,Rd ). Suppose further there exists C > 0 such that∥h∥C∞(B1) +∥a∥C∞(B1) +∥c∥C∞(B1,Rd ) ≤C .Then u ∈C∞loc(B1). Moreover, for every k ∈N and every multi-index α with |α| = k, we havesupB7/6|Dαu| ≤C(1+∥u∥W 2,q (B1)).Claudemir Alcantara, Edgard Pimentel and José Miguel Urbano 539Proof. Fix a direction i ∈ {1, . . . ,d} and definev := ∂u∂xi.Clearly, v solves∆2v = g (x)+a(x)v(x)+ c(x) ·Dv,with g given byg (x) := ∂∂xih(x)+ ∂∂xia(x)u(x)+ ∂∂xic(x) ·Du(x).One easily notices that∣∣g (x)+a(x)v(x)+ c(x) ·Dv∣∣≤C(1+∥u∥W 2,q (B1) + ∣∣v(x)∣∣+ ∣∣Dv(x)∣∣).Hence, Theorem 7 implies v ∈ C 2,σloc (B1). Because the direction i is arbitrary, we concludeu ∈C 3,σloc (B1). An induction argument on the order of differentiation completes the proof. □Declaration of interestsThe authors do not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and have declared no affiliations other than their researchorganizations.References[1] L. A. Caffarelli and A. Friedman, “The obstacle problem for the biharmonic operator”, Ann.Sc. Norm. Super. Pisa, Cl. Sci. (4) 6 (1979), no. 1, pp. 151–184.[2] D. Danielli and A. Haj Ali, “A two phase boundary obstacle-type problem for the bi-Laplacian”, Nonlinear Anal., Theory Methods Appl. 214 (2022), article no. 112583 (26 pages).[3] Y. Deng and Y. Li, “Regularity of the solutions for nonlinear biharmonic equations in RN ”,Acta Math. Sci., Ser. B, Engl. Ed. 29 (2009), no. 5, pp. 1469–1480.[4] G. Di Fratta and A. Fiorenza, “A short proof of local regularity of distributional solutions ofPoisson’s equation”, Proc. Am. Math. Soc. 148 (2020), no. 5, pp. 2143–2148.[5] L. C. Evans, “Some new PDE methods for weak KAM theory”, Calc. Var. Partial Differ. Equ.17 (2003), no. 2, pp. 159–177.[6] L. C. Evans, “Further PDE methods for weak KAM theory”, Calc. Var. Partial Differ. Equ. 35(2009), no. 4, pp. 435–462.[7] J. Frehse, “On the regularity of the solution of the biharmonic variational inequality”,Manuscr. Math. 9 (1973), pp. 91–103.[8] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Reprintof the 1998 edition, Classics in Mathematics, Springer, 2001, pp. xiii+517.[9] S. Mayboroda and V. Maz’ya, “Regularity of solutions to the polyharmonic equation ingeneral domains”, Invent. Math. 196 (2014), no. 1, pp. 1–68.[10] A. P. S. Selvadurai, Partial differential equations in mechanics 2. The biharmonic equation,Poisson’s equation, Springer, 2000, pp. xviii+698.

Hessian regularity in Hölder spaces for a semi-linear bi-Laplacian equation

Diem Hang T. Le

Phuong Le

Liouville theorems for $p$-Laplace equations with critical Hardy–Hénon exponents

Jared Marx-Kuo

Comptes RendusMathématiqueJared Marx-KuoThe isospectral problem for p-widths: an application of Zoll metricsVolume 363 (2025), p. 565-570Online since: 5 June 2025https://doi.org/10.5802/crmath.708This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/C EN T R EMER S ENN EThe Comptes Rendus. Mathématique are a member of theMersenne Center for open scientific publishingwww.centre-mersenne.org — e-ISSN : 1778-3569Comptes Rendus. Mathématique2025, Vol. 363, p. 565-570https://doi.org/10.5802/crmath.708Research article / Article de rechercheGeometry and Topology / Géométrie et TopologieThe isospectral problem for p-widths: anapplication of Zoll metricsLe problème isospectral pour les p-largeurs : uneapplication des métriques de ZollJaredMarx-Kuo aa 450 Jane Stanford Way, Stanford, United States of AmericaE-mail: jmarxkuo@gmail.comAbstract. We pose the isospectral problem for the p-widths: is a Riemannian manifold (Mn+1, g ) uniquelydetermined by its p-widths,{ωp (M , g )}∞p=1? We construct many counterexamples on S2 using Zoll metricsand 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 pardes 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. IntroductionIn [15], Gromov introduced the p-widths, {ωp }, of a Riemannian manifold, (M n+1, g ), as ananalogue 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 inthe definition of ωp – see Section 2, as well as [10,16,17,19] for more background. The p-widthshave proven to be extremely useful due to the Almgren–Pitts/Marques–Neves Morse theoryprogram for the area functional (see [2,3,25–27,30,34,38]). In [24], Liokumovich–Marques–Nevesdemonstrated 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 minimalsurfaces. 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 existinfinitely many embedded minimal hypersurfaces.ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/566 Jared Marx-KuoWe 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 resultfor non-generic metrics. We also note the work of Li [23] who showed the existence of infinitelymany immersed minimal surfaces with optimal regularity when n + 1 ≥ 8. We refer the readerto the introduction of [9] for a more thorough overview of the history and applications of thep-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 discretespectrum, {λk }?Informally, this is known as “can you hear the shape of a drum?” and was made famousby Kac [22]. In [33], Milnor constructed two 16-dimensional tori which are isospectral but notisometric, providing the first resolution of Problem 2. Gordon–Webb–Wolpert [12] constructedsimpler examples within the class of polygons in R2. By contrast, Berger [6] showed that S2 isspectrally 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 valuesωp (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 Zollmetrics 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 thespace 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) werecomputed explicitly in [9], and this was the first example of the p-width spectrum being knownfor all p. Corollary 5 now gives uncountably more manifolds where the full p-width spectrum isknown.The proof of Theorem 4 uses the defining properties of Zoll metrics on S2, which are metricsfor which all geodesics (with multiplicity one) are simple and of length 2π (note this choice oflength, 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 ) isachieved 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 theresults 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. BackgroundWe 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 subcomplexJared Marx-Kuo 567of the k-dimensional unit cube (cf. [29, Section 2.2]), and let Zn(M ;Z2) denote the set of n-cyclesmod Z2 – informally, a cycle is the boundary of a nice (Caccioppoli) open set. In [2], Almgrenshowed that Zn(M ;Z2) is weakly homotopic to RP∞. We let λ ∈ H 1(Zn(M ;Z2),Z2) ∼= Z2 denotethe generator of this group, so that λpgenerates 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 Radonmeasure, 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∗(λp) ̸= 0 ∈ H p (X ).Definition 7 ([28, Section 3.3]). A map Φ : X → Zn(M ;Z2) is said to have no concentration ofmass iflimr→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 cubicalsubcomplex X , with no concentration of mass. The p-width of (M , g ) isωp (M , g ) := infΦ∈Ppsup{M(Φ(x)): x ∈ domain(Φ)}.To see the analogy between ωp and λp , consider the following characterization of λp (M) asλp (M) := infV ⊆C∞(M)dimV =p+1supf ∈V \{0}∫M |∇ f |2∫M f2.Note the parallels between the min-max definitions of ωp ,λp , with Φ∗(λp) ̸= 0 being analogousto 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 regularitytheory 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, theVi 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–Mantoulidisused the Allen–Cahn equation to prove that the p-widths can always be achieved by a union ofclosed 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 thatωp (M , g ) =N (p)∑j=1m 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 closedand of length 2π.Zoll [39], Funk [11], Guillemin [18], and Gromoll–Grove [14] each contributed significantlyto the theory of Zoll metrics. In particular, Gromoll–Grove showed that all of the primitivegeodesics on a Zoll surface are simple. Furthermore, the combined work of Guillemin and Funk568 Jared Marx-Kuodemonstrated that the tangent space of Zoll metrics at the round metric is given by odd functionsf : S2 → R. We also mention a higher dimensional analogue of these metrics constructed byAmbrozio–Marques–Neves [4].Let Z (S2) denote the space of Zoll metrics on S2, Zα, an arbitrary connected component ofZ (S2), and Z0, the connected component containing g0, the unit round metric.3. Proof of main theoremLet M∞(S2) denote the set of all smooth metrics on S2, and let g0 denote the round metric. Forfixed 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 haveωp (S2, g ) =N (p)∑j=1m j ·Lg (σp, j )= 2πN (p)∑j=1m 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, lyinginside 2πN, butωp is continuous in each component due to Lemma 12, thus the map is constanton connected components, proving Theorem 4.4. Open questionsHaving shown a counterexample to p-width spectral rigidity, it would be interesting to know anyof 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-intersectinggeodesics, or two geodesics intersecting may occur. See [1,32] which highlight the dif-ferences between multiplicity, immersion vs. embeddedness, and index bounds of theunderlying 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? Wenote 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 manifoldso that isospectral rigidity holds?AcknowledgmentsThe author thanks Otis Chodosh for his conversations on p-widths and Lucas Ambrozio for hisconversations on Zoll metrics. The author is also thankful to the referee for their in-depth andthoughtful feedback.Jared Marx-Kuo 569Declaration of interestsThe author does not work for, advise, own shares in, or receive funds from any organizationthat could benefit from this article, and has declared no affiliations other than their researchorganizations.References[1] N. S. Aiex, “The width of ellipsoids”, 2016. Online at https://arxiv.org/abs/1601.01032.[2] F. J. Almgren Jr., “The homotopy groups of the integral cycle groups”, Topology 1 (1962),no. 4, pp. 257–299.[3] F. J. Almgren Jr., “The theory of varifolds”, 1965. Online at https://albert.ias.edu/entities/publication/37372941-b4fc-424f-902d-c182299c13af. Mimeographed notes.[4] L. Ambrozio, F. C. Marques and A. Neves, “Riemannian metrics on the sphere with Zollfamilies of minimal hypersurfaces”, 2021. Online at https://arxiv.org/abs/2112.01448.[5] L. Ambrozio, F. C. Marques and A. Neves, “Rigidity theorems for the area widths of Rie-mannian manifolds”, 2024. Online at https://arxiv.org/abs/2408.14375.[6] M. Berger, “Sur le spectre d’une variété riemannienne”, C. R. Math. 263 (1966), A13–A16.[7] A. L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik undihrer Grenzgebiete, Springer, 1978, pp. ix+262.[8] O. Chodosh and C. Mantoulidis, “Minimal surfaces and the Allen–Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates”, Ann. Math. 191 (2020), no. 1,pp. 213–328.[9] O. Chodosh and C. Mantoulidis, “The p-widths of a surface”, Publ. Math., Inst. 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