Annales de l’institut Fourier (AIF)

Voros, André

ANNALES DE L’INSTITUT FOURIER
ANDRÉ VOROS
Corrigendum: Résurgence quantique
Annales de l’institut Fourier, tome 44, no 3 (1994), p. 987
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Ann. Inst. Fourier, Grenoble
44, 3 (1994), 987
CORRIGENDUM
"Résurgence quantique"
par André VOROS
Article paru dans le tome 43 (1993), fascicule 5, pp. 1509-1534
Par suite d'un incident technique au tirage final, la figure 2 se trouvant
à la page 1519 n'a pas été imprimée dans sa totalité, ce qui la rend
incompréhensible par rapport à sa légende.
Voici la figure telle qu'elle doit être :
î Imr
i^(p)A (27^)-lWp^
-2|x| +2|^| -2|x| + 1 x 1
->Rer
-67T -47T -27T
Nous prions Fauteur et nos lecteurs d'accepter nos excuses pour cette
erreur.


Corrigendum: Résurgence quantique

Brunella, Marco

ANNALES DE L’INSTITUT FOURIER
MARCO BRUNELLA
Foliations on the complex projective plane
with many parabolic leaves
Annales de l’institut Fourier, tome 44, no 4 (1994), p. 1237-1242
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Ann. Inst. Fourier, Grenoble
44, 4 (1994), 1237-1242
FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE
WITH MANY PARABOLIC LEAVES
by Marco BRUNELLA
Let T be a holomorphic foliation with isolated singularities on the
complex projective plane CP2. The leaves of F are non compact Riemann
surfaces whose structure can be very complicated and may be analyzed from
various points of view. From the function theoretic point of view, there is
a classical distinction between parabolic and hyperbolic (=not parabolic)
Riemann surfaces [Tsu] : a Riemann surface R is said to be parabolic
if it does not admit a positive non constant superharmonic function, or
equivalently if it does not admit a Green function. Parabolic Riemann
surfaces have a "small" ideal boundary and are, in some sense, the simplest
Riemann surfaces; examples are closed surfaces minus a finite set. On the
other hand, their topology can be arbitrary and so their simplicity is only
from the analytical point of view.
Aim of this paper is to prove that a foliation with "sufficiently many"
parabolic leaves must have a very simple dynamics. We need the following
concept [Suz]. Let V C CP2 \ Sing(.F) be a F - invariant set, then we say
that V has positive capacity if there exists a disc T transverse to F such
that T H V C T has positive (logarithmic) capacity [Tsu]. We also recall
that a singularity p € Sing(.F) is hyperbolic if the linear part at p of a
vector field which generates F near p has eigenvalues different from zero
and with non real ratio. See [Am], §36, for the topology of this kind of
singularities. Clearly, a generic foliation has only hyperbolic singularities.
THEOREM. — Let T be a foliation with hyperbolic singularities on
CP2. Assume that the set of parabolic leaves
P == {p e CP2 \ Sing(^) | Lp is parabolic}
Key words : Holomorphic foliations - Harmonic measures - Parabolic Riemann surfaces.
A.M.S. Classification : 58F23 - 32L30 - 57R30.
1238 MARCO BRUNELLA
has positive capacity. Then F is a linear hyperbolic foliation, i.e. there is
an affine chart C2 C CP2 such that F\c2 is given by xdy + Xydx = 0, for
some A e C \ R.
Let us observe that some hypothesis on Sing^) is necessary to obtain
the conclusion of the theorem. For example, a foliation with a meromorphic
first integral or a foliation which is the pull-back of a linear foliation by a
rational map of CP2 (of degree >, 2) has all its leaves parabolic, but also
has non hyperbolic singularities.
The proof of the theorem is in two steps : first we prove the existence
of an algebraic leaf L € F (i.e., a leaf whose closure is an algebraic curve),
using the theory of harmonic measures [Gar], then we analyze the structure
of L in order to apply the constructions of [CLS2].
1. Existence of an algebraic leaf.
We firstly recall the (well-known) fact that there is only a countable
set of leaves of F with nontrivial holonomy. This is a consequence of the
finite generation of the holonomy pseudogroup of F, together with the
property that nontrivial fixed points of one dimensional biholomorphisms
are isolated.
For every L € F we denote by L its holonomy covering, i.e. the
covering of L associated to the kernel of the holonomy representation of
L. Because a countable set has zero capacity, the hypothesis that V has
positive capacity means that
V = {p € CP2 \ Sing(^) | Lp is parabolic}
has also positive capacity.
Take now the restriction F* of F to any affine chart C2 C CP2,
with CP2 \ C2 not F- invariant. To every leaf L C F there corresponds
a leaf L* € F" obtained from L by deleting a discrete set, so that the
parabolicity of L or L is equivalent to that of L* or L* [Tsu]. The foliation
F" is defined on a Stein manifold and the leaves with parabolic holonomy
covering form a set of positive capacity. Hence we may apply theorem I of
[Suz] to conclude that every leaf of F* has parabolic holonomy covering. It
follows that every leafofF has parabolic holonomy covering (in particular,
every leaf off is parabolic).
FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 1239
Let pi,...,pfc ^ CP2 be the (hyperbolic) singularities of F. For
every j = !,...,& let (7j be a small round neighborhood of p^, with
Sj = 9(7j real analytic and transverse to T. Let Q be the foliation obtained
by doubling the foliation y\cp2\\Jk_ (7.; it is a transversely holomorphic
foliation without singularities and defined on some compact manifold M
(which is not complex). Moreover, the leaves of Q have a natural complex
structure, obtained by doubling the complex structures of the leaves of
y\cp2\uk_ u- (Schwarz's reflection principle). A theorem of Kusunoki and
Mori [KM] asserts that if R is a parabolic Riemann surface and S C R
is a subsurface with real analytic boundary then the double of S is still
a parabolic Riemann surface. Hence we deduce that every leaf of Q is a
parabolic Riemann surface.
In order to continue the proof, we need a few facts about harmonic
measures [Gar], [Ghy]. Fix a Riemannian metric g on M, such that the
restriction of g to every leaf of Q is hermitian. Let A^ denote the foliated
laplacian associated to (G^g)- A probability measure u, on M is said to be
harmonic if its foliated laplacian is zero, i.e. /i(A^/) = 0 for every function
/ continuous on M and of class C2 along the leaves (we are considering
a measure as a linear functional on the space of continuous functions). It
is established in [Gar] that the set of harmonic measures is non empty;
more precisely, if K C M is a compact G - saturated set then there exists
a harmonic measure whose support is contained in K [Ghy].
The simplest example of harmonic measure is constructed by combin-
ing a holonomy invariant measure (if it exists) with the area form on leaves
of G' Conversely, it is proven in [Ghy] that if fi is a harmonic measure such
that for ft - almost every p € M the leaf Lp € G does not possess a positive
non constant harmonic function, then JLA can be decomposed as a product
of a holonomy invariant measure with the area form on leaves.
All these properties of harmonic measures hold for any foliation on a
compact Riemannian manifold; but in our case the leaves are parabolic
Riemann surfaces and so they do not have positive non constant (su-
per) harmonic functions (recall that the metric g is hermitian on leaves).
Hence a consequence of the previous theory is that for every compact Q-
saturated set K C M there exists a holonomy invariant measure whose
support is contained in K.
Let now K C M be a minimal set of G (possibly K = M) and let p,
be a holonomy invariant measure with supp(/A) C K, that is supp(/^) = K.
Denote still by 5i,..., Sk C M the 3-spheres arising from the boundary of
1240 MARCO BRUNELLA
k k
CP \ U Uj. If KC\ (J Sj = 0, then K corresponds to a compact minimal
j=i j=i
set of .^Icp^Sing^); but this is not possible because such a minimal set
cannot support a holonomy invariant measure [CLS1]. Hence K intersects
some sphere Si, and KnSi is a compact set saturated by the foliation Q\s,,
which is a one - dimensional foliation with two hyperbolic closed leaves as
limit set. The measure p, induces a holonomy invariant measure v for G\Si
with supp(^) == K H Si, it follows that K H Si is contained in the union of
the two closed leaves and hence K is a compact leaf of Q.
Returning to .F, we see that there exists an algebraic leafL € T. The
closure L is an algebraic curve with normal crossings, obtained adding to
L some singular points of F.
2. Holonomy of L and linearization of F.
Because L contains singularities (all hyperbolic), if L is uniformized
by C then L is isomorphic to C* and L either is smooth (diffeomorphic to
a sphere) and contains two singularities, or contains only one singularity
which is a normal crossing. In particular, in both cases the holonomy ofL is
cyclic (because TT^(L) is) with a hyperbolic generator (because singularities
are hyperbolic).
Now assume that L is uniformized by D, i.e. L = D/F where
r c^ TI-I (L) is a finitely generated Fuchsian group. Observe that F contains
parabolic elements, because -L is a closed Riemann surface minus a finite
non empty set. The holonomy covering L of L is isomorphic to D/F, where
r is the normal subgroup of r corresponding to paths with trivial holonomy.
The holonomy group G of L is then identifiable (as an abstract group) with
r/r.
The parabolic elements of r corresponding to paths freely homotopic
to the ends of L have infinite order in the quotient r/r, because the sin-
gularities in L are hyperbolic and hence their separatrices have hyperbolic
(==»• infinite order) holonomy.
Recall that L is a parabolic Riemann surface, and that this is
equivalent to say that T is a Fuchsian group of divergent type [Tsu]. We can
apply a theorem ofVaropoulos [Var] asserting that (because r has parabolic
elements of infinite order in r/r) the group F/F is a finite extension of Z.
FOLIATIONS ON THE COMPLEX PROJECTIVE PLANE 1241
In fact, the holonomy group G ̂  r/F is abelian. To see this, assume
that it is not; then G contains a hyperbolic germ h and also a germ g
tangent to but different from the identity, and clearly id 7^ /i71 7^ ^m -^ id
Vn,m € Z \ {0}. But this is impossible, because a finite extension of Z
cannot contain two infinite cyclic subgroups H\^ H^ with H^C\ H^ = {id}.
Because G is abelian and contains hyperbolic germs (even when L ̂
C*), we may apply the methods of [CLS2] to construct in a neighborhood of
L a closed meromorphic 1-form uj which defines .F, and then to extend uj to
all of CP2 by a Levi type theorem. The polar divisor (uj)oo ofcj is composed
by irreducible algebraic curves Fi, ...,Fn C CP2, and the hyperbolicity of
n
Sing(^7) implies that |j Fj is an algebraic curve with normal crossings
j=i
n
(singularities of (J Fj are also singularities of.77).
j=i
Fix an affine chart C2 C CP2 such that CP2 \ C2 does not contain
singularities of the foliation. Let /i,..., fn be irreducible polynomials which
define Fi, ...,r\ in C2, then we have [CLS2]
ij E \ ^3c2 = ^-rj=i 'f,
where the complex numbers Aj are related to the eigenvalues at the
n
singularities and satisfy ^ \jdeg{fj) = 1. Again the hyperbolicity of the
j=i
singularities (Aj's are pairwise rationally independent) implies that every
Fj is smooth (i.e. without selfintersections), and that n > 3.
n
The foliation F does not have singular points outside (J Fj, because
j=i
outside this set uj is holomorphic (and closed). Hence the cardinality of
Sing^) is given by
aSing(^) = ̂  deg(/,)deg(/,) = ^ fe de^)) - D^^
,\<; " v-,—1 / -,—1
1
0
i>j I \J=  / J=l
2 n
^ ) o E^g -) ^(^g^-))2
On the other hand, ^F\c2 is generated by a polynomial vector field of degree
f "d = ^ deg(/j) — 1 (^ 2) and hence (because the line at infinity is not
[j=i
invariant)
aSing(JF) = (d - I)2 + (d - 1) + 1 = d2 - d + 1.
1242 MARCO BRUNELLA
Comparing the two expressions :
^(degCf,))2^-^^-!
j==i
and, taking into account that d > 2 and n >, 3, the only possibility is d = 2,
n = 3, deg(fj) = 1 Vj, from which it follows easily the conclusion of the
theorem.
Acknowledgements : I thank C. Camacho, E. Ghys, A. Lins Neto and
P. Sad for many interesting conversations, and IMPA for its hospitality
during the elaboration of this paper.
BIBLIOGRAPHY
[Arn] V. I. ARNOL'D, Chapitres supplementaires de la theorie des equations
differentielles ordinaires, Mir, Moscou (1980).
[CLS1] C. CAMACHO, A. LINS NETO, P. SAD, Minimal sets of foliations on complex
projective spaces, Publ. IHES, 68 (1988), 187-203.
[CLS2] C. CAMACHO, A. LINS NETO, P. SAD, Foliations with algebraic limit sets,
Ann. of Math., 136 (1992), 429-446.
[Gar] L. GARNETT, Foliations, the ergodic theorem and brownian motion, Jour. of
Punct. Anal., 51 (1983), 285-311.
[Ghy] E. GHYS, Topologie des feuilles generiques, preprint ENS de Lyon (1993)
[KM] Y. KUSUNOKI, S. MORI, On the harmonic boundary of an open Riemann
surface, I, Jap. Jour. of Math., 29 (1959), 52-56
[Suz] M. SUZUKI, Sur les integrales premieres de certains feuilletages analytiques
complexes, Seminaire Norguet, Springer Lect. Notes, 670 (1977), 53-79.
[Tsu] M. TSUJI, Potential theory in modern function theory, Maruzen, Tokyo (1959).
[Var] N. Th. VAROPOULOS Random walks on groups. Applications to Puchsian
groups, Ark. for Math., 23 (1985), 171-176
Manuscrit recu Ie 9 decembre 1993,
revise Ie 18 fevrier 1994.
Marco BRUNELLA,
Dipartimento di Matematica
Piazza di Porta S. Donato 5
40127 Bologna (Italia).


Foliations on the complex projective plane with many parabolic leaves

Feffermann, Charles

Narasimhan, Raghavan

On the polynomial-like behaviour of certain algebraic functions

Putinar, Mihai

On dense ideals in spaces of analytic functions

Dries, Lou van den

Miller, Chris

Extending Tamm's theorem

Cartwright, Donald I.

Młotkowski, Wojciech

Steger, Tim

Property $(T)$ and $\tilde{A}_2$ groups

Languasco, Alessandro

Perelli, Alberto

ANNALES DE L’INSTITUT FOURIER
ALESSANDRO LANGUASCO
ALBERTO PERELLI
On Linnik’s theorem on Goldbach numbers in
short intervals and related problems
Annales de l’institut Fourier, tome 44, no 2 (1994), p. 307-322
<http://www.numdam.org/item?id=AIF_1994__44_2_307_0>
© Annales de l’institut Fourier, 1994, tous droits réservés.
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Ann. Inst. Fourier, Grenoble
44, 2 (1994), 307-322
ON LINNIICS THEOREM ON GOLDBACH NUMBERS
IN SHORT INTERVALS AND RELATED PROBLEMS
by A. LANGUASCO and A. PERELLI
1. Introduction.
Define a Goldbach number (G-number) to be an even number rep-
resentable as a sum of two primes, and write L = log N. The first result
concerning the existence of G-numbers in short intervals is due to Lin-
nik [6] who proved, assuming the Riemann Hypothesis (RH), that for any
e > 0 and N sufficiently large, the interval [TV, N + L3"^] contains a G-
number. Linnik's result was improved by Katai [4] and, independently, by
Montgomery-Vaughan [7] who showed that the interval [N, N + CL2} con-
tains a G-number provided C and N are sufficiently large.
Linnik used the circle method in the proof of his result, while Katai
and Montgomery-Vaughan exploited the connection between G-numbers
and primes in short intervals. Indeed, Katai and Montgomery-Vaughan's
result follows easily from the following estimate, due to Selberg [9] under
RH,
/.N
(1) J ( N , H ) = ^ { x ^ - H ^ - ^ x ^ - H ^ d x ^ N H L 2 .
J i
Estimate (1) has been proved by other methods by Saffari- Vaughan [8] and
Gallagher [1]. We remark that the slightly weaker estimate, still under RH,
(2) J(7v, H) < NHL3
may be obtained in a straightforward way using the explicit formula for
^(x). Estimate (2) corresponds, in a sense, to Linnik's result. The method
Key words : Goldbach problem.
A.M.S. Classification : 11P32.
308 A. LANGUASCO, A. PERELLI
of Saffari-Vaughan for the proof of (1) is based on a preliminary ingenious
averaging technique which makes more efficient the use of the explicit
formula.
In 1990, Goldston [2] pointed out that the result of Katai and
Montgomery-Vaughan can also be obtained by the circle method and hence,
in a way, closer to Linnik's approach. However, Linnik's argument needs no
use of (1) as opposed to Goldston who reduces the problem to this estimate,
via an application of Gallagher's lemma.
The principal aim of this paper is to show that a variation of Linnik's
original approach is capable of proving the result of Katai and Montgomery-
Vaughan without the use of estimate (1). This is obtained first by inserting
the Saffari-Vaughan technique into the machinery of the circle method and
then by avoiding the use of ParsevaPs identity in a critical part of the unit
interval.
Throughout this paper, we will formulate our arguments in terms of
the infinite exponential sum
S(a) = ̂ A^e-^e^a),
n=l
as Linnik himself did. However, completely analogous results may be
obtained by using, instead, the finite exponential sum
S{a) = ̂  K(n)e(na).
n<,N
The modifications required in using S(a) in place of S (a) are based on the
explicit formula for ^(rc).
THEOREM 1. — Assume RH and let z = — — 2ma. For N sufficiently
large and 0 < $ < . we have
f S{af - 4 da < N^L2 + N^L.
We remark that the estimate in Theorem 1 appears to be in contrast
with Goldston's statement at the beginning of sect. 4 of [2]. In fact, the
loss of a factor L in Linnik's work is due to his treatment of the explicit
GOLDBACH NUMBERS IN SHORT INTERVALS 309
formula for 5(0;), which does not use the smoothing technique of Saffari-
Vaughan, and not to the fact that S(a) does not truncate at N. The above
loss corresponds to the loss of a factor L in (2), compared with (1).
From Theorem 1 we deduce the following
COROLLARY 1. — Assume RH. There exists a constants C > 0 such
that, for N >_ 2, the interval [N, N + CL2} contains a G-number.
Essentially, our method can be used to obtain G-numbers in intervals
[TV, N + H] for those H for which an estimate of the form
/*-* S(a)2 da < cN
holds, where c > 0 is a suitable constant. A simple consequence of Theorem
1 is the following
COROLLARY 2. — Assume RH. For N sufficiently large and 0 < ^ < -
we have
f S(a) 2 da = N- arctan 27r7v$ + 0(A^L2) + O^N^^LY^
1-fiJ-(- 7T
Corollary 2 should be compared with the result provided by the
Parseval identity, i.e.
r1/2 ,^ ,2 MT.NL/1/-Z-1/2 S(a)\ da(3)
Hence Corollary 2 may be regarded as a conditional truncated version of
(3). However, note that taking ^ = - in Corollary 2 only gives the weaker
Zf
result
V2 ^ 2/ S(a) da<^NL2.-1/2
A sharper version of Theorem 1, and hence of Corollary 1 and 2, may
be obtained by assuming the Montgomery pair correlation conjecture in
addition to RH.
In 1959, Lavrik [5] proved that
(a)|2da= b—a-NL+0(N\og2L)
310 A. LANGUASCO, A. PERELLI
if 0 < & — a < 1. An unconditional result concerning truncations of
ParsevaPs identity, which improves Lavrik's result, is the following
THEOREM 2. — Let 0 <: b - a <; 1 and N be sufficiently large. Then
{ S{o) 2 da = b-^a NL + 0(N(L(b - a))173) + 0(N).
Ja 2
We remark that Theorem 2 is essentially best possible, in the sense
that one cannot replace the term 0(N) by o{N).
An application of Theorem 2 can lead, under suitable circumstances,
to a sharpening of results which involve the use of ParsevaPs identity. For
example, the use of Theorem 2 instead of ParsevaPs identity in Linnik's
original arguments allows one to remove the e in Linnik's result. This
should be compared with Goldston's comments on the Fourier polynomial
V{a) in sect. 4 of [2]. In the same way, Theorem 2 may replace the partial
integration argument in the proof of Corollary 1.
We finally remark that Theorem 2 enables one to deduce the order
^ 1^ J2 , , , . . 1/: 5'(a) da in the whole range 0 < ^ <: -. Writingof magnitude of - 2 '
f(x) x g(x) for g(x) < f(x) < g(x), we have
COROLLARY 3. — Let N be sufficiently large. Then
^
i-n
N^ if0^^
•^<-<za)\ dax < N
A^ ifj^^j.
Corollary 3 may be regarded as a truncated version of ParsevaPs
identity.
2. Proof of Theorem 1.
We use the following explicit formula
S(a)=--^z-^(p)+0(L3)
GOLDBACH NUMBERS IN SHORT INTERVALS 311
where z = _ _ - 2ma and p = . + ry runs over the non-trivial zeros of
C(5), see [6]. Hence
w-^« E^^) + -E^"^)
(4)
say. Since
we have
(5)
(6)
(7)
and
(8)
+L3 ^z-"r(p) +L3 1 +£6=i?l+...+i^5,
- < min f N, -z \ a ,
/•( ^
/ Asda < ̂ L6,
J-i,
f R^da < A^1/^3,
J-S
^ Asda « <ev2£3 f /s filda)
•/-C V-S /
1/2
/.$ / ^ ~ \ 1/2
/ fizdo' < AT1/2 / R^da
J-^ \J-^ )
Since z~(' = |2;|-pexp(-^parctan27^A?'a), by Stirling's formula we
have that
^ ̂ Hp) < ̂  l^l"172 exp (7 arctan 27r^Va - 7r l^l).
If 70; ^ 0 or |Q:| ^ — we get
E^'rCoXAr1/2,
where, in the first case, p runs over the zeros with 70 ^ 0.
312 A. LANGUASCO, A. PERELLI
Hence
(9) ^ -\ Rida < N^
J-f.
if 0 < ^ < . and
/*^ - r^ 2 r-l/N
(10) / Jlida</ V;^r(p) da+/ Y z-^(p)
J-^ JI/N ̂  J^ ^ da+N^
if ^ > xr • We will treat only the first integral on the right hand side of
(10), the second being completely similar.
Clearly
(n) / | E ̂ 0°)2da = E f E ̂ -pr^)2da + °(1)
t/l/^ ^>o fc-i ^»7 /v^n^/^s^ A^l177? 7>0
where r ] = r ] k = ^ , — . < r ] < ^ and JC is a suitable integer satisfying
J<r = 0(L). Writing arctan27rA^a = -- - arctan——— and using the
c ff • ^T ^ . u • i 2 27rA^a 6Sanan-Vaughan technique we have
r^ 2 /-2 / /.2^ 2 \
/ \Y^z-^{p) da< ( ^z-^(p) da]d6
^ 7>0 ^ W ^ /
(12)
where
== E Er^i^^e^^^^J
7l>0'y2>0
/•2 / f2S^ \
J = J(N,T],^,^) = [ A(a)/2(a) da d6,
Ji \J^ j
/i(a) = \z\-l-i^-^ and f^a) = exp (-(71 + 73) arctan —!—V
\ 27rNa )
Now we proceed to the estimation of J . Integrating twice by parts
and denoting by Fi a primitive of /i and by Gi a primitive of -Fi, we get
J = — (Gi(4?7)/2(47?) - Gi(27?)/2(277))
GOLDBACH NUMBERS IN SHORT INTERVALS 313
-^)^)-c.(j)/.(j))
-2^2Gl(2^)/2(2^)^+2/'2Gl(^)/2(^)^
/•2 / -2^ \
(13) + / / Gi(a)f^a)da} d£.
Ji \J^ ]
If a > — we haveN
^<W}f^
'^^{W-W1}^
hence from (13) we get
'"'J< i. î̂ it1" (2!^)2)exp(-c(2•^))•
where c > 0 is a suitable constant.
In order to estimate Gi (a) we use the substitution
/ 1 \ i/2
(15) u=u(a)=^^+4^2a2) ,
thus getting
Fi(a)=-1- fu-^-^ du
^J ("2-^)1/2 '
By partial integration we have
(16)
F,(a) = 1 [ ^l•^(71•72) + Li-.(..-y.) ^^ ^
lv ' 27r(l-^(7l-^))t(u2-^)l/2+7u (^2_^)3/2j-
Prom (15) and (16) we get
(13) Gl(a) = 2.(i-^-^) {^+ y^"}'
314 A. LANGUASCO, A. PERELLI
where
i r ^2-1(71-72)
^ = 2. j ̂ ^- du
and
/ ^2-1(71-72)^^ (.2-^)3/^-
Again by partial integration we obtain
A(a) - 1 J ̂ 3•^(71•72) + 2 L^-^ ^ IAw-2^3-i^-^))\ u^-^ ' i ^ u ("2-^)2J
and
Bfo) - 1 J ^3-^("l-72) , 3 [^(^) ^ \
Bw ~ 3-z(7i -72) I (K2 - ̂ )3/2 ' 7 (u2 - ̂ )5/2 J •
[ ^7 1Hence by (15) we have for a € _ , 477 that
W ^)«T^^r«TT|^i.
(") B(»)«IT^.
where A(a) and B(a) satisfy A^) = ^ ( ^ ) =0,and from (17) - (19)
we obtain
P«) °.(")«in r̂f
for a6 [j^j-
From (14) and (20) we get
^\^L^(^p\\
l+|7i-72|2 \ \ NT] ) )
hence from (12) and Stirling's formula we have
(21)
f|E-r(^« E E ̂ ^(-c^)).
^ '^>0 ' 7i>072>0 r 171 721 v v ^ / /
GOLDBACH NUMBERS IN SHORT INTERVALS 315
But
L.^Tl+T^2 ! / f^+^\\ ( C ^ \M-^rJ j^rl-^r)^^'^-^)'
hence (21) becomes
<22' ^(-^SrrCT^2
since the number of zeros p2 = ^ + ^72 with n < |7i - 72! < ri + 1 is
0(log(n+|7i|)).
From (9)-(11) and (22) we get
^ -(23) / Aida<A^L2,
J-f.
and Theorem 1 follows from (4)-(8) and (23).
3. Proof of Corollaries 1 and 2.
Assume that H, N e N, H < N, and define
H ^ H
L(a) = ^ e{—ma) = V^ a(m)e(—ma)y
m=l m=—H
where a(m) = H — |m|,
R(n)= ^ A{h)A(k) and R{a) = S(a)2 - -^,
h+k=n z
where z = — — 2ma. We have
N+H ^
^ a(n - N^e-^RW = / S(a)2L(a)e(-Na)da
n=N-H " - k
(24) = ( 2 L(^)e(-^a)da+ f 2 R(a)L{a)e(-Na)da = h + h.
J-^ z t/-i
say.
^16 A. LANGUASCO, A. PERELLI
We evaluate Ji using the residue theorem. We have
N+H 1
ii = ^ a^n
n=N-H
/9r\ ,- v-^ / Ar\ / 2 e(—na)(zo) ^i > (n-N) I v , / ,h E -N)r ̂
^  J-k y
-da.
If r > . we get
r \ da r ̂/ n? < / -T^- / l^l2 7i a25 <1,
hence
(26) f
2 e(-na) . /'T e(-na)
j_^-^da=j_^^-lda+0^
uniformly for T ^ -. But
(27) /^-^r^.
Let r denote the left half of the circle s - — = 27i-r. By the residue
theorem we get
g-n/TV /*-^+27riT
27T% 7_ _2^r 5^/:::eT)——^/ex^-
(28) ^ne-^+ofiV
Letting T -^ oo, from (26)-(28) we get
(29) r^ e(^a)da = ne-^ 4- 0(1)
uniformly for n < 2N. Hence from (25) and (29) we obtain that
N-{-H
Ii= ^ a(n - N^e-^ + 0(H2)
n=N-H
(30) =7V ^ a(n - TV) + 0(^3) = H^L^O{H3Y
6 n=N-H e
GOLDBACH NUMBERS IN SHORT INTERVALS 317
We have
r-n ~ /-5 „
(31) J^/ |-R(a)|L(a)da+/ |A(a)|£(a)da.
J — -77 J -77-k -IT!
Since
W^mmfH2,——},(32)
from Theorem 1 we get
/•*
(33) /" ^{a^Ua^da^HNL^+H^NL.
J--h1-n
From (32) we obtain
/'5 ~ t^ ~ da
(34) / |fi(a)|L(a)ria« / |J?(a)|-^,
•'•ff •'•? "
and by partial integration and Theorem 1 we get
/'? ~ /7
(35) y ^ |fi(a) | ̂  < ff^VL2 + ff3/2^^.
H
Hence from (31) and (33)-(35) we have
(36) J2 < HNL2 + H^NL,
and from (24), (30) and (36) we finally get
N-\-H H2N(n) = ——e(37) ^ a(n-^)e-
n/N^(n   +0 (^3 + ̂ A^L2 + H^NL).
n=N-H
Choosing ^f = CL2, C > 0 sufficiently large, from (37) we have that
N-{-H
^ a{n-N)e-n/NR(n)^H2N
n=N-H
and Corollary 1 follows.
In order to prove Corollary 2 we observe that
1 2 -1S(a) 2 = .— + 0 (l?i + ... + R^
\z\
318 A. LANGUASCO, A. PERELLI
see (4), and
^ da N
I —pr = — arctan27r7V^.
J-fi M2 TT
Corollary 2 follows then from (5)-(8) and (23).
4. Proof of Theorem 2 and Corollary 3.
We recall the properties of Vinogradov's auxiliary function, see [10]
p. 196. Let r be any non-negative integer and /z, 77, A be such that
(38) 0 < A < j and A < 7 7 - / ^ < 1 - A .
There exists a function ^(a) = ̂ ^rj^(a) periodic of period 1 such that
i) ^(a)=l i^+^ <c^<rj-^
ii) 0<^(a ) < l i f / ^ - -̂ - < . a < / A + ^-,77- , < a < 7 7 + A
iii) ^(a) =0 i f 7 7 + — < a < l + / ^ - —z z
+00
iv) ^(a) = 77 - ̂  + V^ a(m)e(ma), where
7n=—CX3
m^O
/Q0\ I ^ M ̂  • ( 1 ^ r+ l^ r^(39) a(m)^mm 7 7 - ^ , — — — — - .
\ Trm \7T7nA/ /
Define
r 1 ^ 2
J(7V,^77,A)= / 5(a) ^,A(a)da.
Jo
We need the following
LEMMA. — Let ^ 77, A satisfy (38). Then
W/^,A) = (77-^) f^ +0(7V)) +0 f^V f^V^V
v ^ / V ^ 7 /
GOLDBACH NUMBERS IN SHORT INTERVALS 319
Proof. — By iv) we have
(40)
00 +00
I(N,^l,r|,A)=(r)-^^A2(n)e-2n/N+ ^ a(m)^(N,m),
n=l m=—oo
m^O
where
00
$(A^m)= ^ A(/l)A(AOe-^+^/N.
/i,fc=i
/i—A;=?n
From the prime number theorem with remainder term we get
00 1\FT
(41) ^A2(r^)e-2n/JV=-^-+OW,
n=l
and from Theorem 3.11 of [3] and partial summation we get
(42) $(A^m)<7V6(|m|)
uniformly for m 7^ 0, where
w-nO-^)!!^,1!)-
|̂m| v ^ / / pt|m| v ^ /
Hence from (40)-(42) we obtain
(43)
/NT \ ( +00 ^
J(7V^,^A)=(r7-/.)(^-+0(JV)J+0 ^V ^ |a(m)|©(|m|) .
v / \ TM=—00 /
\ m^O /
Now we use the estimate
(44) ^ @(n) < x,
n<x
see e.g. [2]. From (39) and the choice r = 1 we have that
[ j ] - [ i if|m|^(A(77-^))- l /2
(45) a{m) < < 1 . , , . , . 1 / 2
Im^A ^H^^-^)) / -
320 A. LANGUASCO, A. PERELLI
hence from (44), (45) and partial summation we obtain
(46)
+oo , ^ \ 1/2
^ |a(m)|6(|m|)«(^) ,
m=—oo \ — /
m^O
and the lemma follows from (43) and (46).
Suppose first that
(47) 0 < A < - and 0 < & - a < l - 2 A ,4 — — ?
and choose p, == a - — and T] = b+ —. Hence (38) is satisfied and we have
/ S(a)\a =I(N^^ A)
J a
[a - 1 2 y^^ _< ^ ,2
W ^,r?,A(o:)ria- / S(a)\ ^,A(a)da- / S(a)
Ja-A 1 Jb
(^ =J(^,^77,A)-Ji-J2,
say. By i) and ii) we have
(49) A<W// ,7/ ,A)
where ^/ = a- . A and 77' = a+ — . It is easy to verify that ^', T/, A satisfy
(38). An analogous upper bound holds for 1^ too.
Choosing
(50) A= b — a
~L2~
1/3
from (48), (49) and the lemma we obtain
(51) [ S(a)2da=b—aNL+0(N(L(b-a))l/3)+0(N),
J a "
provided that (47) holds.
If
(52)
then
l - 2 A < 6 - a < l ,
(53) r -
 2 r - 2 /'i-y 5(a) da= y 5(a) da- \S{(a) | da
GOLDBACH NUMBERS IN SHORT INTERVALS 321
with |J| < 2A. Hence we may treat fj S(a) da in a way similar to the
treatment of Ji, thus getting
(54) y I S(a) 2 da < A7VL + N.
Since b - a = 1 + 0(A) we have
(55) / S{a)
Jo
i o i,
'^oOl da= -^7VL+0(A7VL)+0(AO,
hence from (50) and (53)-(55) we obtain (51) under condition (52), and
Theorem 2 follows.
The proof of Corollary 3 runs as follows. If 0 < ^ < —, by Stirling's
formula and the zero-free region of (^(s) we have
^z-p^(p)<^|2:|-/3|7|^-iexp(7arctan27^7Va- ̂ l) = o{N),
P P 2 /
where p = /3 + 27 runs over the non-trivial zeros of €(«). Hence, arguing as
in the proof of Corollary 2, we get
/^ ^ 2
/ 5(a) da^N2^.
J-^
^ - 2
Since / S(a) da is an increasing function of ^, from the previous result
J-^
we have that
/ S(a) 2 da^>N
J-^
for -- < ^ < — . The corresponding upper bound follows from Theorem 2.
Corollary 3 follows then arguing in a similar way in the range — < ^ < -.
L 2
322 A. LANGUASCO, A. PERELLI
BIBLIOGRAPHY
[1] P.X. GALLAGHER, Some consequences of the Riemann hypothesis, Acta Arith.,
37 (1980), 339-343.
[2] D.A. GOLDSTON, Linnik's theorem on Goldbach numbers in short intervals,
Glasgow Math. J., 32 (1990), 285-297.
[3] H. HALBERSTAM, H.-E. RICHERT, Sieve Methods, Academic Press, 1974.
[4] I. KATAI, A remark on a paper of Ju.V. Linnik (Hungarian), Magyar Tud. Akad.
Mat. Fiz. Oszt. KozL, 17 (1967), 99-100.
[5] A.F. LAVRIK, Estimation of certain integrals connected with the additive prob-
lems (Russian), Vestnik Leningrad Univ., 19 (1959), 5—12.
[6] Yu. V. LINNIK, Some conditional theorems concerning the binary Goldbach
problem (Russian), Izv. Akad. Nauk SSSR, Ser. Mat., 16 (1952), 503-520.
[7] H. L. MONTGOMERY, R. C. VAUGHAN, The exceptional set in Goldbach's
problem, Acta Arith., 27 (1975), 353-370.
[8] B. SAFFARI, R. C. VAUGHAN, On the fractional parts of x / n and related
sequences II, Ann. Inst. Fourier, 27-2 (1977), 1-30.
[9] A. SELBERG, On the normal density of primes in small intervals, and the
difference between consecutive primes, Arch. Math. Naturvid., 47 (1943), 87—105.
[10] I.M. VINOGRADOV, Selected Works, Springer Verlag, 1985.
Manuscrit recu Ie 7 juin 1993,
revise Ie 21 octobre 1993.
A. LANGUASCO, A. PERELLI,
Dipartimento di Matematica
Via L.B. Alberti 4
16132 Genova (Italy).
e-mail perelli @dima.unige.it
e-mail languasco @dima.unige.it


On Linnik's theorem on Goldbach numbers in short intervals and related problems

Haraoka, Yoshishige

Finite monodromy of Pochhammer equation

Helein, Frédéric

ANNALES DE L’INSTITUT FOURIER
FRÉDÉRIC HELEIN
Inégalité isopérimétrique et calibration
Annales de l’institut Fourier, tome 44, no 4 (1994), p. 1211-1218
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Ann. Inst. Fourier, Grenoble
44, 4 (1994), 1211-1218
INEGALITE ISOPERIMETRIQUE
ET CALIBRATION
par Frédéric HELEIN
Pour tout domaine fl. dans le plan IR2, dont la frontière 90, est
régulière, l'inégalité isopérimétrique relie l'aire de fî, |n|, au périmètre Î2,
\9SÎ\ de la façon suivante :
I^^TrIQI.
Cette inégalité est optimale, et l'égalité a lieu si et seulement si fl. est un
disque. Des inégalités isopérimétriques similaires sont vraies également si
n est un domaine de la sphère 52,
W^^TT- |W|,
ou si fî est un domaine de l'espace hyperbolique H2
|^|2^(47r+|^|)|0|,
avec à chaque fois égalité si et seulement si îî est un disque. Dans le cas du
plan, cette inégalité semble être connue depuis longtemps. Un des plus vieux
exemples est la ville de Carthage, fondée par les phéniciens. Les fouilles
archéologiques relèvent que les plus anciennes fondations de Carthage
s'étendaient sur un territoire contenu entre la côte méditérannéenne et
un arc de cercle. Selon la légende, il fut accordé à Didon, fondatrice de
Carthage, autant de terres qu'elle pouvait clôturer avec la peau d'une vache.
Didon, fort astucieusement fit tailler dans la peau d'une vache une fine
lanière avec laquelle elle encercla un territoire semi-circulaire.
Il existe de nombreuses démonstrations pour l'inégalité isopérimétri-
que. Le lecteur intéressé pourra consulter [l], [2], [5], [6].
Notre propos ici est de donner une preuve directe reposant sur une
idée inspirée des calibrations. Le principe, utilisé par exemple par Fédérer
Mots-clés : Inégalités isopérimétriques - Calibrations - Lagrangiens nuls.
Classification A.M.S. : 49B05 - 49C05 - 51M25 - 53C42.
1212 FRÉDÉRIC HELEIN
[Fe], et largement exploré par Harvey et Lawson [Hl], en est simple :
supposons que l'on sache définir sur une variété riemannienne M une p-
forme (3 telle que df3 = 0 et
|/3| ^ 1 sur M.
Soit Af une sous-variété orientée de dimension p de M. On dit que Af
est calibrée par f3 si la restriction fS\^ de f3 à M est exactement la forme
volume sur Af. Alors toute sous-variété U calibrée par /? est minimale parmi
les variétés qui sont homologues à Af. En effet, pour toute sous-variété ./V',
homologue à A/", on a
1^1= / /?= / fs^m,j^ JM' .^JM
en vertu de la formule de Stokes. La stratégie que nous mettons en place
ici est une variante.
Notons Ec l'espace de dimension 2 de courbure constante égale à
c e {-l,0,l}(E_i = ff2, Eo = R2, Ei = S^.UEc sera le fibre des
vecteurs unitaires tangents au-dessus de Ec, c'est-à-dire la sous-variété du
fibre tangent TEç définie par
UEc={^y)çTEc/\t\=l}^
où y est un point de Ec et t ç TyEc. Introduisons la variété Me égale au
produit
UEc x Ec = {^ ?/, x ) / ( t , y) ç UEc, x e EJ.
À tout domaine régulier Sî de Ec, nous allons associer dans Me la
sous-variété à bord Af définie de la façon suivante. En tout point y de 90,
on note t(y) le vecteur unitaire, tangent à 9fl en y tel que si n(y) est le
vecteur normal extérieur à 9fî en y , (n(y),t(y)) soit un repère direct de
TyEc. Nous noterons alors U^QSÏ le "relèvement" de <9îî dans UEc donné
par
U^Qfl = {(t(y),y) e UEc/y C 9SÏ}.
Enfin nous poserons
M = U^QÇl x îî c Me.
Notre méthode utilise de façon cruciale un champ de vecteur qui, à
tout (t,y,x) e Me, associe un vecteur V(t,y,x) dans T^Ec, de norme 1,
INÉGALITÉ ISOPÉRIMÉTRIQUE ET CALIBRATION 1213
pour x 7^ y. Autrement dit, il s'agit d'une section du fibre
UEc x UEc -^ Me
((^2/)»(^)) 1—^2/^)î
définie sur Mc\^ où A = {(t,x,y) ç. M/x = y}. Pour tout (t,y,x) e
.Mc\A, on considère l'unique cercle de Ec contenant y et x, et tangent
à t en y. Ce cercle est orienté par t. Alors V(ty y , x) est l'unique vecteur
unitaire dans T^Eci qui est tangent à ce cercle, et a même orientation que
t. Enfin si (., .)c désigne le produit scalaire riemannien sur Ec, on considère
la deux-forme sur Ec
a=(V(t,y,x),dx)c A(t,A/)c.
Nous avons alors le
THÉORÈME 1. — Avec les notations précédentes, on a
(i) ;^a=(47r-c|W|,
(ii) J^c^W,
(iii) SQ^-O. = |ôn|2 si et seulement si Q est un disque dans Ec.
COROLLAIRE. — L'inégalité îsopérimétrîque est alors une conséquence
immédiate de la formule de Stokes
\9fî\2^ I a= / da=(47r-c|îî|)|Q|.
JM JQ^
Preuve du théorème. — L'inégalité ii) découle immédiatement du fait
que V(t, î/, x) est unitaire. En effet,
/ Q= \ / <V(t,y,x),dx>c<t,dy>c
JQN J {t,y}ç.U^9Q. JxeQfî
= ( \dy\ ( <V{t,y,x),dx>c
JyçQfl Jxç.9^1
^ ( \dy\ I \dx\ = \9fl\2.
Jyç.Q^Î JXÇ.QÇI
Nous remarquons dans cette inégalité qu'il y a égalité dans ii) si et
seulement si V ( t ^ y ^ x ) est tangent à 90 en x pour tout (t^y) e U^-Q^Ï
et tout x e ôfl\{y}. Cela n'est possible que si 90, est un cercle, ce qui
entraîne iii).
1214: FRÉDÉRIC HELEIN
II ne reste plus qu'à démontrer i). Nous avons besoin pour cela de
préciser un système de coordonnées sur Ec et Me, afin d'y exprimer a et
da.
Cas c = 0. £'o = R est équipé du produit scalaire canonique
^Oo^C'-K'C2,
et de la norme [$|o =< $,$ >^2 . On note
(^o^c'-eV
le produit vectoriel de deux vecteurs de R2.
Cas c = 1. Nous représentons £?i = 52 par son plongement standard dans
R3, équipé du produit scalaire
^Oi^C'+^C'+^C3,
de la norme |$|i =< ç,ç >^2, et du produit mixte
(^C^)=det(^C^).
Ainsi
52 = {y e R^lyli = 1},
US2 = {(t,y) ç R3 x ^/[(li = 1,< t,y >i= 0}.
Cas c = -1. De même, nous représentons .ELi = H2 par son
plongement standard dans E,3 muni du produit scalaire minkowskien
^c^i^c^c2-^:3,
de la pseudo-norme |^|_i =< $,ç >_i, et du produit mixte
(^,C,??)=det($,C,î7).
On a ainsi
H2 = {y £ R3/ < y, y >_i= -l,y3 > Q^
(/a-2 = {(t, y ) ( = R3 x fl-2/ < t, t >_i= 1, < t, y >_i= 0}.
Un calcul élémentaire montre que pour c = —1,0,1,
„, , 2 < x - y , t > c ,
Vc(t, y, x) = ————ï-——c- (x - y) -1.
\x — i/pc
INÉGALITÉ ISOPÉRIMÉTRIQUE ET CALIBRATION 1215
Nous devons calculer f^-da. Pour cela, il n'est pas nécessaire de calculer
da mais seulement
/3=^ ̂ dxa^< ——,dx>c\f\<t,dy>c,
a v /
car da — (3 s'annule sur J\f = U^-Qfl, x fl..
Cas c = 0. Dans ce cas, on obtient
/ ?=2^~^ <t,dy>oMx1 r\dx2.
1^ ~ 2/10
Remarque. — Cette dernière expression se montre simplement en
remarquant que cette relation ne dépend pas du système de coordonnées
choisies, et en la vérifiant pour y = 0 et t = ( ) .
On peut maintenant calculer
f da == / (3 = ( dx^dx2 ( 2 (y ~ xit^ < t,dy >o
JAT JM Jxç^î J(t,y)çU+ôÇî \y~x\0
= 1 dxwS 2^-^y.
Jxç^î JyçQfî \y ~ ̂ lo
Et comme pour x fixé
r2(p^t^^^
L \y~x\o J
il vient grâce à la formule de Stokes
/ ^(p1^/' 4.M^lAd,2=4.,
Jyçôn W ~ ̂ lo Jyçfl
puisque x ç fî. On en déduit donc que
/ da=47r|^|.
JM
Cas c = ±1. Le calcul est similaire. Ici (3 est une 3-forme définie a priori
sur M3 x M3 x M3, mais on ne s'intéresse qu'à la restriction de (3 à M.c'
D'après le Lemme 1, on a
^ = ̂ ^^^^ < ̂ v>c l̂
1216 FRÉDÉRIC HELEIN
où (jj(x) = xldx2 A dx3 -+- x^dx^ A dx1 -h a^cte1 A dx2 est une deux-forme
sur E,3, dont la restriction à Ec est la forme volume sur Ec. D' où
f da= f ( 3 = 2 c f ^ ( x } ! (xlyît-<t,dy>^
JM Jjv Jxesî Jye9Sî I3'- y\c
^[ ^f ^^.
^eîî JyçQ^Î \X-y\Ï
Or, pour a; fixé, on a (voir Lemme 2) :
^^^l^-^^-^^^'L î  y\c j
d'où, par la formule de Stokes,
2c/ ^y^=( (4^)-cMy)
^2/ean 1^-2/ lc ^eîî
= 47r - c|n|.
On conclut finalement que
/ da = ! (47r - c^dx^dx2 = (4^ - c|^|)|Q|.
JM JxÇfl
II ne reste plus maintenant qu'à prouver les deux lemmes suivants.
LEMME 1. — Pour c = ±1, on a
S^A< ̂ ,&>.k-2c^l^)k.
LEMME 2. — Pour c = d=l, et à x 6xé, on a
^^ l̂'̂ M-cM»).
Preuve du Lemme 1. — Comme le résultat cherché ne dépend pas du
système de coordonnées orthonormées directes utilisées, on peut choisir
(°\ f'\y= 1 0 \ e U = 0 .
M/ \0 /
INÉGALITÉ ISOPÉRIMÉTRIQUE ET CALIBRATION 1217
Alors
ex1 x
1 \ /!'
a =
Vc = ———, x2 - 0
1-'3 1.3-1; ^
,2 _,,,1,J,..3|f/c(a;
1)2 \ , ca;^2 , , „]
KÏ-^ ~ 1) dx + T^^ ~ x dx j A < ̂  >c •
Et
/3=c[(^^^2A^3+(c+(^)^A^
x2 1
+ 3 cte1 A dx2 A <t,dy>c .
D'autre part, il existe une fonction fc(a;) et une 1-forme l(x) telles que
Va; e Ec,
f3=k{x)uj(x)^l(x)^d(^Y
Comme on ne s'intéresse qu'à /3|^, on a besoin que de k(x). Ce dernier
coefficient peut être obtenu en faisant le produit extérieur de {3 par
( j/y|2 \d M£)'ca^
w2
d ( la^ ) A (3 = k(x)dx1 A dx2 A dx3,
pour x € Ec. D'où, par identification
x2 2cx2k(x)
1-x3 \x- y]2 '
et
2cx2
^M-= T,—i?'^2^ < ̂ ^y >c |̂ (c
l2' — i/lc
=2c|^la;(a;)A<^^>cl^.
Preuve du Lemme 2. — Ici également, on peut utiliser un système de
coordonnées tel que
( " }x- ? •
1218 FRÉDÉRIC HELEIN
Alors
{x, y , dy) _ yldy2 - y2dyl
^\x-y\ï - 1-2/3 '
et à a; fixé, on a pour tout y ^ x
J'^^^I^.^^A^--^
L \x-y\^ \ (1-î/3)2 (1-2/3)2
+1-^3 ̂ 1A^2-
Et un raisonnement analogue à celui du Lemme 1 donne pour x fixé, et
pour tout y ^ x,
^2c(x,y,dy)'\
'l^^irj,,'-0'^-
On conclut alors aisément que, au sens des distributions
.^[2•l^]ll..-[4^)-cM,,).
BIBLIOGRAPHIE
[1] M. BERGER, Géométrie, chapitre 12, Tome 2, Nathan, 1990.
[2] W. BLASCHKE, Kreis und Kugel, Chelsea.
[3] H. FEDERER, Géométrie Measure Theory, Springer-Verlag, New York, 1969.
[4] R. HARVEY, H.B. LAWSON, Calibrated geometries, Acta Mathematica, 148 (1982),
47-157.
[5] R. OSSERMAN, Bonnensen-styie isoperimetric inequalities, Amer. Math. Monthly,
86 (1979), 1-29.
[6] R. OSSERMAN, Thé isoperimetric inequality, Bull. Amer. Math. Soc., 84 (1978),
1182-1238.
Manuscrit reçu le 11 octobre 1993.
Frédéric HELEIN,
ENS de Cachan et CNRS URA 1611
Centre de Mathématiques et
de leurs Applications
61 avenue du Président Wilson
F-94235 Cachan Cedex.


Inégalité isopérimétrique et calibration

Stolovitch, Laurent

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