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Comptes Rendus
Mathématique
Xucheng Zhang
Moduli of rank 2 vector bundles over a curve
Volume 362 (2024), p. 1915-1918
Online since: 16 December 2024
https://doi.org/10.5802/crmath.628
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
2024, Vol. 362, p. 1915-1918
https://doi.org/10.5802/crmath.628
Research article / Article de recherche
Algebraic geometry / Géométrie algébrique
Moduli of rank 2 vector bundles over a curve
Espace de modules des fibrés vectoriels de rang 2 sur une
courbe
Xucheng Zhang ,∗,a
a Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
E-mail: zhangxucheng@mail.tsinghua.edu.cn
Abstract. We show that among rank 2 vector bundles over a curve only semistable and simple objects admit
a good moduli space.
Résumé. Nous montrons que parmi les fibrés vectoriels de rang 2 sur une courbe, seuls les objets semistables
et simples admettent de bons espaces de modules.
Keywords. Vector bundles, Moduli space, Algebraic stacks.
Mots-clés. Fibrés vectoriels, Espaces de modules, Champs algébriques.
2020 Mathematics Subject Classification. 14D20, 14D23.
Manuscript received 3 May 2023, revised 5 March 2024, accepted 28 February 2024.
1. Main result
It is a classical result that semistable vector bundles and simple vector bundles over a curve form
a moduli space. The purpose of this note is to show that they are the only possibilities in rank 2
case. To be precise
Theorem 1 (Corollary 5 and Lemma 6). Let C be a smooth projective connected curve of genus
gC > 1 over an algebraically closed field k of characteristic 0. Denote by Bundn the moduli stack of
vector bundles of rank n and degree d over C .
(1) If (2,d) = 1, i.e., d is odd, then the open substack Bund ,simple2 ⊆ Bund2 of simple vector
bundles is the unique maximal open substack that admits a good moduli space.
(2) If (2,d) ̸= 1, i.e., d is even, then the open substack Bund ,simple2 ⊆ Bund2 of simple vector
bundles and the open substack Bund ,ss2 ⊆Bund2 of semstable vector bundles are the only
maximal open substacks that admit a good moduli space.
Similar results no longer hold in higher rank, as observed in [3, Theorem C] that there are other
open substacks that admit (even separated) good moduli spaces.
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1916 Xucheng Zhang
2. Preliminary
The key ingredient to prove Theorem 1 is the following existence criteria for algebraic stacks to
admit good moduli spaces.
Theorem 2 ([1, Theorem 4.1]). Let X be an algebraic stack, locally of finite type with affine
diagonal over a quasi-separated and locally noetherian algebraic space. Then X admits a good
moduli space if and only if
(1) every closed point of X has linearly reductive stabilizer;
(2) X isΘ-reductive; and
(3) X has unpunctured inertia.
Let U ⊆Bundn be an open substack of vector bundles. Since automorphism groups of vector
bundles (or in general, coherent sheaves) are connected, the open substack U has unpunctured
inertia by [1, Proposition 3.55]. So the condition (3) is automatic for U . Furthermore, [3,
Proposition 3.6] gives a characterization for U to be Θ-reductive, which implies that in rank 2
case any Θ-reductive open substack cannot contain direct sum of line bundles of different
degrees (see [3, Proposition 4.2]). Therefore, it remains to consider condition (1), i.e., we need
to understand vector bundles with linearly reductive automorphism groups. In rank 2 case we
have a complete classification of such vector bundles.
Proposition 3. Let E ∈Bund2 (k) be a point such that Aut(E ) is linearly reductive. Then one of the
following occurs:
(1) E is simple, in which case Aut(E ) =Gm .
(2) E ∼= L1 ⊕L2 for some line bundles L1 ≇ L2 with Hom(L1,L2) = 0 = Hom(L2,L1), in
which case Aut(E ) =G2m .
(3) E ∼=L ⊕2 for some line bundle L , in which case Aut(E ) = GL2.
Proof. In characteristic 0, linear reductivity is equivalent to reductivity.
Suppose E is indecomposable. By [2, Proposition 16] the set of nilpotent endomorphisms
Endnil(E ) ⊆ End(E ) forms a k-subalgebra and
End(E ) = k · idE ⊕Endnil(E ).
In particular we see idE +Endnil(E ) is a unipotent normal subgroup of Aut(E ), so Endnil(E ) = 0
and then End(E ) = k. This is case (1).
Suppose E = L1 ⊕L2 for some line bundles L1,L2. If L1 ∼= L2, then Aut(E ) = GL2. This is
case (3).
If L1 ≇L2, then any compositions L1 →L2 →L1 and L2 →L1 →L2 are zero. Using this it
is straightforward to see that the following subset{(
idL1 φ12
φ21 idL2
)
:φi j ∈ Hom(Li ,L j )
}
⊆ Aut(E )
is a unipotent normal subgroup of Aut(E ), so Hom(L1,L2) = 0 = Hom(L2,L1) and then Aut(E ) =
G2m . This is case (2). □
3. Proof
Now we are able to prove Theorem 1. Let U ⊆Bund2 be an open substack, the road map is:
• If every closed point of U has linearly reductive stabilizer and U isΘ-reductive, then U
is contained in Bund ,ss2 ∪Bun
d ,simple
2 (see Lemma 4).
Xucheng Zhang 1917
• If U ⊈ Bund ,ss2 and U ⊈ Bun
d ,simple
2 , then U supports a degeneration family from
a semistable vector bundle to an unstable vector bundle, which is not allowed by Θ-
reductivity (see [3, Corollary 3.9]).
Thus, to obtain a good moduli space we must have U ⊆Bund ,ss2 or U ⊆Bun
d ,simple
2 .
Lemma 4. If U ⊆ Bund2 is an open substack that admits a good moduli space, then U ⊆
Bund ,ss2 ∪Bun
d ,simple
2 .
Proof. Any point E ∈ U (K), for some field K/k, specializes to a closed point E0 ∈ U (k) and we
know Aut(E0) is linearly reductive by Theorem 2. Using Proposition 3
(1) either E0 is simple. Then E is also simple by the openness of simpleness.
(2) or E0 ∼=L1⊕L2 for some line bundles L1,L2. By [3, Proposition 4.2] theΘ-reductivity of
U forces that deg(L1) = deg(L2). Then E0 is semistable and hence E is also semistable
by the openness of semistability.
This shows that E ∈Bund ,ss2 (K)∪Bun
d ,simple
2 (K), as desired. □
Corollary 5. If (2,d) = 1, i.e., d is odd, then Bund ,simple2 ⊆ Bund2 is the unique maximal open
substack that admits a good moduli space.
Proof. In this case Bund ,ss2 =Bund ,s2 ⊆Bun
d ,simple
2 and we are done by Lemma 4. □
Lemma 6. If (2,d) ̸= 1, i.e., d is even, then Bund ,simple2 ⊆Bund2 and Bund ,ss2 ⊆Bund2 are the only
maximal open substacks that admit a good moduli space.
Proof. If U ⊆Bund2 is an open substack that admits a good moduli space, then U ⊆Bund ,ss2 ∪
Bund ,simple2 by Lemma 4. It suffices to show U ⊆ Bund ,ss2 or U ⊆ Bun
d ,simple
2 . Suppose
otherwise, then
(1) U ⊈Bund ,ss2 implies that U contains an unstable vector bundle.
Let E ∈ U (k) be an unstable vector bundle. Denote by 0 → L1 → E → L2 → 0 its
Harder–Narasimhan filtration.
(2) U ⊈Bund ,simple2 implies that U contains a direct sum of line bundles.
Indeed, if U doesn’t contain any direct sum of line bundles, then by Proposition 3
every point in U specializes to a simple vector bundle, thus U ⊆ Bund ,simple2 by the
openness of simpleness, a contradiction.
Let V1 ⊕ V2 ∈ U (k) be a direct sum of line bundles, then V1 ⊕ V2 is semistable by
Lemma 4, i.e., deg(V1) = deg(V2) = d/2. Consider the direct sum morphism
⊕ : Picd/2×Picd/2 →Bund2
ThenΩ :=⊕−1(U ) ⊆ Picd/2×Picd/2 is open and dense since it contains (V1,V2).
From these data, we can construct a degeneration family in U , over some DVR R over k with
fraction field K and residue field κ, from a semistable vector bundle GK (as an extension of line
bundles of the same degree, using the open subset Ω constructed in (2)) to an unstable vector
bundle Gκ (using the unstable vector bundle E in (1)). Then any Jordan–Hölder filtration G •K
of GK with the associated graded sheaf gr(G •K ) ∈ U (K ) cannot even extend to a filtration of
subbundles of Gκ (otherwise Gκ will be semistable as well), contradicting to the Θ-reductivity
of U . This concludes the proof.
Let us make this precise. By the same proof of [3, Lemma 4.7], the following subset
∆ := {L ∈ Pic0 : ∃ [e] ∈ Ext1(L ⊗L2,L ⊗L1) such that E ([e]) ∈U }
is open and dense in Pic0, where E ([e]) is the extension vector bundle associated to [e]. To finish
the proof, we claim that
1918 Xucheng Zhang
Claim 7. There exist a line bundle L ∈ ∆ and an effective divisor D ∈ Diva(C ) such that
(L (−D)⊗L1)⊕ (L (D)⊗L2) ∈U (k), where a := deg(L1)−d/2 > 0.
Indeed, for any line bundle L ∈ ∆ and effective divisor D ∈ Diva(C ) in Claim 7, by definition
of ∆ there exists a class [e] ∈ Ext1(L ⊗L2,L ⊗L1) such that E ([e]) ∈U (k). The extension
[e] : 0 →L ⊗L1 → E ([e]) →L ⊗L2 → 0
yields a twisted one
[eD ] : 0 →L (−D)⊗L1 → E ([e]) → (L ⊕OD )⊗L2 → 0.
Then by definition the quadruple
(L (−D)⊗L1,L (D)⊗L2,D, [eD ])
∈ Picd/2(k)×Picd/2(k)×Diva(C )×Ext1((L ⊕OD )⊗L2,L (−D)⊗L1)
satisfies (L (−D)⊗L1)⊕ (L (D)⊗L2) ∈ U (k) and E ([eD ]) ∈ U (k). Then U is not Θ-reductive
by [3, Corollary 3.9], a contradiction.
Proof of Claim 7. Consider the morphism
λ : Pic0×Diva(C ) → Picd/2×Picd/2 mapping (L ,D) 7→ (L1 ⊗L (−D),L2 ⊗L (D))
and it is surjective since for any (N1,N2) ∈ Picd/2×Picd/2 we have(
(N1 ⊗N2 ⊗L ∗1 ⊗L ∗2 )1/2, (N ∗1 ⊗N2 ⊗L1 ⊗L ∗2 )1/2
) 7→ (N1,N2).
Then λ−1(Ω) ⊆ Pic0×Diva is an open dense subset of Pic0×Diva and therefore
λ−1(Ω)∩ (∆×Diva(C )) ̸= ;.
By definition any point (L ,D) in this intersection satisfies the required condition. □
Acknowledgments
The work presented in this paper is extracted from my thesis carried out at Universität Duisburg-
Essen under the supervision of Jochen Heinloth, to whom I am deeply indebted. Thanks to
Yichen Qin for his help in translating the title and abstract into French.
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] J. Alper, D. Halpern-Leistner and J. Heinloth, “Existence of moduli spaces for algebraic
stacks”, Invent. Math. 234 (2023), no. 3, pp. 949–1038.
[2] M. F. Atiyah, “Complex analytic connections in fibre bundles”, Trans. Am. Math. Soc. 85
(1957), pp. 181–207.
[3] D. Weissmann and X. Zhang, “A stacky approach to identify the semistable locus of bundles”,
2023. https://arxiv.org/abs/2302.09245.


Moduli of rank 2 vector bundles over a curve

Antje Dabeler

Emilie Mai Elkiær

Maria Gerasimova

Tim de Laat

Comptes Rendus
Mathématique
Antje Dabeler, Emilie Mai Elkiær, Maria Gerasimova and Tim de Laat
Unitary Lp+-representations of almost automorphism groups
Volume 362 (2024), p. 245-249
Online since: 2 May 2024
https://doi.org/10.5802/crmath.549
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
2024, Vol. 362, p. 245-249
https://doi.org/10.5802/crmath.549
Research article / Article de recherche
Representation theory / Théorie des représentations
Unitary Lp+-representations of almost
automorphism groups
Représentations unitaires Lp+ de groupes de
presqu’automorphismes
Antje Dabeler a, Emilie Mai Elkiær b, Maria Gerasimova a and Tim de Laat ∗,a
a University of Münster, Mathematical Institute, Einsteinstraße 62, 48149 Münster,
Germany.
b Department of Mathematics, University of Oslo, Norway
E-mails: antje.dabeler@uni-muenster.de (A. Dabeler), elkiaer@math.uio.no
(E. M. Elkiær), maria.gerasimova@uni-muenster.de (M. Gerasimova),
tim.delaat@uni-muenster.de (T. de Laat)
Abstract. Let G be a locally compact group with an open subgroup H with the Kunze–Stein property, and
let π be a unitary representation of H . We show that the representation π̃ of G induced from π is an Lp+-
representation if and only if π is an Lp+-representation. We deduce the following consequence for a large
natural class of almost automorphism groups G of trees: For every p ∈ (2,∞), the group G has a unitary
Lp+-representation that is not an Lq+-representation for any q < p. This in particular applies to the Neretin
groups.
Résumé. Soit G un groupe localement compact avec un sous-groupe ouvert H ayant la propriété de Kunze–
Stein, et soitπune représentation unitaire de H . Nous montrons que la représentation π̃de G induite parπ est
une représentation Lp+ si et seulement si π est une représentation Lp+. Nous en déduisons la conséquence
suivante pour une grande classe naturelle de groupes de presqu’automorphismes G d’un arbre : pour tout
p ∈ (2,∞), le groupe G a une représentation unitaire Lp+ qui n’est pas une représentation Lq+ pour tout
q < p. Ceci s’applique en particulier aux groupes de Neretin.
Funding. AD, MG, and TdL were supported by the DFG – Project-ID 427320536 – SFB 1442, and under
Germany’s Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics–Geometry–Structure.
Manuscript received 27 April 2023, accepted 21 July 2023.
1. Induction of unitary Lp+-representations
Let G be a locally compact group with a closed subgroup H . Given a unitary representation π of
H , it can be induced to G in order to get a unitary representation of G . Technically, this procedure
is much easier if H is not only closed, but also open in G . In that case, it is well known that the
restriction π̃|H (restricted to H as a function) contains the representation π.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
246 Antje Dabeler, Emilie Mai Elkiær, Maria Gerasimova and Tim de Laat
We briefly recall the construction of induced representations, and we refer to [4,10] for further
details. Let H be an open subgroup of G , and let {ti }i ∈ I be a maximal set in G consisting of
left coset representatives. Let (π,V ) be a unitary representation of H . Let Ṽ = { f : G → V |
f (g h) = π(h−1) f (g ) ∀ g ∈ G ∀ h ∈ H and ∑i ∈ I ∥ f (ti )∥2 <∞}. With respect to the inner product
〈 f , f ′〉 = ∑i ∈ I 〈 f (ti ), f ′(ti )〉, this is a Hilbert space. The induced representation (π̃,Ṽ ) is defined
by (π̃(g ) f )(g ′) = f (g−1g ′).
Given a unitary representation (π,V ) of G , a function of the form πv,w : g 7→ 〈π(g )v, w〉,
where v, w ∈ V , is called a matrix coefficient of π. This short note is concerned with the
behaviour of integrability properties of such matrix coefficients under the procedure of induction
of representations from open subgroups.
Definition 1. Let p ∈ [1,∞). A unitary representation (π,V ) of a locally compact group G is an
Lp -representation if there is a dense linear subspace V0 ⊂ V such that for all v, w ∈ V0, we have
πv,w ∈ Lp (G). The representation π is an Lp+-representation if it is an Lp+ε-representation for all
ε> 0.
The property of being an Lp -representation (or Lp+-representation) is preserved under induc-
tion of representations. We briefly recall the proof, which is based on [21, Theorem 2.4], in which
Wiersma shows this fact in the setting of discrete groups. The same proof works in our setting
and we claim no originality here.
Proposition 2. Let G be a locally compact group, H an open subgroup of G, and p ∈ [1,∞).
Let (π,V ) be a unitary representation of H and (π̃,Ṽ ) the associated induced representation. If
π is an Lp -representation (resp. Lp+-representation), then π̃ is an Lp -representation (resp. Lp+-
representation).
Proof. Let V0 be a dense linear subspace of V such that for all v, w ∈ V0, the matrix coefficient
πv,w lies in Lp (H). Let {ti }i ∈ I be as above, and let Ṽ0 be the subspace of Ṽ consisting of functions
f for which f (ti ) ∈V0 for all i ∈ I and such that f (ti ) is non-zero for only finitely many i ∈ I . Then
Ṽ0 is a dense linear subspace of Ṽ .
For each i ∈ I , denote by Ṽ i0 the linear subspace of Ṽ consisting of all functions f with support
in ti H and such that f (ti ) ∈ V0. Then Ṽ0 consists of finite linear combinations of functions from
(different) spaces Ṽ i0 . For i , j ∈ I and f ∈ Ṽ i0 and f ′ ∈ Ṽ
j
0 , consider the matrix coefficient π̃ f , f ′ . Set
w = f (ti ) ∈ V0 and w ′ = f ′(t j ) ∈ V0, and denote the modular function on G (and its restriction to
H) by ∆. Then∥∥π̃ f , f ′∥∥pp = ∫
G
∣∣∣〈π̃(g ) f , f ′〉∣∣∣p dµG (g ) = ∫
G
∣∣∣ ∑
k ∈ I
〈
f
(
g−1tk
)
, f ′(tk )
〉∣∣∣p dµG (g )
=
∫
G
∣∣∣〈 f (g−1t j ) , w ′〉∣∣∣p dµG (g ) =∆(t−1i )∫
H
∆
(
h−1
) ∣∣∣〈 f (ti h), w ′〉∣∣∣p dµH (h)
=∆(t−1i )∫
H
∣∣∣〈π(h)w, w ′〉∣∣∣p dµH (h) =∆(t−1i ) ∥∥πw,w ′∥∥pp <∞.
It follows that π̃ is an Lp -representation. □
In general, it may happen that the induced representation has better integrability properties
than the representation from which one induces, in the sense that the induced representation
of an Lp -representation may be an Lq -representation for some q < p (even if one considers the
“optimal” value of p). Indeed, this for example happens if one induces Lp -representations of
F2 for large p to higher-rank simple Lie groups, since for every higher rank simple Lie group G ,
there exists p(G) < ∞ (depending on the group) such that every nontrivial unitary irreducible
representation of G is an Lp(G)-representation [6, Théorème 2.4.2]. This phenomenon is not well
understood and we do not know whether it can actually happen in the case of induction from an
open subgroup.
Antje Dabeler, Emilie Mai Elkiær, Maria Gerasimova and Tim de Laat 247
The main point of this note is to show that in a specific case, namely for representations of a
locally compact group G induced from representations of an open subgroup H with the Kunze–
Stein property, we know that the (optimal) Lp -integrability of a representation is preserved under
induction. A locally compact group G has the Kunze–Stein property if for every p ∈ [1,2), the
convolution product on Cc (G) extends to a bounded bilinear map Lp (G)×L2(G) → L2(G). This
property originated from the work of Kunze and Stein [12], who proved it for SL(2,R).
Theorem 3. Let G be a locally compact group, H an open subgroup of G, and p ∈ [1,∞). Let (π,V )
be a unitary representation of H and (π̃,Ṽ ) the associated induced representation of G. If H has the
Kunze-Stein property, thenπ is an Lp+-representation of H if and only if π̃ is an Lp+-representation
of G.
Proof. One direction is Proposition 2. For the other direction, suppose that π̃ is an Lp+-
representation of G . Then π̃|H (restriction as a function) is an Lp+-representation of H . Because
π is contained in π̃|H and H has the Kunze–Stein property, it follows from Proposition [13,
Proposition 4.3] that π is itself an Lp+-representation. □
If we do not assume H to have the Kunze–Stein property, the restriction π̃|H might theoretically
have better integrability properties than being an Lp+-representation. It would be interesting to
have a better understanding of the relation between Lp+-representations of a group and Lp+-
representations of its subgroups (also in the case of closed subgroups).
2. Application to representations of almost automorphism groups of trees
For d ≥ 3, let Td be a d-regular tree and Aut(Td ) the automorphism group of Td , which carries
a natural non-discrete, totally disconnected, locally compact group topology. In [16], Neretin
introduced the group Nd of almost automorphisms of Td . The Neretin groups Nd also carry a
natural non-discrete, totally disconnected, locally compact group topology, and Aut(Td ) embeds
into Nd as an open subgroup. It is known that the groups Nd are simple [11] and that they do not
have any lattices [3]. Recently, the Neretin groups have attracted increasing interest, also from
the point of view of unitary representation theory (see e.g. [17]). Notably, it was recently shown
that the Neretin groups are not of type I [5] (see also [2]).
More generally, let H be a non-compact, closed subgroup of Aut(Td ) acting transitively on the
boundary ∂Td of Td . Then H satisfies the Kunze–Stein property [15]. In [14, Section 2], Lederle
constructs the group F (H) of H-honest almost automorphisms, i.e. almost automorphisms of Td
that “locally look like” elements of H . She also shows that this group can be viewed as the topo-
logical full group of H acting on ∂Td . If it is moreover assumed that H satisfies Tits’ independence
property, then the group F (H) carries a unique non-discrete, totally disconnected, locally com-
pact group topology such that H embeds into F (H) as an open subgroup [14, Proposition 2.22].
A special class of such groups F (H) is the class of coloured Neretin groups.
The unitary representation theory of Aut(Td ) goes back to [18] (see also [7]), and more
generally, for subgroups H of Aut(Td ) as above, it was described by Amann [1]. It follows from
Theorem 3 and the above discussion that every unitary representation of H can be induced to
F (H) retaining the optimal Lp+-integrability of the representation.
We now make this more precise for the spherical complementary series of unitary represen-
tations under the additional assumption that the action of H on Td is transitive. From the point
of view of Lp+-representations, this part of the representation theory is most interesting. Indeed,
for the groups H under consideration, all non-spherical unitary irreducible representations are
tempered (i.e. they are L2+-representations).
Let H be a non-compact, closed subgroup of Aut(Td ) acting transitively on Td and on ∂Td
and satisfying Tits’ independence property. In what follows, we will use Amann’s description [1,
248 Antje Dabeler, Emilie Mai Elkiær, Maria Gerasimova and Tim de Laat
Theorem 2] of the unitary dual and refer to that text for details. The unitary irreducible spherical
representations are in one-to-one correspondence with the positive-definite spherical functions,
and because the latter are diagonal matrix coefficients with respect to a cyclic vector, Lp -
integrability (for some p) of the spherical function implies that the corresponding representation
is an Lp -representation. As follows from (the proof of) [1, Theorem 2], the positive-definite
spherical functions associated with non-tempered representations are parametrised by the set
P = [0, 12 )+ i {0, πlog(d−1) } through a bijection
P →S P (K \G/K ), z 7→ϕγo (z).
Let πγ0(z) denote the (unique) spherical unitary irreducible representation associated withϕγo (z).
For an explicit description of the spherical function ϕγo (z), we refer to [1]; we just recall its L
p -
integrability. Outside the setting of Lie groups, the Lp -integrability of spherical functions was
first considered for spherical functions on the free group Fn [8, Corollary of Theorem 2]. By the
K -bi-invariance of spherical functions, this leads to the following consequence in the case of a
subgroup H < Aut(Td ) as above: For p ∈ (2,∞) and p ′ ∈ (1,2) with 1p + 1p ′ = 1, the representation
πγo (z) is an L
p+(G)-representation if and only if Re z ∈ [ 1p , 1p ′ ]. We also refer to [9, Lemma 4.11] for
a direct proof of this fact.
Theorem 4. Let H be a non-compact, closed subgroup of Aut(Td ) acting transitively on Td and
on ∂Td and satisfying Tits’ independence property. Let p ∈ (2,∞) and p ′ ∈ (1,2) be such that
1
p + 1p ′ = 1. Then the representation of F (H) induced from πγ0(z) is an Lp+-representation if and
only if Re z ∈ [ 1p , 1p ′ ]. As a consequence, for every p ∈ (2,∞), the group F (H) has a unitary Lp+-
representation that is not an Lq+-representation for any q < p. This gives an uncountable family
of pairwise inequivalent non-tempered unitary representations of F (H).
A version of the above theorem can also be formulated if the action of H on Td is not transitive.
However, due to an inaccuracy in the description of the unitary dual in [1] in that case (as pointed
out in e.g. [19]), the parametrisation of the spherical complementary series of H requires some
more care.
3. Remark on Lp+-group-C∗-algebras
In the previous section, we constructed Lp+-representations of F (H) that are not Lq+-
representations for q < p. It is interesting to point out that by C∗-algebraic methods, it fol-
lows that the Lp+-group-C∗-algebras C∗Lp+ (F (H)) of F (H) must be pairwise canonically distinct
for p ∈ [2,∞). More precisely, since H is an open subgroup of F (H), it follows from Proposi-
tion 2, [20, End of Section 3] and [9, Theorem A] that for every 2 ≤ q < p < ∞, the canonical
quotient map C∗Lp+ (F (H)) → C∗Lq+ (F (H)) is not injective. Although this gives some information
about the existence of Lp+-representations, this method does not give information about the op-
timal Lp+-integrability of specific representations (such as induced representations).
Acknowledgements
We thank Pierre-Emmanuel Caprace and Sven Raum for useful comments and for pointing
out [2]. We also thank Pierre-Emmanuel Caprace for pointing out the inaccuracy in [1].
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.
Antje Dabeler, Emilie Mai Elkiær, Maria Gerasimova and Tim de Laat 249
References
[1] O. É. Amann, “Groups of tree-automorphisms and their unitary representations”, PhD Thesis, ETH Zürich, Switzer-
land, 2003.
[2] R. Arimoto, “On the type of the von Neumann algebra of an open subgroup of the Neretin group”, Proc. Am. Math.
Soc. 9 (2022), p. 311-316.
[3] U. Bader, P.-E. Caprace, T. Gelander, S. Mozes, “Simple groups without lattices”, Bull. Lond. Math. Soc. 44 (2012),
no. 1, p. 55-67.
[4] B. Bekka, P. de la Harpe, Unitary representations of groups, duals, and characters, Mathematical Surveys and
Monographs, vol. 250, American Mathematical Society, 2020.
[5] P.-E. Caprace, A. Le Boudec, N. Matte Bon, “Piecewise strongly proximal actions, free boundaries and the Neretin
groups”, 2022, https://arxiv.org/abs/2107.07765.
[6] M. G. Cowling, “Sur les coefficients des représentations unitaires des groupes de Lie simples”, in Analyse harmonique
sur les groupes de Lie, II, Lecture Notes in Mathematics, vol. 739, Springer, 1979, p. 132-178.
[7] A. Figà-Talamanca, C. Nebbia, Harmonic analysis and representation theory for groups acting on homogeneous trees,
London Mathematical Society Lecture Note Series, vol. 162, Cambridge University Press, 1991.
[8] A. Figa-Talamanca, M. A. Picardello, “Spherical functions and harmonic analysis on free groups”, J. Funct. Anal. 47
(1982), p. 281-304.
[9] D. Heinig, T. de Laat, T. Siebenand, “Group C∗-algebras of locally compact groups acting on trees”, 2020, https:
//arxiv.org/abs/2011.11265.
[10] E. Kaniuth, K. F. Taylor, Induced representations of locally compact groups, Cambridge Tracts in Mathematics, vol.
197, Cambridge University Press, 2013.
[11] Ch. . Kapoudjian, “Simplicity of Neretin’s group of spheromorphisms”, Ann. Inst. Fourier 49 (1999), no. 4, p. 1225-
1240.
[12] R. A. Kunze, E. M. Stein, “Uniformly bounded representations and harmonic analysis of the 2× 2 real unimodular
group”, Am. J. Math. 82 (1960), p. 1-62.
[13] T. de Laat, T. Siebenand, “Exotic group C∗-algebras of simple Lie groups with real rank one”, Ann. Inst. Fourier 71
(2021), no. 5, p. 2117-2136.
[14] W. Lederle, “Coloured Neretin groups”, Groups Geom. Dyn. 13 (2019), no. 2, p. 467-510.
[15] C. Nebbia, “Groups of isometries of a tree and the Kunze–Stein phenomenon”, Pac. J. Math. 133 (1988), no. 1, p. 141-
149.
[16] Y. A. Neretin, “On Combinatorial analogues of the group of diffeomorphisms of the circle”, Izv. Math. 41 (1993), no. 2,
p. 337-349.
[17] ——— , “On spherical unitary representations of groups of spheromorphisms of Bruhat–Tits trees”, 1906.12197,
2019.
[18] G. I. Ol’šanskĭı, “Classification of the irreducible representations of the automorphism groups of Bruhat–Tits trees”,
Funkts. Anal. Prilozh. 11 (1977), no. 1, p. 32-42.
[19] L. Semal, “Irreducibly represented Lie groups and Nebbia’s CCR conjecture on trees”, PhD Thesis, Université
Catholique de Louvain, Belgique, 2023.
[20] M. Wiersma, “Lp -Fourier and Fourier–Stieltjes algebras for locally compact groups”, J. Funct. Anal. 269 (2015), no. 12,
p. 3928-3951.
[21] ——— , “Constructions of exotic group C*-algebras”, Ill. J. Math. 60 (2016), no. 3-4, p. 655-667.


Unitary $L^{p+}$-representations of almost automorphism groups

Jason McCullough

Comptes Rendus
Mathématique
Jason McCullough
Prime Ideals and Three-generated Ideals with Large Regularity
Volume 362 (2024), p. 251-255
Online since: 2 May 2024
https://doi.org/10.5802/crmath.544
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
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Comptes Rendus. Mathématique
2024, Vol. 362, p. 251-255
https://doi.org/10.5802/crmath.544
Research article / Article de recherche
Algebra / Algèbre
Prime Ideals and Three-generated Ideals with
Large Regularity
JasonMcCullough a
a Iowa State University, Department of Mathematics, Ames, IA, USA
E-mail: jmccullo@iastate.edu (J. McCullough)
Abstract. Ananyan and Hochster proved the existence of a function Φ(m,d) such that any graded ideal I
generated by m forms of degree at most d in a standard graded polynomial ring satisfies reg(I ) ≤ Φ(m,d).
Relatedly, Caviglia et. al. proved the existence of a functionΨ(e) such that any nondegenerate prime ideal P of
degree e in a standard graded polynomial ring over an algebraically closed field satisfies reg(P ) ≤Ψ(deg(P )).
We provide a construction showing that both Φ(3,d) andΨ(e) must be at least doubly exponential in d and
e, respectively. Previously known lower bounds were merely super-polynomial in both cases.
2020 Mathematics Subject Classification. 13D02, 13D05, 13P20.
Funding. The author was support by National Science Foundation grant DMS–1900792.
Manuscript received 10 May 2023, revised 5 July 2023, accepted 21 July 2023.
1. Introduction
Let k be a field and let S =k[x1, . . . , xn] be a standard graded polynomial ring. Let I = ( f1, . . . , fm)
be a graded ideal of S. The (Castelnuovo-Mumford) regularity of I is
regS (I ) = min
{
j − i ∣∣TorSi (I ,k) j ̸= 0} .
Originally defined for ideal sheaves via cohomology vanishing by Mumford, this version is due
to Eisenbud and Goto [11]. Since the maximal degree of an element in a minimal generating set
of I is at most reg(I ), finding upper bounds on the regularity of graded ideals is a very active area
of research. Moreover, Bayer and Stillman showed that in the reverse lexicographic order and
in generic coordinates, the regularity of an ideal matches that of its generic initial ideal; thus,
regularity is a proxy for computation complexity.
For arbitrary ideals, regularity can be doubly exponential in terms of the degrees of the
generators and number of variables. In contrast, regularity is well-behaved for defining ideals of
smooth projective varieties [2, Theorem 3.12], [4, Corollary 4], or those with mild singularities [8,
Theorem 0.1]. For reduced and irreducible but possibly singular varieties, defining ideals were
thought to also have well-behaved regularity, as predicted by the Eisenbud–Goto Conjecture,
which holds for curves [13], smooth surfaces [19, 26], three-folds with mild singularities [23], and
arithmetically Cohen–Macaulay varieties [11]. Further results for smooth three-folds [16, 17, 27]
supported the conjecture. Work of Noma [24] and Kwak and Park [18] shows that the regularity
of the structure sheaf of any smooth variety satisfies the bound predicted by the Eisenbud–Goto
Conjecture, and as a result, the conjecture holds for smooth and projectively normal varieties.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
252 Jason McCullough
However, counterexamples to the conjecture were constructed by Peeva and the author in [20,
Theorem 1.9]. More precisely, they showed that the regularity of a nondegenerate prime ideal
over an algebraically closed field was not bounded by any polynomial function of its degree (i.e.
multiplicity). Further counterexamples were constructed in [7, Theorem 3.3], [9, Theorem 1.2],
and [14, 3.3, 3.6, 3.8]. Yet, Caviglia et. al. [7, Corollary 5.3] proved the existence of a function
Ψ(e) bounding the regularity of all nondegenerate primes of degree e. It is easy to see that such
a function does not exist even for primary ideals, thus determining the asymptotic growth rate
of the optimal functionΨ(e) is an interesting open problem. In this short note, we show how to
sharpen the best known lower bound forΨ(e), proving that it grows at least doubly exponentially
in e.
Relatedly, Stillman [25, Problem 3.15] conjectured a bound Φ(m,d) on the regularity of a
graded ideal generated by m forms of degree at most d without fixing the number of variables.
The existence of such a bound was first proved by Ananyan and Hochster [1, Theorem D] and
later by Erman, Sam, and Snowden [12, Theorem 4.12]. The previous best lower bounds in the
three-generated case for Φ(3,d) were quadratic [6] or super-polynomial [9] in d . We show that
Φ(3,d) must also be at least doubly exponential in d .
2. Main Results
Three-generated ideals capture almost all of the pathologies seen in arbitrary resolutions, as is
made precise by Bruns’s Theorem [5, Korollar 1]. It is known that three-generated ideals can
have exponentially large projective dimension in terms of the degrees of the generators [3, Corol-
lary 3.6]. For many years, a construction due to Caviglia [6, Example 4.2.1] of a three-generated
ideal with quadratic growth of regularity was the best known lower bound for Φ(3,d). As Eisen-
bud writes in [10, p. 62], “It would be interesting to have more and stronger examples with
high regularity.” The author conjectured that a certain three-generated ideal family exhibited
super-polynomial growth of regularity relative to the degrees of the three generators [21, Con-
jecture 13.9]. Recently, Choe [9, Theorem 1.2] constructed a different such example. Below we
construct a three-generated ideal with doubly exponential growth of its regularity in terms of the
common degree of its minimal generators.
The following is a slight generalization of [9, Proposition 5.1], which in turn generalized [7,
Proposition 7.1]. It shows how to compute invariants of three-generated analogues of arbitrary
graded ideals. We include a proof for completeness.
Proposition 1. Let I = (g0, . . . , gm) be a graded ideal of a standard graded polynomial ring
S = k[x1, . . . , xn] with di = deg(gi ) nondecreasing and d = dm . Let R = S[y, z] and pick integers
s, t ≥ m +d −d0 +1. Set
J =
(
y s , z t ,
m∑
i=0
gi y
d−di+i zm−i
)
⊆ R.
Then J is a graded ideal with generators of degree s, t ,m +d. Moreover,
(1) regR (J ) ≥ regS (I )+ s + t −2,
(2) pdR (J ) ≥ pdS (I )+2,
(3) deg(J ) = tm.
Proof. Note that R/J is a finitely generated S-module and that y s−1z t−1(R/J ) ∼= (S/I )(−s − t +2)
as an S-direct summand of R/J . Thus
regR (R/J ) ≥ regS
(
(S/I )(−s − t +2))= regS (S/I )+ s + t −2,
or equivalently,
regR (J ) ≥ regS (I )+ s + t −2.
Jason McCullough 253
Similarly, suppose pdS (S/I ) = p. Modding out by a regular sequence of linear forms (extending
the field if necessary), we may assume that depth(S/I ) = 0 so that S/I has a nonzero socle element
w . Clearly y s−1z t−1w is a nonzero socle element in R/J . The result follows by the Auslander–
Buchsbaum Theorem.
For the degree computation, simply note that since
p
J = (y, z) = p, we have deg(J ) = λ(Sp/Jp)
and Sp/Jp has Sp/pSp-basis y i z j for 0 ≤ i ≤ m −1 and 0 ≤ j ≤ t −1. □
As an immediate corollary, we see that finding an explicit answer to Stillman’s Conjecture
reduces to the three-generated case.
Corollary 2. To find an explicit functionΦ(m,d) bounding the regularity of ideals generated by m
forms of degree at most d, it suffices to find an explicit bound Φ(3,d) in the three-generated ideal
case.
We also get a larger lower bound on the possible value of Φ(3,d). Recall that a function
f :N→N isΩ(g (n)) if there is a constant c such that f (n) ≥ c · g (n) for n ≫ 0.
Theorem 3. Let k be a field and fix an integer r ≥ 2. Then there is a standard graded polynomial
ring R over k and a graded ideal J of R generated by 3 forms of degree 22r − 2 with reg(J ) ≥
22
r−1 +44r −6. In particular,Φ(3,d) is 22Ω(d) .
Proof. By [15], there is a polynomial ring S and graded ideal I generated by 22r −3 quadrics with
reg(I ) ≥ 22r−1 . Applying Proposition 1 with m = 22r − 4 and s = t = 22r − 2, we get an ideal J
generated by three forms of degree 22r −2 and reg(J ) ≥ 22r−1 +44r −6. □
We turn now to constructing prime ideals with large regularity. In [20], Peeva and the author
employed Rees-like algebras to create counterexamples of the Eisenbud–Goto Conjecture. After
homogenizing the defining ideals of such algebras, they computed the invariants of the new
standard graded prime ideals in terms of the arbitrary starting ideal. We record the essential
part of their theorem here.
Theorem 4 (McCullough–Peeva [20]). Let k be any field. Let I = ( f1, . . . , fm) be a graded ideal
in the standard graded polynomial ring S = k[x1, . . . , xn]. There is a nondegenerate graded prime
ideal P in the standard graded polynomial ring R = S[y1, . . . , ym ,u1, . . . , um , z, v] satisfying
(1) regR (P ) = regS (I )+2+
∑m
i=1 deg( fi ),
(2) degR (R/P ) = 2
∏m
i=1(deg( fi )+1).
The trick to achieving more extreme counterexamples to the Eisenbud–Goto Conjecture is to
feed the Mayr–Meyer ideals, such as Koh’s version [15], into Proposition 1, and moreover to do it
twice, before constructing a Rees-like algebra.
Theorem 5. Over any field k and for any integer r , there exists a standard graded polynomial ring
S and nondegenerate prime ideal P ⊆ S with
deg(P ) = 396r and reg(P ) ≥ 22r−1 +66r +6.
In particular,Ψ(e) is 22
Ω(e)
.
Proof. Begin with the ideal J from Theorem 3 which has three generators of degree 22r −2 and
reg(J ) ≥ 22r−1 + 44r − 6. Applying Proposition 1 again with m = 2 and s = t = 3, we get an ideal
J ′ generated by forms of degrees 3,3,22r with reg(J ′) ≥ 22r−1 +44r −2. Applying Theorem 4 to J ′
yields a standard graded prime ideal P with deg(P ) = 396r and reg(P ) ≥ 22r−1 +66r +6. □
In particular, we get counterexamples to the Eisenbud–Goto Conjecture for r ≥ 5.
254 Jason McCullough
We recall that the existence of a functionΦ(m,d) bounding the regularity of an ideal generated
by m forms of degree at most d is equivalent to a function Ψ(e) bounding the regularity of
nondegenerate prime ideals of degree e. GivenΦ(m,d), it follows from [7] that
Ψ(e) ≤ max{e2,Φ(e,e2)+1} .
Similarly, given an explicit functionΨ(e), it follows from [20] that
Φ(m,d) ≤Ψ(2(d +1)m) .
There is a parallel version of Stillman’s Conjecture [25] which posits a boundΦpd(m,d) on the
projective dimension of graded ideals generated by m forms of degree at most d . Caviglia showed
that the existence ofΦ(m,d) is equivalent to that ofΦpd(m,d). In fact, he showed that
Φ(m,d) ≤ (2d)2Φpd(m,d)−2
and
Φpd(m,d) ≤Φ(m,d) ·
Φ(m,d)∑
i=0
m2i .
See [22, Theorem 4] for details. There is yet a fourth functionΨpd(e) bounding the projective
dimension of nondegenerate prime ideals of degree e over an algebraically closed field. Given
Φpd(m,d), it follows from [7, Theorem 5.2] that
Ψpd(e) ≤ max
{
e,Φpd
(
e,e2
)−1} .
Similarly, givenΨpd(e), it follows from [20, Theorem 1.6] that
Φpd(m,d) ≤Ψpd
(
2(d +1)m) .
Thus finding explicit formulas for any of Ψ(e),Ψpd(e),Φ(m,d),Φpd(m,d) will yield explicit
formulas for all of the others. Currently the best lower bound estimate forΦpd(3,d) is
p
d
p
d−1
[3,
Corollary 3.6]. Using Theorem 1, we can increase this lower bound as well.
Theorem 6. For any integer r ≥ 1, there is a standard graded ideal I with three generators in
degree 4r +1 with pd(I ) ≥ r 2r . In particular,Φpd(3,d) is (d/5)Ω(d).
Proof. By [3, Corollary 3.7], there is an ideal I with 2r + 1 generators of degree 2r + 1 and
pdS (S/I ) ≥ r 2r . Applying Proposition 1 with m = 2r and s = t = 4r +1 yields an ideal J generated
by three forms of degree 4r +1 and pdR (J ) ≥ r 2r +1. □
Acknowlegements
This work was completed while preparing for the Workshop on Regularity and Syzygies hosted
at the University of Illinois Chicago and organized by Wenliang Zhang. The author thanks Paolo
Mantero, Matt Mastroeni, and Lance Edward Miller for feedback on an earlier draft of this paper.
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.
Jason McCullough 255
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Kansas, Lawrence, USA, 2004, published by ProQuest LLC, 101 pages.
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[8] M. Chardin, B. Ulrich, “Liaison and Castelnuovo–Mumford regularity”, Am. J. Math. 124 (2002), no. 6, p. 1103-1124.
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137.


Prime Ideals and Three-generated Ideals with Large Regularity

Yisheng Tian

A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation

Yujiro Kawamata

Deformations over non-commutative base

Nicolas Burq

Ivan Moyano

Comptes Rendus
Mathématique
Nicolas Burq and Ivan Moyano
Simultaneous control for the heat equation with Dirichlet and Neumann boundary
conditions
Volume 362 (2024), p. 937-942
Online since: 4 November 2024
https://doi.org/10.5802/crmath.535
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
2024, Vol. 362, p. 937-942
https://doi.org/10.5802/crmath.535
Research article / Article de recherche
Control theory / Théorie du contrôle
Simultaneous control for the heat equation
with Dirichlet and Neumann boundary
conditions
Contrôle simultané de l’équation de la chaleur avec
conditions aux limites de Dirichlet et Neumann
Nicolas Burq a and IvanMoyano ∗,b
a Université Paris-Saclay, Mathématiques, UMR 8628 du CNRS, Bât 307, 91405 Orsay Cedex, France,
and Institut Universitaire de France
b Université de Nice Sophia-Antipolis Parc Valrose, Laboratoire J.A. Dieudonné, UMR 7351 du CNRS
06108 Nice Cedex 02 France
E-mails: nicolas.burq@universite-paris-saclay.fr, ivan.moyano@unice.fr
Abstract. It is well known that both the heat equation with Dirichlet or Neumann boundary conditions are
null controlable as soon as the control acts in a non trivial domain (i.e. a set of positive measure). In this
article, we show that for any couple of initial data (u0, v0) we can achieve the null control for both equations
(Dirichlet and Neumann boundary conditions respectively) simultaneously with the same control function
for both equations.
Résumé. Il est bien connu que l’équation de la chaleur avec aussi bien la condition au bord de Dirichlet que
la condition de Neumann est contrôlable à 0 dès que le contrôle agit sur un domain non trivial (i.e. de mesure
positive). Dans cet article, nous montrons que pour tout couple de données initiales (u0, v0) ∈ L2, le contrôle
à 0 peut être réalisé simultanément avec la même fonction de contrôle pour les deux équations.
Funding. The research of N. Burq was partially supported by Institut Universitaire de France and the Euro-
pean Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme
(Grant agreement 101097172 - GEOEDP).
Manuscript received 4 February 2023, revised 29 June 2023, accepted 6 July 2023.
1. Introduction and main results
Let us consider a smooth bounded domainΩ⊂Rd and let ω⊂Ω be a subset of positive measure
|ω| > 0 and the following internal simultaneous controlability problem{
(∂t −∆)u = f 1(0,T )×ω, u |∂Ω= 0, u |t=0= u0,
(∂t −∆)v = f 1(0,T )×ω, ∂νv |∂Ω= 0, v |t=0= v0. (1)
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
938 N. Burq and I. Moyano
Definition 1. We shall say that the heat equation in Ω is simultaneously null controlable with
Dirichlet and Neumann boundary conditions if for any (u0, v0)∈L2(Ω) there exists f ∈L2((0,T )×ω)
such that the solution of the system (1) satisfies
u |t>T = 0, v |t>T = 0.
The question of simultaneous controllability of various partial differential equations has been
raised in the literature (see for example [1, 3, 4, 6, 8] and the recent work [2] for more references
on the subject), especially when the system involves some transmission mechanisms between
the equations allowing to reduce the number of commands. The interest of our problem lies
on the fact that both heat equations in (1) exhibit relatively independent dynamics and yet they
can be steered to zero using exactly the same control. As no coupling exists between these two
equations, this simultaneous controlability is at first glance counter intuitive. Yet, by considering
the two new unknowns w1 = u + v, w2 = u − v , the simultaneous controlability reduces to the
controlability of (w1, w2), with a control acting only on the w1 component of the system. Notice
that the system (w1, w2) is now coupled at the boundary by the transmission conditions
∂νw1 |∂Ω= ∂νw2 |∂Ω, w1 |∂Ω=−w2 |∂Ω,
and we need to show that this coupling is sufficient. However, our strategy will follow a more
direct path (the double manifold) and will not study per se this transmission problem.
1.1. Simultaneous controllability
Our first result is the following
Theorem 2. Let T > 0, ω ⊂ Ω of positive measure |ω| > 0. Then the heat equation in Ω is
simultaneously null controlable with Dirichlet and Neumann boundary conditions.
Remark 3. It is classical that both the heat equation with Dirichlet or with Neumann conditions
are null controlable. The novelty in Theorem 2 lies precisely on the fact that the null controlability
can be achieved for any initial data (u0, v0) ∈ L2(Ω)×L2(Ω) with the same control for both.
Remark 4. The proof we give below relies on the doubling manifold approach from [5]. This
approach is very robust and allows rough domains (of class W 2,∞, for instance) and rough space-
dependent Laplace operators (Lipschitz coefficients).
∆= 1
κ(x)
∑
i , j
∂xi g
i , j (x)κ(x)∂x j ,
where we assume that the coefficients κ, g are Lipschitz and that g is uniformly elliptic. As will
appear clearly, the proof (which is very simple once the results in [5] were established) shows
that all the control results from [5] are true with the same control functions for the Dirichlet and
Neumann heat equations.
Remark 5. It is an interesting question whether similar results might hold for the wave equation.
We plan to address this question in a forthcoming paper. However, in this case, the analysis is
much more involved and we do not expect to get such a general answer.
1.2. Simultaneous controllability and spectral inequalities
Let (eD
λ
)λ be the spectral family associated to the Laplacian inΩwith Dirichlet conditions, i.e.,
−∆eDλ =λ2eDλ , eDλ |∂Ω = 0 (2)
N. Burq and I. Moyano 939
and let (eNµ )µ be the spectral family associated to the Laplacian inΩwith Neumann conditions,
−∆eNµ =µ2eNµ , ∂νeNµ |∂Ω = 0. (3)
Letω⊂Ω be a non-empty subset. In the spirit of [7], one can show that the spectral families (eD
λ
)λ
and (eNµ )µ enjoy a concentration property on the subsetsω as long as they are not too small. In [5]
the authors shows the following: given ω with |ω| > 0, there exist constants C ,D such that all the
spectral truncations
ΠDΛu :=
∑
λ≤Λ
uλe
D
λ , u ∈ L2(Ω), Λ> 0,
satisfy the estimate (cf. [5, Theorem 1]):
∥ΠDΛu∥L∞(Ω) ≤CeCΛ∥1ωΠDΛu∥L1(Ω), ∀ u ∈ L2(Ω). (4)
The analogous estimate also holds for the spectral truncations of (eNµ )µ, defined by
ΠNΛu :=
∑
µ≤Λ
uµe
N
µ , u ∈ L2(Ω), Λ> 0.
In this note we show that these spectral inequalities also hold simultaneously (i.e. we can estimate
each spectral projector by the sum on arbitrary small set of positive measures).
Theorem 6. Let ω⊂Ωwith |ω| > 0. There exist C ,D > 0 such that for anyΛ> 0, we have
∥ΠDΛu∥L∞(Ω) +∥ΠNΛ v∥L∞(Ω) ≤CeCΛ∥1ω(ΠDΛu +ΠNΛ v)∥L1(ω), ∀ u, v ∈ L2(Ω). (5)
2. Double manifold and spectral estimates
In this section we recall a result from [5], which allows to glue any given manifold M with a copy
of itself along its boundary, in order to produce a double manifold without boundary. This will be
a crucial point in the analysis below.
2.1. The double manifold
Let (M , g ) be a compact Riemannian manifold of class C 1 ∩W 1,∞.
Let ∆ be the Laplace–Beltrami operator on M and let (ek ) be a family of eigenfunctions of −∆,
with eigenvalues λ2k →+∞ forming a Hilbert basis of L2(M).
−∆eD ;Nk =λ2k ek , eDk |∂M= 0 (Dirichlet condition) or ∂νeNk |∂M= 0 (Neumann condition).
Let be M̃ the double space made of two copies of M
M̃ = M × {−1,1}/∂M ,
where we identified the points on the boundary, (x,−1) and (x,1), x ∈ ∂M . In the double manifold
M̃ we have the following result.
Theorem 7 (The double manifold, [5, Theorem 7]). Let g be given. There exists a W 2,∞ structure
on the double manifold M̃, a metric g̃ of class W 1,∞ on M̃, and a density κ̃ of class W 1,∞ on M̃ such
that the following holds.
• The maps
i± : x ∈ M → (x,±1) ∈ M̃ = M × {1,−1}/∂M
are isometric embeddings.
• The density induced on each copy of M is the density κ,
κ̃ |M×{1,−1}= κ.
940 N. Burq and I. Moyano
• For any eigenfunction e with eigenvalue λ2 of the Laplace operator −∆ = − 1κ div g−1κ∇
with Dirichlet boundary conditions on M, there exists an eigenfunction ẽ with the same
eigenvalue λ2 of the Laplace operator −∆=− 1κ̃ div g̃−1κ̃∇ on M̃ such that
ẽ(x,1) = e(x), ẽ(x,−1) =−e(x) (6)
i.e. gluying e and −e along the boundary (notice that it makes sence since e |∂M= 0) gives
an eigenfunction of the Laplace operator on the double manifold
• For any eigenfunction e with eigenvalue λ2 of the Laplace operator −∆ = − 1κ div g−1κ∇
with Neumann boundary conditions on M, there exists an eigenfunction ẽ with the same
eigenvalue λ2 of the Laplace operator −∆=− 1κ̃ div g̃−1κ̃∇ on M̃ such that
∀ x ∈ M , ẽ(x,1) = e(x), ẽ(x,−1) = e(x) (7)
i.e. gluying two copies of e along the boundary gives an eigenfunction of the Laplace
operator on the double manifold
• Conversely, there exists a Hilbert basis of L2(M̃) composed of eigenfunctions of the Laplace
operator ∆̃ which are either such extensions of Dirichlet Laplace eigenfunctions in M or
such extensions of Neumann Laplace eigenfunctions in M.
Remark 8. The last property was not stated explicitely in [5, Theorem 7], but it is straightforward
as the vector space generated by such eigenfunctions is clearly dense in L2(M̃).
2.2. Spectral projector on the double manifold and proof of Theorem 6
Let us denote by Π̃Λ the spectral projector on the manifold M̃ . Let u, v ∈ L2(M) and define the
function
ũ(x,1) = (u + v)(x), ũ(x,−1) = (−u + v)(x). (8)
Clearly if
u =∑
k
uk e
D
k , v =
∑
k
vk e
N
k ,
we get
ũ =∑
k
uk ẽ
D
k + vk ẽNk .
According to the reflection principle of the previous section, we can link the Dirichlet and
Neumann spectral projectors on M and the spectral projector on M̃ by the relation
Π̃Λ(ũ)( · ,1) =ΠDΛ (u( · ))+ΠNΛ (v( · )), Π̃Λ(ũ)( · ,−1) =−ΠDΛ (u( · ))+ΠNΛ (v( · )). (9)
Theorem 9 ([5, Theorem 1]). Let ω̃⊂ M̃ with positive Lebesgue measure. Then, there exists C > 0
such that for anyΛ> 0 and any ũ ∈ L2(M̃), we have
∥Π̃Λũ∥L∞(M̃) ≤CeCΛ∥1ω̃Π̃Λũ∥L1(ω̃). (10)
We can now prove Theorem 6. Indeed, letω⊂ M of positive Lebesgue measure. Let ω̃=ω×{1}.
According to Theorem 9 and (9), we get for any u, v ∈ L2(M),
∥ΠDΛu∥2L∞(M) +∥ΠNΛ v∥2L∞(M) ≤ ∥ΠDΛu +ΠNΛ v∥2L∞(M) +∥ΠDΛu −ΠNΛ v∥2L∞(M)
= ∥Π̃Λũ∥2L∞(M̃) ≤Ce
CΛ∥1ω̃Π̃Λũ∥2L1(ω̃)
=CeCΛ∥1ω×{1}Π̃Λũ∥2L1(ω̃) =CeCΛ∥1ωΠDΛu +ΠNΛ v∥2L1(ω). (11)
N. Burq and I. Moyano 941
2.3. Control and the double manifold
To prove our control result, we could just apply the spectral projector estimate we just proved and
some functional analysis. Here we prefered to prove the result directly on the double manifold.
We start with
Theorem 10 ([5, Theorem 2]). Let ω̃⊂ M̃ be a measurable set with |ω̃| > 0. Then, for every T > 0
and every ũ0 ∈ L2(M), there exists f̃ ∈ L2((0,T )× ω̃) such that the solution to the heat equation on
M̃ satisfies
ũ|t≥T = 0.
We can now prove Theorem 2. For any u, v ∈ L2(M), let us define ũ by (8), and for any ω ⊂ M
of positive measure, let ω̃ = ω× {1}. According to Theorem 10, for every T > 0, there exists
f̃ ∈ L2((0,T )× ω̃) such that
(∂t − ∆̃)Ũ = f̃ 1(0,T )×ω, Ũ |t=0= ũ, Ũ |t≥T = 0. (12)
Let us define next
u(t , x) = Ũ (t , x,1)−Ũ (t , x,−1), v(t , x) = Ũ (t , x,1)+Ũ (t , x,−1),
where Ũ is defined by (12). Notice that u clearly satisfies the Dirichlet boundary condition while
v satisfies the Neumann boundary condition. This second condition is not obvious but comes
from the construction of the double manifold in [5]. Indeed, in our construction, we defined
normal coordinate system near any point in the boundary of M such that M = {xn > 0}, and then
we glued the two copies defined by M × {1} = {xn > 0}, M × {−1} = {xn < 0} by the relation
(x,1) = (xn , x ′,1), (x,−1) = (−xn , x ′,−1),
which implies
∂νv = ∂xn (Ũ (t , x,1)+Ũ (t , x,−1)) |xn=0= ∂xn (Ũ )(t , x,1))−∂xn (Ũ )(t , x,−1)) = 0.
Now, by definition of u and v we have
(∂t −∆)u = f̃ (t , x,1)1(0,T )×ω− f̃ (t , x,−1)1(0,T )×ω̃ = f (t , x)1(0,T )×ω,
as f̃ (t , x,−1)1(0,T )×ω̃ = 0 by the choice of ω̃=ω× {1}. By the same token, we have
(∂t −∆)v = f̃ (t , x,1)1(0,T )×ω+ f̃ (t , x,−1)1(0,T )×Ω̃ = f (t , x)1(0,T )×ω.
As a consequence, u and v solve (1) with control f 1(0,T )×ω. Finally, using (12), we get
u|t≥T = 0, v |t≥T = 0,
which ends 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.
942 N. Burq and I. Moyano
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Simultaneous control for the heat equation with Dirichlet and Neumann boundary conditions

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