ANNALES
DE LA FACULTÉ DES SCIENCES
Mathématiques
GRAHAMSMITH
An Arzela-Ascoli theorem for immersed submanifolds
Tome XVI, no4 (2007), p. 817-866.
<http://afst.cedram.org/item?id=AFST_2007_6_16_4_817_0>
© Université Paul Sabatier, Toulouse, 2007, tous droits réservés.
L’accès aux articles de la revue « Annales de la faculté des sci- ences de Toulouse Mathématiques » (http://afst.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://afst.cedram.
org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
pp. 817–866
An Arzela-Ascoli theorem for immersed submanifolds
(∗)Graham Smith(1)
ABSTRACT.—The classical Arzela-Ascoli theorem is a compactness re- sult for families of functions depending on bounds on the derivatives of the functions, and is of invaluable use in many fields of mathematics. In this paper, inspired by a result of Corlette, we prove an analogous com- pactness result for families of immersed submanifolds which depends only on bounds on the derivatives of the second fundamental forms of these submanifolds. We then show how the result of Corlette may be obtained as an immediate corollary.
R´ESUM ´E.—La version classique du th´eor`eme d’Arzela-Ascoli, qui est d’une tr`es grande importance dans plusieurs domaines math´ematiques, nous donne un r´esultat de compacit´e pour des familles de fonctions en ter- mes de majorations des d´eriv´ees de ces fonctions. Dans cet article, inspir´e par un r´esultat r´ecent de Corlette, on montre un r´esultat de compacit´e analogue pour des familles de sous-vari´et´es immerg´ees ne d´ependant que de majorations sur les d´eriv´ees des secondes formes fondamentales de ces sous-vari´et´es. On montre ensuite comment obtenir le r´esultat de Corlette comme un corollaire imm´ediat.
1. Introduction
In [1], Cheeger proved his famous “Finiteness Theorem” which states that, given positive real numbers K, , R ∈ R+, there exist only finitely many homeomorphism classes of complete manifolds of a given dimension with sectional curvature bounded above in absolute value byK, injectivity radius bounded below by and diameter bounded above by R. Gromov later showed how this result may be viewed in terms of (more precisely, as a corollary of) a compactness result for the family of such manifolds (see, for
(∗) Re¸cu le 3 janvier 2006, accept´e le 7 janvier 2006
(1) Equipe de topologie et dynamique, Laboratoire de math´ematiques, Bˆatiment 425, UFR des sciences d’Orsay, UMR 8628 du CNRS, 91405 Orsay cedex (France).
example [3] and [4]). Viewed from this perspective, the conditions imposed by Cheeger become quite satisfying to our intuition. The curvature bound is probably the most fundamental. Indeed, curvature reflects the “derivatives”
of the manifold (how fast it turns), and whenever derivatives are bounded, following the philosophy of the classical Arzela-Ascoli theorem, one is enti- tled to expect to find a compactness result. The other two conditions reflect more geometrical considerations. The lower bound on the injectivity radius excludes “pinching” (one can consider a sequence of cylinders of ever smaller radius: the only intrinsic data that informs us of degeneration is the injectiv- ity radius which tends to zero), and the bound on the diameter excludes the possibility of adding components indefinitely without introducing very high curvatures (to see what happens without this condition, one can consider a sequence of surfaces of ever increasing genus).
In [2], Corlette proved an analogous finiteness theorem for immersed submanifolds of a given Riemannian manifold. He proves that, given positive real numbers K, R ∈ R+ and a given compact manifold M, there exists only finitely many C1 isotopy classes of complete immersed submanifolds of M with second fundamental form bounded above in norm by K and diameter bounded above byR. This result no longer requires the condition on the injectivity radius of the immersed submanifold since such a lower bound is now a product of the bounds on the second fundamental form of the submanifold and the curvature of the ambient manifold (one may consider again the example of a cylinder in Euclidean space: its injectivity radius cannot become small without its second fundamental form becoming large). Following the philosophy of Cheeger’s finiteness theorem, one would expect this result to also arise from a compactness result for immersed submanifolds.
The compactness result thus obtained bears a perfect analogy to the classical Arzela-Ascoli theorem, being, in certain aspects, a generalisation of this result, and it is for this reason that we have chosen to name it thus. An example of an application of this result may be found in [5]. The statement of this theorem requires the following definition:
Definition 1.1. — Let (M, g) be a Riemannian manifold. Let X = (Y, i)be an immersed submanifold inM. Let∇ibe the Levi-Civita covariant derivative generated over Y by the immersion i into (M, g). Let A(X) be the second fundamental form of X. For all k2, we defineAk(X) using the following recurrence relation:
A2(X) =A(X),
Ak(X) =∇iAk−1(X)∀k3.
We now defineAk(X)for allk2 by:
Ak(X) = k i=2
Ai(X).
The principal result of this paper is now given by the following theorem:
Theorem 1.2. — Letk2, mn∈Nbe positive integers.
Let(Mi, pi)i∈N,(M∞, p∞)be complete pointed Riemannian manifolds of dimensionnand of class at leastCk such that(Mi, pi)i∈Nconverges towards (M∞, p∞)in the pointed Ck Cheeger/Gromov topology.
For alli ∈N, let Σi = (Si, qi) be an m-dimensional pointed immersed submanifold of Mi of type at least Ck such thati(qi) =pi.
Suppose that for allR >0, there existsB such that, for alln:
AkΣn(q)B ∀q∈BR(qn).
Then, there exists a pointed complete immersed submanifold Σ∞ = (S∞, q∞) of M∞ of type Ck−1,1 such that i(q∞) = p∞ and that, after extraction of a subsequence,(Σi, qi)i∈Nconverges towards(Σ∞, q∞)in the pointed weakCk−1,1 Cheeger/Gromov topology.
The terms used in this theorem are explained in section 2 and appendix A of this paper. What we call the Cheeger/Gromov topology is essentially the canonical topology that one would expect to use for immersed submanifolds.
In particular, when k = 2, the condition on the submanifolds amounts to a bound on the norms of the second fundamental forms of the immersed submanifolds.
We would like to underline that in [2], Corlette clearly states that he anticipates that his finiteness result should arise from a compactness result in a way analagous to the Cheeger/Gromov case. We consider however that, given the importance of this result to our own work, it was necessary to properly unearth this theorem and to state and prove it in its own right.
In the second section, we introduce various concepts associated to im- mersed submanifolds, and we describe the Cheeger/Gromov topology for pointed Riemannian manifolds and immersed submanifolds. In the third section, we study the manner in which immersed submanifolds may locally
be described in terms of graphs over the tangent space to each point. The objective here being to bound from below the radius of the disk over which the submanifold is a graph, and to bound from above the derivatives of the function of which the submanifold is a graph. In the fourth section, using the technical results of the third section, we prove theorem 1.2 and we then prove Corlette’s result [2] as a corollary.
In this paper, we will be working in theCk,α category. Since we geome- ters are not in general in the habit of using mappings of this type, appendix A reviews the properties required of a class of functions for one to be able to construct a theory of manifolds out of it, and we then show how the class of Clock,αmappings satisfies these properties. Finally, in appendixB, we provide a proof of the now classical compactness theorem of Riemannian geometry in a form that is most appropriate for our uses. We also hope (perhaps vainly) that we have provided here a slightly more accessible proof of what is an important, but technically challenging result.
I would like to thank Fran¸cois Labourie for introducing me to the subject and drawing my attention to the utility of this result.
2. Convergence of manifolds 2.1. Immersed submanifolds
We review the basic definitions from the theory of immersed submani- folds and establish the notations that will be used throughout this article.
Let M be a smooth manifold. An immersed submanifold is a pair Σ = (S, i) where S is a smooth manifold and i : S → M is a smooth immersion. Let g be a Riemannian metric on M. We give S the unique Riemannian metric i∗g which makes i into an isometry. We say that Σ is completeif and only if the Riemannian manifold (S, i∗g) is.
We introduce the following definition:
Definition 2.1. — Let (M, g) be a Riemannian manifold. Let X = (Y, i)be an immersed submanifold inM. Let∇ibe the Levi-Civita covariant derivative generated over Y by the immersion i into (M, g). Let A(X) be the second fundamental form of X. For all k2, we defineAk(X) using the following recurrence relation:
A2(X) =A(X),
Ak(X) =∇iAk−1(X)∀k3.
We now defineAk(X)for allk2 by:
Ak(X) = k i=2
Ai(X).
2.2. The Cheeger/Gromov topology
A pointed Riemannian manifoldis a pair (M, p) whereM is a Rie- mannnian manifold andpis a point inM. If (M, p) and (M, p) are pointed manifolds then amorphism(ormapping) from (M, p) to (M, p) is a (not necessarily even continuous) function fromM toMwhich sendsptopand isC∞in a neighbourhood ofp. The family of pointed manifolds along with these morphisms forms a category. In this section, we will discuss a notion of convergence for this family. It should be borne in mind that even though this family is not a set, we may consider it as such. Indeed, since every manifold may be plunged into an infinite dimensional real vector space, we may discuss, instead, the equivalent family of pointed finite dimensional submanifolds of this vector space, and this is a set.
Let (Mn, pn)n∈N be a sequence of complete pointed Riemannian mani- folds. For all n, we denote bygn the Riemannian metric over Mn. We say that the sequence (Mn, pn)n∈Nconvergesto the complete pointed manifold (M0, p0) in theCheeger/Gromov topologyif and only if:
(i) for alln, there exists a mappingϕn : (M0, p0)→(Mn, pn),
such that, for every compact subsetK of M0, there exists N ∈Nsuch that for allnN:
(i) the restriction ofϕn toK is aC∞ diffeomorphism onto its image, and (ii) if we denote byg0 the Riemannian metric overM0, then the sequence of metrics (ϕ∗ngn)nN converges tog0 in theC∞topology over K.
We refer to the sequence (ϕn)n∈Nas a sequence ofconvergence map- pingsof the sequence (Mn, pn)n∈N with respect to the limit (M0, p0). The convergence mappings are trivially not unique. However, two sequences of convergence mappings (ϕn)n∈Nand (ϕn)n∈Nare equivalent in the sense that there exists an isometryφof (M0, p0) such that, for every compact subset K ofM0, there existsN ∈Nsuch that fornN:
(i) the mapping (ϕ−n1◦ϕn) is well defined overK, and
(ii) the sequence (ϕ−n1◦ϕn)nN converges toφin theC∞topology overK.
One may verify that this mode of convergence does indeed arise from a topological structure over the space of complete pointed manifolds. More- over, this topology is Hausdorff (up to isometries).
Most topological properties are unstable under this limiting process.
For example, the limit of a sequence of simply connected manifolds is not necessarily simply connected. On the other hand, the limit of a sequence of surfaces of genusk is a surface of genus at mostk (but quite possibly with many holes).
LetM be a complete Riemannian manifold. Apointed immersed sub- manifoldinM is a pair (Σ, p) where Σ = (S, i) is an immersed submanifold in M andpis a point inS.
Let (Σn, pn)n∈N= (Sn, pn, in)n∈Nbe a sequence of complete pointed im- mersed submanifolds inM. We say that (Σn, pn)n∈Nconvergesto (Σ0, p0) = (S0, p0, i0) in the Cheeger/Gromov topology if and only if (Sn, pn)n∈N
converges to (S0, p0) in the Cheeger/Gromov topology, and, for every se- quence (ϕn)n∈N of convergence mappings of (Sn, pn)n∈N with respect to this limit, and for every compact subsetK ofS0, the sequence of functions (in◦ϕn)nN converges to the function (i0◦ϕ0) in theC∞topology overK.
As before, this mode of convergence arises from a topological structure over the space of complete immersed submanifolds. Moreover, this topology is Hausdorff (up to isometries).
2.3. The Class of Ck,α pointed manifolds
We define the space ofClock,α mappings as in appendixA, and we define the following topological structure over the spaceCk,α:
Definition 2.2. — LetΩ⊆Rn be an open set. Let(fn)n∈N, f be func- tions overΩof typeCk,α. We say that(fn)n∈Nconverges tof in theweak Ck,α topology if and only if, for allβ∈(0, α):
(fn−fCk,β(Ω))n∈N→0.
If(fn)n∈N, f are functions of typeClock,αoverΩ, then we say that(fn)n∈N
converges tof in theweakClock,αtopologyif and only if for allp∈Ωthere exists a neighbourhood V of p in Ω such that f ∈ Ck,α(V), fn ∈ Ck,α(V) for alln and(fn)n∈N converges to f in the weakCk,α topology overV.
We show in appendixA that fork1, addition, multiplication, compo- sition and inversion ofClock,αfunctions are continuous operations with respect to the weak Clock,α topology.
We show in appendixA how the class ofClock,α functions for k1 has sufficient structure for us to construct the class ofCk,αmanifolds. We thus make the following definition:
Definition 2.3. — Fork∈Nandα∈(0,1], aCk,αRiemannian man- ifold is a triplet(S,A, g)where:
(1)S is a connected topological manifold,
(2)Ais aCk,αatlas (i.e. all the transition mappings are of typeClock,α) and (3) g is a Clock−1,α metric over (S,A) (i.e. g is described locally in every chart by aClock−1,α function).
We define pointed Ck,α manifolds and immersed submanifolds as for smooth manifolds. We may also define for such manifolds a topology anal- ogous to the Cheeger/Gromov topology. In the case of convergence ofCk,α manifolds, the convergence mappings are all of type Ck,α and the metrics converge in the weak Clock−1,α topology. We call the resulting topology the weak Ck,α Cheeger/Gromovtopology.
2.4. Uniqueness of the Cheeger/Gromov limit
Let U ⊆ Rn be an open subset of Rn. Let g be a Riemannian metric over U. Let T U be the tangent bundle over U. Let π : T U → U be the canonical projection. Let T T U be the tangent bundle overT U. LetV T U be the vertical subbundle ofT T U:
V T U = Ker(π).
LetHT U be the horizontal subbundle of T T U associated to the Levi- Civita connection ofg. We have:
HT U⊕V T U=T T U.
We know that:
HT U, V T U ∼=π∗T U.
Let us denote byiH (resp. iV) the canonical isomorphism which sends HT U (resp. V T U) into π∗T U. We define the metric T g over T T U such that:
T g|HT U =i∗Hπ∗g, T g|V T U =i∗Vπ∗g, HT U ⊥T g V T U.
We callT gtheLevi-Civita liftingofgoverT U. Since the Levi-Civita connection ofg depends on the first derivative ofg, aCk,αbound ongand g−1 yields aCk−1,α bound onT gandT g−1.
We have the following elementary result:
Lemma 2.4. — Let U,V be open subsets of Rn. Let ϕ : U → V be a diffeomorphism. Let g andhbe Riemannian metrics overU andV respec- tively. LetM andN be the matrices representing g andhrespectively with respect to Euclidean metric over Rn. There exists a function C0 such that if ϕ∗h=g, then:
DϕC0(U)C0(MC0(U),M−1C0(U),NC0(U),N−1C0(U)).
Proof. —Sinceϕ∗h=g, we have:
DϕtN Dϕ=M.
Since the mapping A → A2 defines a diffeomorphism of the space of symmetric positive definite matrices onto itself, we may write:
(N1/2Dϕ)t(N1/2Dϕ) =M.
For any matrixA, we have:
AtA=A2. Thus:
N1/2Dϕ2=M. Consequently:
Dϕ=N−1/2N1/2DϕN−1/2 M1/2. The result now follows.
Using the Levi-Civita lifting of the metric, we may generalise this result to higher derivatives of ϕ:
Lemma 2.5. — LetU,V be open subsets ofRn. Let ϕ:U →V be a dif- feomorphism. Letgandhbe Riemannian metrics overU andV respectively.
Let M andN be the metrics representing g and hrespectively with respect to the Euclidean metric over Rn. For all k∈N, there exists a functionCk
such that ifϕ∗h=g, then:
Dk+1ϕC0(U)Ck(MCk(U),M−1Ck(U),NCk(U),N−1Ck(U)).
Proof. —We proof this by induction. By lemma 2.4, the result is true whenk= 0. Suppose that the result is true for allk < m. LetTmϕbe the m’th jet of ϕ sending TmU into TmV. Let Tmg and Tmh be the m-fold Levi-Civita liftings ofg andhrespectively. We have:
(Tmϕ)∗Tmh=Tmg.
Let Mm and Nm be the matrices of Tmg and Tmh respectively with respect to the Euclidean metric. Since TmU (resp.TmV) is a bundle over U (resp.V), forR∈R+, we may consider the subbundleBRmU (resp.BRmV) of balls of radius R (with respect to the Euclidean metric) in TmU (resp.
TmV). Since the result is true for all k < m, there existsR which depends only onMCm(U),M−1Cm(U),NCm(U),N−1Cm(U) such that:
Tmϕ(B1m(U))⊆BmR(V).
There exist functions ˆCm,R and ˆcm,R such that:
Nm−1C0(BR(V)) ˆcm,R(NCm(V),N−1Cm(V)), MmC0(B1(U)) Cˆm,R(MCm(U),M−1Cm(U)).
Finally, there is a functionDm+1such that:
Dm+1ϕC0(U)Dm+1(DTm+1ϕC0(B1(U))).
The result now follows fork =m by lemma 2.4, and the result follows for allkby induction.
In particular, we obtain the following corollary:
Corollary 2.6. — Let U,V be open subsets of Rn. Let (gn)n∈N,g be Riemannian metrics overU such that:
(gn−gCk(U))n∈N→0.
Let(hn)n∈N,hbe Riemannian metrics overV such that:
(hn−hCk(V))n∈N→0.
Let gEuc be the Euclidean metric over Rn and suppose that there exists Λ∈R+ such that, for alln:
1
ΛgEucgn, hn ΛgEuc.
Let(ϕn)n∈N:U →V beCk+1 mappings such that, for alln:
ϕ∗nhn =gn, ϕn(0) = 0.
Then, there exists aCk,1 mapping ϕ0:U →V such that, after extrac- tion of a subsequence (ϕn)n∈N converges to ϕ0 in the weak Ck,1 topology.
Moreover:
ϕ∗0h=g, ϕ0(0) = 0.
Proof. —This follows from the preceeding lemma and the classical Arzela- Ascoli theorem.
Remark. —This result yields the uniqueness of Cheeger/Gromov limits.
3. Immersed submanifolds locally as graphs 3.1. Immersed submanifolds as graphs
It is trivial that every immersed submanifold may be described every- where locally as a graph over a ball of a given radius in the tangent space to the submanifold at each point. In this section, we will show how to obtain a bound from below for the radius of such a ball in terms of the norm of the second fundamental form in a neighbourhood of the given point. We then
show how bounds on the derivatives of the function of which the subman- ifold is a graph may be obtained in terms of bounds on the derivatives of the second fundamental form of the submanifold.
In this section we will only considerC∞ manifolds, although the same reasoning remains valid forCk,α manifolds. Let (S, i) be an immersed sub- manifold in Rn =Rm⊕Rn−m. Let Ω be an open subset of S and letV be an open subset ofRm. We say that (Ω, i) is a graphoverV if and only if there exists a diffeomorphism α: V → Ω and a function f : V → Rn−m such that for allx∈V:
i◦α(x) = (x, f(x)).
We callf thegraph functionof Σ and we callαthegraph reparametri- sationof Ω.
We have the following result:
Lemma 3.1. — Let Σ = (S, i) be an immersed submanifold of Rn = Rm⊕Rn−m. Let U1, U2 be open subsets of Rm. Suppose that there exist open subsets Ω1,Ω2 of S such that, for each k (Ωk, i) is a graph over Uk. For eachk, letαk :Uk →Ωi be the graph reparametrisation of Ωk and let fk :Ui→Rn−m be the graph function.
Suppose that there existsp∈U1∩U2 such that:
α1(p) =α2(p),
then, for everyq in the connected component of U1∩U2 containingp:
α1(q) =α2(q), f1(q) =f2(q).
Remark. —In other words, for a given pair of open sets (U,Ω), the graph function and the graph reparametrisation are locally unique.
Proof. —Let V be the connected component of U1∩U2 containing p.
Let us defineX by:
X ={q∈V s.t.α1(q) =α2(q)}.
The setX is closed. Letπ:Rn →Rmbe the canonical projection. For allq∈V:
(π◦i)◦α1(q) = (π◦i)◦α2(q) =q.
Since (π◦i) is locally invertible, it follows that ifα1(q) =α2(q) then, for allq in a neighbourhood Ω ofq:
α1(q) =α2(q).
Consequently,X is open. The result now follows.
We have the following definition:
Definition 3.2. — Let (S, i) be an immersed submanifold in Rn. Let K be a closed subset of S, and letΩ be an open subset ofRn. We say that (K, i) is complete with respect to Ω if and only if for every Cauchy sequence (xn)n∈N inS such that(i(xn))n∈Nconverges in Ω, there exists x0
inS such that(xn)n∈N converges tox0.
Remark. —If Ω is relatively compact, then this definition is independant of the Riemannian metric chosen over a neighbourhood of Ω.
Let Σ = (S, i) be an immersed submanifold ofRn. For∈R+, suppose that Σ is a complete with respect to the ball B(0). Let pbe a point inS and suppose that:
i(p) = 0, TpΣ =i∗TpS=Rm× {0}.
We define E to be the set of all η ∈ (0,∞) such that there exists a neighbourhood Ω ofpsuch that (Ω, i) is a graph overBη(0). We defineη0
by:
η0= Sup(E).
By the inverse function theorem,E is non-empty, and consequently η0
is well defined. Moreover, by lemma 3.1, η0 ∈ E. We obtain the following result:
Lemma 3.3. — Letr > >0be positive real numbers, and suppose that the closed ball of radius r about p in S is complete with respect to B(0).
There exists a function µgraph(, r) such that one of the following must be true:
(1) η0/2,
(2) Sup{f(p) s.t. p∈Bη0(0)}/2, or (3) Sup{Df(p) s.t.p∈Bη0(0)}µgraph(, r).
Remark. —There are three ways for the submanifold Σ to stop being a graph:
(1) The submanifold leavesB(0).
(2) The submanifold leavesBr(p).
(3) The graph becomes vertical.
The first two conditions in the lemma take into account the first form of degeneration, whereas the last condition simultaneously takes into account the other two.
Proof. —We defineµgraph(, r) by:
µgraph(, r) = 4r2
2 −1.
We will assume the contrary in order to obtain a contradiction. Let Ω be a neighbourhood of p, α:Bη0(0)→Ω a diffeomorphism and f :Bη0(0)→ Rn−ma mapping such that, for allq∈Bη0(0):
i◦α(q) = (q, f(q)).
For allq∈Bη0(0), we have:
Df(q)µgraph(, r).
Since its derivative is bounded, f extends to a continuous function on Bη0(0).
Let us denote bygmthe Euclidean metric overRm. Using the bound of Df, we find that there existsδ2>0 such that:
(i◦α)∗gn 4r2
2 −δ2
gm.
It follows that there existsδ3>0 such that for allq∈Bη0(0):
α(q)∈Br−δ3(p).
Moreover, if (qn)n∈Nis a Cauchy sequence inBη0(0) then (α(qn))n∈Nis a Cauchy sequence in Ω. Since the closed ball of radius r about pin Σ is
complete with respect to B(0), it follows that αextends to a continuous function overBη0(0).
Now, there existsδ1>0 such that, for all q∈∂Bη0(0):
(i◦α)(q) =q, f(q) q+f(q) −δ1.
Consequently, for allq∈Bη0(0):
(q, f(q)) = (i◦α)(q)∈B(0).
In other words, for allq∈Bη0(0), i(α(q)) is contained inB(0).
Letπ:Rn→Rm×Rn−m be the canonical projection. Letpbe a point in ∂Bη0(0). Since, for allq∈Ω:
Det(Tq(π◦i)) = Det(δij+∂if, ∂jfα−1(q))−1/2 (1 +µgraph(, r)2)−n2
It follows that:
Det(Tα(p)(π◦i))= 0.
Thus, by the inverse function theorem, there exists ηp ∈ R+, a neigh- bourhood Ωpofα(p) inS, a diffeomorphismα:Bηp(p)→Ωpand a function fp:Bηp(p)→Rn−m such that for allqin Bηp(p):
(i◦αp)(q) = (q, fp(q)).
Sinceαp(p)∈Ω, it follows that Ω∩Ωp=∅. Letqbe a point in Ω∩Ωp. We have:
(α◦(π◦i))(q) = (αp◦(π◦i))(q) =q.
Since (π◦i)(q)∈Bη0(0)∩Bηp(p) it follows by lemma 3.1 thatαcoincides withαp overBη0(0)∩Bηp(p).
By compactness, there exists a finite set of pointsp1, ..., pk ∈∂η0(p) and δ∈R+ such that:
Bη0+δ(0)⊆Bη0(0)∪ k
i=1∪ Bηpi(pi)
.
Suppose that Bηpi(pi)∩Bηpj(pj) =∅. Since the straight line joining pi
andpj, which is contained inBη0(0) intersectsBηpi(pi)∩Bηpj(pj) non triv- ially, we have:
Bηpi(pi)∩Bηpj(pj)∩Bη0(0)=∅.
The mappings αpi and αpj coincide in this set. Thus, sinceBηpi(pi)∩ Bηpj(pj) is connected, it follows by lemma 3.1 that αpi and αpj coincide over this set.
It thus follows that there existsδ∈R+ such that we may extendαand f toCk,α mappings defined onBη0+δ(0). Moreover, since:
(π◦i)◦α= Id,
it follows thatαis a diffeomorphism onto its image. Consequently:
η0+δ∈E.
We thus obtain a contradiction and the result now follows.
3.2. A bound from below of the radius of definition
Let Ω be an open subset of Rn. Let g be a Riemannian metric over Ω.
LetM : Ω→End(Rn) be a smooth function taking values in the space of positive definite symmetric matrices such that for allVp∈TpΩ:
g(Vp, Vp) =Vp, M·Vp.
For everyp∈Ω, let Exppbe the exponential mapping of Ω with respect tog defined in a neighbourhood of 0∈Tp(Ω). We recall the following facts concerning this application:
Lemma 3.4. — For allp∈Ω:
DExpp(0) =M−11/2, DExpp(0)−1 =M1/2. There exist functionsµ1exp, µ−exp1 such that:
D2Expp(0) µ1exp(M,M−1,DM), D2(Expp)−1(p) µ−exp1(M,M−1,DM).
Next, we have the following result concerning the transformation of the second fundamental form:
Lemma 3.5. — There exists a continuous function µtransformII such that if:
(1) U,V are open sets,
(2) ϕ:U →V is a diffeomorphism,
(3) Σ = (S, i)is an immersed hypersurface and,
(4)II (resp.II) is the (Euclidean) second fundamental form of(S, i)(resp.
(S, ϕ◦i)),
then, for allp∈S:
II(p) µtransformII (II(p),Dϕ(p),Dϕ−1(p),D2ϕ(p)), II(p) µtransformII (II(p),Dϕ(p),Dϕ−1(p),D2ϕ(p)).
In particular, if Dϕ,Dϕ−1 andD2ϕ are bounded, then a bound onII yields a bound onII.
Proof. —Since translations and rotations are isometries, we may assume that Σ = (S, i) (resp. Σ = (S, ϕ◦i)) is the graph of a functionf (respf) such that f(0) = df(0) = 0 (resp. f(0) = df(0) = 0). In this case, II (resp.II) coincides withD2f (resp.D2f) and the result now follows from a direct calculation.
By combining lemmata 3.4 and 3.5 we now obtain the following result:
Lemma 3.6. — There exists a continuous function µcompareII such that if Σ is an immersed submanifold in Ω and IIg (resp. II) is the second fundamental form of Σ with respect to g (resp. the Euclidean metric over Ω) then:
IIµcompareII (IIg,M,M−1,DM).
In particular, ifM,M−1 andDM are bounded, then a bound on IIg yields a bound onII.
Proof. —Letpbe a point inS. By applying (Expi(p))−1, we may work in an exponential chart abouti(p). At the origin in such a chart, the second fundamental form of an immersed submanifold with respect to (Expi(p))∗g
coincides with the Euclidean second fundamental form. Using lemma 3.5, we may thus bound II in terms of IIg and the derivatives of Expi(p) atp. The result now follows by lemma 3.4.
We now obtain:
Lemma 3.7. — There exists a functionµII(K, B, ,M,M−1,DM) /2 such that if:
(1)δµII, (2)B(0)⊆Ω,
and if f :Bδ(0)→Rn−m is a function such that:
(1)f(0), df(0) = 0, and
(2) the norm of the second fundamental form of the graph off with respect toM is bounded above byK,
then, for allq∈Bδ(0) :
f(q) /2, Df(q) B.
Remark. —In other words, we start by studying the graph of a function f such that f(0), df(0) = 0 (these conditions reflect the fact that, in the sequel, we will be studying immersed submanifolds in terms of graphs over the tangent space at each point). Then, given a boundKon the second fun- damental form of the graph, and given a desired boundB on the derivative of f, we find a radius δ, depending only on B and K (and various other variables nonetheless independant off), over which this bound is satisfied (provided, of course, thatM◦f is defined over this radius, henceδ < /2).
Finally, for no extra cost, we also obtain a bound for f over this radius which will be useful in the sequel.
Proof. —By lemma 3.6, the norm of the second fundamental form of the graph of f with respect to the Euclidean metric is bounded above by K=µcompareII (K,M,M−1,DM). For alli, we denote ˆ∂i and ˆNi by:
∂ˆi= ∂i
Df·∂i
, Nˆi=
Dft·∂i
−∂i
.
( ˆ∂1, ...,∂ˆn) is a basis of tangent vectors to the graph off and ( ˆN1, ...,Nˆn−m) is a basis of normal vectors to the graph of f. Let II be the second fun-
damental form of the graph off with respect to the Euclidean metric. We have: II( ˆ∂i,∂ˆj),Nˆk =D∂ˆi
Nˆk,∂ˆj
=∂i∂jfk Thus: ∂i∂jfk K∂ˆi∂ˆjNˆk
K(1 +Df2)3/2
⇒ D2f K(1 +Df2)3/2
Solving this differential inequality with the intial conditionsf(0) = 0, Df(0) = 0, we obtain the desired result.
We now obtain the following result as an immediate corollary:
Lemma 3.8. — For r > > 0, there exist functions ∆(K, r, ,M, M−1, DM) /2, and B(r, ,M,M−1) such that if (Σ, p) = (S, i, p)is a pointed immersed submanifold of B(0)such that:
(1) the closed ball of radiusraboutpinΣis complete with respect toB(0), (2) i(p) = 0, and
(3) if II is the second fundamental form ofΣ with respect tog, then:
IIK,
then, for all δ∆, there exists a unique neighbourhoodU of p inS such that(U, i)is a graph over a Euclidean ball of radiusδabout the origin, and if f is the graph function ofΣoverBδ(0), then:
f/2, DfB.
Proof. —The ball of radius M−1/2raboutpinS with respect to the Euclidean metric over Rn is contained within the ball of radius r about p in S with respect to the metricg. We thus choose:
B=µgraph(,M−11/2r).
We now define:
∆ =µII(K, B, ,M,M−1,DM).
The result now follows from lemmata 3.3 and 3.7.
3.3. Bounds on the higher derivatives of the graph function In this section, we aim to show how bounds on the higher derivatives of a function may be obtained in terms of bounds on the higher derivatives of the second fundamental form of its graph. The results of this section are essentially trivial, but we include them for completeness and clarity.
Let Ω be an open subset of Rn. For every positive integer k, and for everyr∈R+, we defineBrk(0) to be the ball of radiusrabout the origin in Rk. Let mbe a positive integer not greater than n. Let∈R+ be a small positive real number and let f : Bm(0) → Rn−m be a smooth function whose graph is contained in Ω. Let us denote the graph off by Σ.
Let ·,· be the Euclidean metric over Rn. Let g be a metric over Ω and let M : Ω →Symm(Rn) be the matrixrepresenting g relative to the Euclidean metric. Thus, for all Vp∈TΩ:
g(Vp, Vp) =Vp, M(p)·Vp.
Let∂1, ..., ∂n be the canonical basis ofRn. For all 1im, we define
∂ˆ1, ...,∂ˆm by:
∂ˆi = (0, ...,1, ...,0, ∂ifm+1, ..., ∂ifn)t. Form+ 1in, we define ˆEm+1, ...,Eˆn by:
Eˆi= (−∂1fi, ...,−∂mfi,0, ...,1, ...,0)t. We define ˆNm+1, ...,Nˆn by:
Nˆi=M−1Eˆi.
We find that ( ˆ∂1, ...,∂ˆm) defines a moving (non-orthonormal) frame for TΣ. Similarly, ( ˆNm+1, ...,Nˆn) defines a moving (non-orthonormal) frame for the normal bundle to Σ relative to metric g. We define the matrix B by:
B= ( ˆ∂1, ...,∂ˆm,Nˆm+1, ...,Nˆn).
Let∇and D be the Levi-Civita covariant derivatives ofg and the Eu- clidean metric respectively. Let II be the second fundamental form of Σ, the graph off.
In the sequel, for alli, j, k, we denote byF(Jif, JjM, JkM−1) any func- tion depending only on the derivatives of f, M andM−1 up to orders i,j andkrespectively.
We obtain the following result:
Lemma 3.9. — For all i, j, k:
g(II( ˆ∂i,∂ˆj),Nˆk) =∂i∂jfk+F(J1f, J1M, J1M−1).
Proof. —For alli, j, k, we have:
g(II( ˆ∂i,∂ˆj),Nˆk) =∇∂ˆi
∂ˆj, MNˆk
=D∂ˆi∂ˆj,Eˆk+F(J1f, J1M, J1M−1)
=∂i∂jf +F(J1f, J1M, J1M−1).
The result now follows.
Letπbe the orthogonal projection ontoTΣ with respect tog. We have:
Lemma 3.10. — For allVp∈TΣ:
π(Vp) = n
i=1
(B−1)t∂i, Vp∂ˆi.
Proof. —Since (B∂1, ..., B∂n) is a basis forTpΩ, there existsa1, ..., an∈ Rsuch that:
Vp = n i=1
aiB∂i.
Moreover, since ( ˆ∂1, ...,∂ˆm) = (B∂1, ..., B∂m) is a basis forTΣ and since ( ˆNm+1, ...,Nˆn) = (B∂m+1, ..., B∂n) is a basis forTΣ⊥, we have:
π(Vp) =m i=1aiB∂i
=m i=1ai∂ˆi. However, for 1im, we have:
(B−1)t∂i, Vp =∂i, B−1n
j=1ajB∂j
=∂i,n
j=1aj∂j
=ai. The result now follows.
