On Tonelli periodic orbits with low energy on surfaces

0.73 MB·

by  Luca Asselle

ON TONELLI PERIODIC ORBITS
WITH LOW ENERGY ON SURFACES
6 LUCAASSELLEANDMARCOMAZZUCCHELLI
1
0
Abstract. We prove that, on a closed surface, a Lagrangian system defined
2
byaTonelliLagrangianLpossessesaperiodicorbitthatisalocalminimum

g of the free-period action functional on every energy level belonging to the
u low range of energies (e0(L),c0(L)). We also prove that almost every energy
A level in (e0(L),cu(L)) possesses infinitely many periodic orbits. These state-
ments extend two results, respectively due to Taimanov and Abbondandolo-
6 Macarini-Mazzucchelli-Paternain, known for the special case of electromag-
1 neticLagrangians.

] Contents
S
D 1. Introduction 1
. Acknowledgments 3
h 2. Localminimizersofthefree-periodactionfunctional 4
at 2.1. Uniquefree-timelocalminimizers 4
m 2.2. Embeddedglobalminimizersofthefree-periodactionfunctional 7
2.3. Compactnessofminimizingsequences 13
[ 3. Free-perioddiscretizations 25

3.1. Uniqueactionminimizers 25
2
3.2. Thediscretefree-periodactionfunctional 27
v
3.3. Thediscreteleastactionprinciple 28
2
3.4. TheHessianofthefree-periodactionfunctional 29
9
3.5. Theiterationmap 32
6
4. Multiplicityoflowenergyperiodicorbits 36
6
4.1. Apropertyofhighiteratesofperiodicorbits 36
0
4.2. Theminimaxscheme 39
.
1 5. ATonelliLagrangianwithfewperiodicorbitsonlowenergylevels 41
0 6. Tonellisettingwithageneralmagneticform 42
6 References 44
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:
v
i
X
r
a
1. Introduction
Hamiltonian systems on cotangent bundles sometimes admit a dual description
as Lagrangian systems, which are second order dynamical systems on the config-
uration space. This is the case for geodesic flows, which can be viewed as Reeb
flows on unit-sphere cotangent bundles, or equivalently as the family of curves γ
on the configuration space satisfying the geodesic equation ∇ γ˙ ≡ 0. The widest
t
class of fiberwise convex Hamiltonian functions admitting a dual Lagrangian is the
Date:January25,2016. Revised: August16,2016.
2000 Mathematics Subject Classification. 37J45,58E05.
Key words and phrases. TonelliLagrangians,periodicorbits,Man˜´ecriticalvalues.
1
2 LUCAASSELLEANDMARCOMAZZUCCHELLI
Tonelli one: a Hamiltonian H : T∗M → R is said to be Tonelli when its restric-
tion to any fiber H| is a superlinear function with everywhere positive-definite
T∗M
q
Hessian; the class of Lagrangians dual to the Tonelli Hamiltonians is given by the
so-called Tonelli Lagrangians L : TM → R, which satisfy the analogous proper-
ties as H on the tangent bundle. The name of these Lagrangians is due to the
seminal work of Tonelli [Ton34], who established the existence and regularity of
action minimizers joining a given pair of points in the configuration space. More
modernapplicationsof1-dimensionalcalculusofvariations,notablyAubry-Mather
theory [Mat91a, Mat91b, Mat93] and weak KAM theory [Fat97], also concerns
Lagrangians and Hamiltonians of Tonelli type. The background and further appli-
cations of Tonelli systems can be found in, e.g., [Fat08, CI99, Maz12, Sor15]. The
purpose of this paper is to extend to the Tonelli class two important existence re-
sults for periodic orbits, which were proved for the smaller class of electromagnetic
Lagrangians.
GivenaclosedmanifoldM andaTonelliLagrangianL:TM →R,theassociated
Euler-Lagrange flow φt : TM → TM is defined by φt(γ(0),γ˙(0)) = (γ(t),γ˙(t)),
L L
where γ : R → M is a smooth solution of the Euler-Lagrange equation, which in
local coordinates can be written as
dL (γ,γ˙)−L (γ,γ˙)=0. (1.1)
dt v q
The energy function E :TM →R, given by E(q,v)=L (q,v)v−L(q,v), is a first
v
integral for the Euler-Lagrange flow. The periodic orbits of the Euler-Lagrange
flowcontainedintheenergyhypersurfaceE−1(k)areinonetoonecorrespondence
with the critical points of the free-period action functional
S :W1,2(R/Z,M)×(0,∞)→R,
k
(cid:90) 1(cid:104) (cid:105)
S (Γ,τ)=τ L(Γ(t),Γ˙(t)/τ)+k dt.
k
0
We see the pair (Γ,τ) as the τ-periodic curve γ(t) := Γ(t/τ), so that S (Γ,τ)
k
represents the action of γ at level k. In this paper, a crucial role will be played by
the energy values e (L)≤c (L)≤c (L), which are given by
0 u 0
e (L)=maxE(q,0),
0
q∈M
cu(L)=−inf(cid:26)τ1 (cid:90) τL(γ(t),γ˙(t))dt (cid:12)(cid:12)(cid:12)(cid:12) γ :R/τZ−C→∞ M contractible(cid:27),
0
c0(L)=−inf(cid:26)τ1 (cid:90) τL(γ(t),γ˙(t))dt (cid:12)(cid:12)(cid:12)(cid:12) γ :R/τZ−C→∞ M null-homologous(cid:27).
0
In the literature, c (L) and c (L) are known as the Man˜´e critical values of the uni-
u 0
versalcoverandoftheabeliancoverrespectively. Alternativeequivalentdefinitions
of such values will be given in Sections 2.2 and 4.2 (more details can be found in
[Con06, Abb13] and in references therein).
Our first result is the following.
Theorem 1.1. Let M be a closed surface, and L : TM → R a Tonelli Lagrangian.
For every energy level k ∈ (e (L),c (L)), the Lagrangian system of L possesses a
0 0
periodic orbit γ with energy k that is a local minimizer of the free-period action
k
functional S over the space of absolutely continuous periodic curves. Moreover,
k
S (γ )<0 and γ lifts to a simple closed curve in some finite covering of M.
k k k
ON TONELLI PERIODIC ORBITS WITH LOW ENERGY ON SURFACES 3
This result was first established by Taimanov [Tai92b, Tai91] for electromag-
netic Lagrangians. His beautiful observation is that the free-period action func-
tional, which is not bounded from below in the energy range concerned by the
statement, becomes bounded from below when restricted to a suitable space of
null-homologous, embedded, periodic multicurves. This latter space is not com-
pact, but a clever argument of Taimanov shows that the global minima of the free
period action functional are contained in the interior of the space, and as such
are critical points. Unfortunately, the papers [Tai92b, Tai91] are rather short and
some crucial details are only sketched. An alternative proof based on a regularity
theorem for almost minimal currents in the sense of Almgren, a deep result from
geometric measure theory, was provided later by Contreras, Macarini, and Pater-
nain [CMP04]. Our proof of Theorem 1.1, which will be carried out in Section 2,
fills the details in Taimanov’s arguments, while at the same time generalizes the
result to the class of Tonelli Lagrangians.
Our second result is the following.
Theorem 1.2. Let M be a closed surface, and L : TM → R a Tonelli Lagrangian.
Foralmosteveryenergylevelkintheinterval(e (L),c (L))theLagrangiansystem
0 u
of L possesses infinitely many periodic orbits with energy k and negative action.
For the special case of electromagnetic Lagrangians, this result was proved by
the second author together with Abbondandolo, Macarini, and Paternain in the
recent paper [AMMP14] (see also [AMP15] for a previous preliminary result in
this direction). The proof involves a highly non-trivial argument of local critical
point theory for the free-period action functional. Such functional, in the classical
W1,2 functional setting, is C∞ for the case of electromagnetic Lagrangians, but is
not C2 for general Tonelli Lagrangians. This poses major difficulties, as the local
arguments from critical point theory involve in an essential way the Hessian of the
functional. Our way to circumvent this lack of regularity is to develop a finite
dimensional functional setting, which may have independent interest, inspired by
Morse’s broken geodesics approximation of path spaces [Mil63, Section 16]. The
free-period action functional in such finite dimensional setting, which we will call
the discrete free-period action functional, is C∞. This will allow to carry out
the proof of Theorem 1.2 along the line of [AMMP14]. The construction of the
finite dimensional functional setting will be given in Section 3, and the proof of
Theorem 1.2 in Section 4
Both Theorems1.1 and 1.2concern energy levels above e (L). One maywonder
0
whether the assertions of the theorems are still valid below e (L). This is not case.
0
Indeed, in Section 5 we will provide an example of Tonelli Lagrangian L such that
e (L)>minE and, for all energy values k >minE sufficiently close to minE, the
0
correspondingLagrangiansystempossessesonlytwoperiodicorbitswithenergyk,
and none of such orbits is a local minimum of the free-period action functional.
Finally, Theorems 1.1 and 1.2 can be extended to the case where the Euler-
Lagrange dynamics, or actually the dual Hamiltonian dynamics, is modified by a
non-exact magnetic term. The precise setting and the statements will be provided
in Section 6.
Acknowledgments. TheauthorsaregratefultoGabrieleBenedettiforpointingout
an inaccuracy in the first draft. L.A. is partially supported by the DFG grant AB
4 LUCAASSELLEANDMARCOMAZZUCCHELLI
360/2-1 “Periodic orbits of conservative systems below the Man˜´e critical energy
value”. M.M.ispartiallysupportedbytheANRprojectsWKBHJ(ANR-12-BS01-
0020) and COSPIN (ANR-13-JS01-0008-01). Part of the work presented in this
paper was carried out in November 2015 at the Ruhr-Universit¨at Bochum. M.M.
wishes to thank Alberto Abbondandolo for the kind hospitality, and the Mathe-
matics Department in Bochum for providing a stimulating working environment.
2. Local minimizers of the free-period action functional
2.1. Uniquefree-timelocalminimizers. LetM beaclosedmanifoldequippedwith
anarbitraryRiemannianmetricg,whichinducesadistancedist:M×M →[0,∞).
Withoutlossofgenerality,wecanassumethatM isorientable: ifthisisnotthecase,
we replace it by its orientation double covering. We consider a Tonelli Lagrangian
L : TM → R. We recall that such L is a smooth function whose restriction to
any fiber of TM is superlinear with everywhere positive-definite Hessian. We are
interested in the Euler-Lagrange dynamics, that is, in the solutions γ : R → M
of the Euler-Lagrange equation (1.1). This type of dynamics is conservative: the
energy E(q,v) = L (q,v)v−L(q,v) is constant along the lift t (cid:55)→ (γ(t),γ˙(t)) of a
v
solution γ of the Euler-Lagrange equation.
For all q ,q ∈ M, we denote by AC(q ,q ) the space of absolutely continuous
0 1 0 1
curves connecting q with q , that is the space of all γ : [0,τ] → M such that
0 1
τ ∈ (0,∞), γ(0) = q , and γ(τ) = q . For any constant k ∈ R, the action of one
0 1
such curve γ with respect to the Lagrangian L+k is given by
(cid:90) τ
S (γ)= L(γ(t),γ˙(t))dt+kτ.
k
0
It is well known that the space AC(q ,q ) has the structure of a Banach manifold:
0 1
it can be seen as the product of the interval (0,∞) with the space of absolutely
continuouscurvesfromtheinterval[0,1]toM joiningq andq . Inthissetting,the
0 1
principleofleastactionguaranteesthattheextremalpointsofS :AC(q ,q )→R
k 0 1
are precisely those γ that are solution of the Euler-Lagrange equation with energy
E(γ,γ˙)≡k. Following Man˜´e [Man˜97, CDI97] we define the Man˜´e action potential
Φ :M ×M →R∪{−∞},
k
(cid:8) (cid:12) (cid:9)
Φk(q0,q1):=inf Sk(γ) (cid:12) γ ∈AC(q0,q1) .
TheMan˜´ecriticalvaluec(L)isdefinedastheinfimumofthesetofk ∈Rsuchthat
the function Φ is everywhere finite. It is easy to see that c(L) is a finite value: it
k
must be larger than or equal to
(cid:0) (cid:1)
e (L):=maxE ·,0 .
0
Ifk ≥c(L),theactionpotentialΦ isaLipschitzfunctionvanishingonthediagonal
k
submanifold of M ×M. If k > c(L), Φ is strictly positive outside the diagonal,
k
and the infimum in its definition is actually a minimum: for any pair of points
q ,q ∈M there exists a curve γ ∈AC(q ,q ) satisfying Φ (q ,q )=S (γ); such a
0 1 0 1 k 0 1 k
γ is necessarily a solution of the Euler-Lagrange equation with energy k.
It will be useful for us to consider also the Hamiltonian characterization of the
Man˜´e critical value, which goes as follows. Let H : T∗M → R be the Tonelli
ON TONELLI PERIODIC ORBITS WITH LOW ENERGY ON SURFACES 5
Hamiltonian dual to L, that is
(cid:104) (cid:105)
H(q,p)= max pv−L(q,v) . (2.1)
v∈TqM
The critical value c(L) can be defined as the infimum of k such that there exists a
smooth function u:M →R that satisfies the Hamilton-Jacobi inequality
(cid:0) (cid:1)
H q,du(q) <k.
It is well known that, for any energy level k > c(L), there exists a diffeomor-
phism of the cotangent bundle T∗M sending the energy hypersurface H−1(k) to
the unit-sphere cotangent bundle of a Finsler metric F on M, see [CIPP98, Corol-
lary 2]; in particular, the solutions of the Euler-Lagrange equation with energy k
are reparametrizations of the geodesics of F, and the following holds.
Lemma 2.1. Let k >c(L). There exists τ >0 such that the following hold:
inj
(i) Every point q ∈M admits an open neighborhood U ⊂M such that the
0 q0
smooth map
ψ :(cid:2)0,τ (cid:1)×(cid:0)E−1(k)∩T M(cid:1)→U
q0 inj q0 q0
given by ψ (τ,v )=π◦φτ(q ,v ) restricts to a diffeomorphism
q0 0 L 0 0
ψ :(cid:0)0,τ (cid:1)×(cid:0)E−1(k)∩T M(cid:1)→U \{q }.
q0 inj q0 q0 0
(ii) For each pair of points q ∈ M and q ∈ U , let (τ,v ) ∈ ψ−1(q ). The
0 1 q0 0 q0 1
embedded curve γ : [0,τ] (cid:44)→ U given by γ(t) := ψ (t,v ) is the unique
q0 q0 0
global minimizer of the action functional S :AC(q ,q )→R.
k 0 1
Proof. We already remarked that, since k > c(L), there exists a Finsler metric F
onM suchthateverysolutionoftheEuler-LagrangeequationofLwithenergyk is
areparametrizationofaunit-speedgeodesicofF. Therefore,point(i)followsfrom
the analogous statement about the exponential maps of closed Finsler manifolds
(see,e.g.,[AP94,BCS00]). Asforpoint(ii),noticethatitisenoughtoestablishits
assertion for all pair of points q ,q ∈ M sufficiently close. The original assertion
0 1
will follow by replacing τ by a smaller positive constant.
inj
For all q ,q ∈ M, we denote by γ : [0,τ ] → M a solution of the Euler-
0 1 q0,q1 q0,q1
Lagrange equation with energy k such that S (γ ) = Φ (q ,q ). In particular,
k q0,q1 k 0 1
γ is a global minimizer of S :AC(q ,q )→R. Notice that
q0,q1 k 0 1
Φ (q ,q )=S (γ )+(k−c(L))τ
k 0 1 c(L) q0,q1 q0,q1
≥Φ (q ,q )+(k−c(L))τ ,
c(L) 0 1 q0,q1
and therefore
Φ (q ,q )−Φ (q ,q )
k 0 1 c(L) 0 1
τ ≤Ω (q ,q ):= . (2.2)
q0,q1 k 0 1 k−c(L)
The function Ω : M × M → [0,∞) is Lipschitz and vanishes on the diagonal
k
submanifold of M×M. We denote by (cid:96) >0 the Lipschitz constant of Ω , so that
k k
Ω (q ,q )=Ω (q ,q )+Ω (q ,q )−Ω (q ,q )≤(cid:96) dist(q ,q ).
k 0 1 k 0 0 k 0 1 k 0 0 k 0 1
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
=0 ≤(cid:96)kdist(q0,q1)
If dist(q ,q ) < τ /(cid:96) , the estimate (2.2) implies that τ < τ , and therefore
0 1 inj k q0,q1 inj
any global minimizer of the action functional S : AC(q ,q ) → R is contained
k 0 1
6 LUCAASSELLEANDMARCOMAZZUCCHELLI
in the open set U . Point (i) implies that such minimizer is unique and is an
q0
embedded curve. (cid:4)
The following lemma shows that on any energy level in (e (L),c(L)] the quali-
0
tativepropertiesoftheEuler-Lagrangedynamicsarelocallythesameasonenergy
levels in (c(L),∞).
Lemma 2.2. If k > e (L), every point of the closed manifold M admits an open
0
neighborhood W and a Tonelli Lagrangian L˜ : TM → R such that L˜| = L|
TW TW
and c(L˜)<k.
Proof. Letq ∈M beagivenpoint. Wechooseasmoothfunctionu:M →Rsuch
0
that du(q )=∂ L(q ,0). This implies that
0 v 0
H(q ,du(q ))=−L(q ,0)≤e (L)<k.
0 0 0 0
We choose an open neighborhood W of q whose closure is contained in an open
0
set W(cid:48) ⊂ M such that H(q,du(q)) < k for all q ∈ W(cid:48). Let χ : M → (0,1] be a
smooth bump function that is identically equal to 1 on W and is identically equal
to a constant smaller than k/max{H(q,du(q))|q ∈ M} outside W(cid:48). We define a
new Tonelli Hamiltonian H˜ : T∗M → R by H˜(q,p) := χ(q)H(q,p), and its dual
TonelliLagrangianL˜ :TM →R. Byconstruction,L˜ coincideswithLonTW,and
we have H˜(q,du(q))<k for all q ∈M. In particular, k >c(L˜). (cid:4)
If k < c(L), the action potential is identically equal to −∞, and in particular
therearenocurvesjoiningtwogivenpointsthatareglobalminimizersoftheaction.
However,ifk >e (L)therearestilllocalminimizersjoiningsufficientlyclosepoints.
0
Lemma2.3. Letk >e (L). Thereexistτ >0andρ >0suchthatthefollowing
0 inj inj
hold:
(i) Everypointq ∈M admitsanopenneighborhoodU ⊂M containingthe
0 q0
compact Riemannian ball B (q ,ρ ) = {q ∈ M|dist(q ,q) ≤ ρ } such
g 0 inj 0 inj
that the smooth map
ψ :(cid:2)0,τ (cid:1)×(cid:0)E−1(k)∩T M(cid:1)→U
q0 inj q0 q0
given by ψ (τ,v )=π◦φτ(q ,v ) restricts to a diffeomorphism
q0 0 L 0 0
ψ :(cid:0)0,τ (cid:1)×(cid:0)E−1(k)∩T M(cid:1)→U \{q }.
q0 inj q0 q0 0
(ii) For each pair of points q ∈ M and q ∈ U , set (τ,v ) := ψ−1(q ).
0 1 q0 0 q0 1
The curve γ : [0,τ] → U , given by γ(t) := ψ (t,v ), is embedded and
q0 q0 0
is the unique global minimizer of the restriction of the action functional
S : AC(q ,q ) → R to the open subset of absolutely continuous curves
k 0 1
contained in U .
q0
In the following, we will say that the curve γ of point (ii) above is the unique
free-time local minimizer with energy k joining q and q .
0 1
Remark 2.4. If k ≤e (L), the assertion of Lemma 2.2 is not valid for any point of
0
M anymore,butonlyforthosepointsq ∈M suchthatE(q ,0)<k. Analogously,
0 0
the assertions of Lemma 2.3 are still valid for all points q such that E(q ,0) < k
0 0
and for some τ >0 depending on q . (cid:4)
inj 0
ON TONELLI PERIODIC ORBITS WITH LOW ENERGY ON SURFACES 7
Proof of Lemma 2.3. By Lemma 2.2, we can find an open cover W ,...,W of M
1 h
and, foralli=1,...,h, aTonelliLagrangianL :TM →Rsuchthatc(L )<k and
i i
L | =L| . Let E :TM →R be the associated energy functions
i TWi TWi i
E (q,v)=∂ L (q,v)v−L (q,v).
i v i i
Wechooser >0largeenoughsothateveryenergyhypersurfaceE−1(k)iscontained
i
in the ball tangent bundle of radius r, i.e.,
g (v,v)<r2, ∀(q,v)∈E−1(k).
q i
We apply Lemma 2.1 to each Lagrangian L , thus obtaining the constant τ > 0
i i
and the maps
ψ :(cid:2)0,τ (cid:1)×(cid:0)E−1(k)∩T M(cid:1)→U .
i,q0 i q0 i,q0
Wedenotebyρ >0theLebesguenumberoftheopencoverW ,...,W . Werecall
leb 1 h
that ρ is such that, for every q ∈ M, the open Riemannian ball B (q ,ρ ) is
leb 0 g 0 leb
contained in some open set W of the cover. Notice that, for all q ∈ M and
i 0
i ∈ {1,...,h}, the open set U is contained in the Riemannian ball B (q ,τ r).
i,q0 g 0 i
We reduce the positive constants τ , so that they all coincide to a same positive
i
constant τ < ρ /r. This implies that, for all q ∈ M, there exists i ∈ {1,...,h}
inj leb 0
such that U is contained in the open set W , above which the Lagrangians L
i,q0 i i
and L coincide; hence, we set U :=U and we define the map
q0 i,q0
ψ :(cid:2)0,τ (cid:1)×(cid:0)E−1(k)∩T M(cid:1)→U ,
q0 inj q0 q0
ψ (τ,v ):=π◦φτ(q ,v )=π◦φτ (q ,v ).
q0 0 L 0 0 Li 0 0
The claims of points (i-ii), except that U ⊃ B (q ,ρ ), follow from Lemma 2.1
q0 g 0 inj
applied to the Lagrangian L . The inclusion B (q ,ρ )⊂U follows if we choose
i g 0 inj q0
the constant ρ such that
inj
0<ρinj <min(cid:8)dist(cid:0)q0,π◦φτLinj(q0,v0)(cid:1) (cid:12)(cid:12) (q0,v0)∈E−1(k)(cid:9). (cid:4)
As a corollary of Lemma 2.3, we reobtain the following well known statement
(see, e.g., [Abb13, Proof of Lemma 7.2] for the original proof).
Corollary 2.5. Every absolutely continuous periodic curve γ : R/τZ → M with
τ > 0 and image contained in a Riemannian ball of diameter ρ is contractible
inj
and satisfies S (γ)>0.
k
Proof. We set q :=γ(0). The curve γ is entirely contained in the Riemannian ball
0
B (q ,ρ ), which in turn is contained in the open set U given by Lemma 2.3.
g 0 inj q0
Since U is contractible, γ is contractible as well. Consider the stationary curve
q0
γ :{0}→{q }. Thecurveγ mightbestationaryaswell,butformallyitisdifferent
0 0
from γ since τ > 0. By Lemma 2.3(ii), γ is the unique global minimizer of the
0 0
restrictionoftheactionfunctionalS :AC(q ,q )→Rtotheopensubsetofcurves
k 0 0
contained in U . Therefore S (γ)>S (γ )=0. (cid:4)
q0 k k 0
2.2. Embedded global minimizers of the free-period action functional. From now
on, we will assume that M is an orientable closed surface, that is,
dim(M)=2.
We will consider the functional S as defined on the space of absolutely continuous
k
periodiccurvesonM witharbitraryperiod. Thelatterspacecanbeidentifiedwith
8 LUCAASSELLEANDMARCOMAZZUCCHELLI
the product AC(R/Z,M)×(0,∞), so that any pair (Γ,τ) in this product defines
theτ-periodiccurveγ(t):=Γ(t/τ). ThefunctionalS willbecalledthefree-period
k
action.
Let us consider the Man˜´e critical value c (L). This can be defined in several
0
equivalent ways: as the minimum of c(L+θ) where θ ranges among the closed 1-
formsonM,astheMan˜´ecriticalvalueoftheliftofLtotheabeliancoveringofM,
or as the minimum k such that the free-period action functional S is non-negative
k
on the connected components of null-homologous periodic curves. It is easy to see
that e (L)≤c (L). From now on, we fix an energy value
0 0
k ∈(e (L),c (L)).
0 0
The proof of Theorem 1.1 will follow Taimanov’s ideas [Tai91, Tai92b, Tai92a]
for the electromagnetic case: the free-period action functional S is bounded from
k
belowonthespaceofembeddedperiodicmulticurvesthataretheorientedboundary
ofacompactregionofM;thissuggeststolookfortheperiodicorbitofTheorem1.1
by minimizing S there. The major difficulty is that such a space of multicurves is
k
not compact, and so might be the minimizing sequences.
Let us introduce a suitable space over which we will perform the minimization.
For each positive integers m, we first define C(m) to be the space of multicurves
γ =(γ ,...,γ ) such that
1 m
(C1) for every i∈{1,...,m}, the curve γ :R/τ Z→M is absolutely continuous
i i
and topologically embedded;
(C2) any two curves γ and γ do not have mutual intersections;
i j
(C3) the multicurve γ is the oriented boundary of a possibly disconnected, em-
bedded, oriented, compact surface Σ ⊂ M whose orientation agrees with
the one of M.
For all positive integers m ≤ n, we then define D(m,n) to be the space of multic-
urves γ =(γ ,...,γ ) such that:
1 m
(D1) each γ :R/τ Z→M is a continuous curve, and there exist
i i
0=τ ≤τ ≤...≤τ =τ
i,0 i,1 i,ni i
such that, for all j =0,...,n −1,
i
(cid:0) (cid:1)
dist γ (τ ),γ (τ ) ≤ρ ,
i i,j i i,j+1 inj
andtherestrictionγ | istheuniquefree-timelocalminimizerwith
i [τi,j,τi,j+1]
energy k joining the endpoints (see the definition after Lemma 2.3);
(D2) n +...+n ≤n.
1 m
We endow both C(m) and D(m,n) with the topologies induced by the inclusions
into the space of absolutely continuous multicurves with m connected components
(which is naturally diffeomorphic to the m-fold product of AC(R/Z,M)×(0,∞)).
Lemma 2.6. The space D(m,n) is compact.
Proof. Let{γ =(γ ,...,γ )|α∈N}⊆D(m,n)beasequenceofmulticurves,
α α,1 α,m
and 0 = τ ≤ τ ≤ ... ≤ τ = τ the time-decomposition for the con-
α,i,0 α,i,1 α,i,nα,i α,i
nectedcomponentγ givenin(D1). Sincen +...+n ≤n,bythepigeonhole
α,i α,1 α,m
principle we can extract a subsequence, which we still denote by {γ | α ∈ N},
α
ON TONELLI PERIODIC ORBITS WITH LOW ENERGY ON SURFACES 9
such that, for some n ,...,n ∈N, we have
1 m
n =n , ∀α∈N, i=1,...,m.
α,i i
SinceM iscompact,uptoextractingafurthersubsequencewecanassumethat,for
each i∈{1,...,m} and j ∈{1,...,n }, the sequence {γ (τ )|α∈N} converges
i α,i α,i,j
to some point q ∈ M. For each i = 1,...,m we set γ : R/τ Z → M to be the
i,j i i
unique continuous curve such that, for some 0 = τ ≤ τ ≤ ... ≤ τ = τ , each
i,0 i,1 i,ni i
portion γ | is the unique free-time local minimizer with energy k joining
i [τi,j,τi,j+1]
q and q . The multicurve γ := (γ ,...,γ ) clearly belongs to D(m,n), and
i,j i,j+1 1 m
γ →γ as α→∞. (cid:4)
α
Finally, the space of multicurves that we will be interested in will be
n
(cid:91)
M(n):= M(m,n),
m=1
where
M(m,n):=D(m,n)∩C(m).
ItreadilyfollowsfromthisdefinitionandLemma2.6thatM(m,n)iscompact,and
so is M(n). With a slight abuse of notation, we will denote by S :M(m,n)→R
k
the free-period action functional with energy k on multicurves, given by
(cid:88)m (cid:88)m (cid:18)(cid:90) τi (cid:19)
S (γ)= S (γ )= L(γ (t),γ˙ (t))dt+τ k .
k k i i i i
i=1 i=1 0
Clearly, S is continuous.
k
We have already seen in Corollary 2.5 that every absolutely continuous periodic
curveγ :R/τZ→M containedinaRiemannianballofdiameterρ iscontractible
inj
andsatisfiesS (γ)≥0. Ontheotherhand,uptopassingtoafinitecoveringofthe
k
configuration space M, we can always find embedded curves with negative action.
Lemma 2.7. There exists a finite covering M(cid:48) → M with the following property.
If we lift the Tonelli Lagrangian L to TM(cid:48), and denote by M(cid:48)(m,n) and S(cid:48) the
k
associated spaces of multicurves and free-period action functional, then for n large
enough there exists a periodic curve γ ∈M(cid:48)(1,n) with S(cid:48)(γ)<0.
k
Proof. Let M be the abelian cover of our configuration space M, that is, the cov-
0
eringspacewhosefundamentalgroupisthekerneloftheHurewiczhomomorphism
π (M)→H (M;Z). WelifttheTonelliLagrangianLtoafunctionL :TM →R,
1 1 0 0
and we denote by M0(m,n) and S0 the associated spaces of multicurves and free-
k
period action functional. Since k < c (L), there exists an absolutely continuous
0
loop γ : R/τZ → M such that S0(γ ) < 0. Let n ∈ N be large enough so that
0 0 k 0
dist(γ (s),γ (t)) < ρ whenever |s−t| ≤ 1/n, where ρ > 0 is the constant of
0 0 inj inj
Lemma2.3(ii). Wereplaceeachportionγ | ofourcurvewiththeunique
0 [i/n,(i+1)/n]
free-timelocalminimizerwithenergyk joiningtheendpoints. Theresultingcurve,
whichwewillstilldenotebyγ :R/τZ→M (possiblyforadifferentperiodτ than
0
before), isapiecewisesolutionoftheEuler-LagrangeequationofL , andinpartic-
0
ular is piecewise smooth. Up to perturbing the vertices γ (i/n), for i=0,...,k−1,
0
we can assume that γ has only finitely many self-intersections, all of which are
0
double points.
10 LUCAASSELLEANDMARCOMAZZUCCHELLI
We claim that we can find such a γ without self-intersections. Suppose that
0
our γ has at least one double point (otherwise we are done). Let t ∈ [0,τ) be
0 0
the smallest time such that γ (t ) is a double point of γ , and let t > t be the
0 0 0 1 0
smallest time such that γ (t ) = γ (t ). We define the piecewise smooth curves
0 1 0 0
γ :R/(t −t )Z→M and γ :R/(τ −t +t )Z→M by
1 1 0 0 2 1 0 0
γ (t)=γ (t +t), ∀t∈[0,t −t ],
1 0 0 1 0
γ (t)=γ (t +t), ∀t∈[0,τ −t +t ].
2 0 1 1 0
Notice that
γ (R/τZ)=γ (R/(t −t )Z)∪γ (R/(τ −t +t )Z),
0 1 1 0 2 1 0
S0(γ )=S0(γ )+S0(γ ),
k 0 k 1 k 2
and that both γ and γ have strictly less number of double points than γ . Since
1 2 0
S0(γ ) < 0, we must have S0(γ ) < 0 or S0(γ ) < 0, say S0(γ ) < 0. If γ has
k 0 k 1 k 2 k 1 1
no double points, we are done. Otherwise, we repeat the whole procedure with the
curve γ . After a finite numbers of iterations of this procedure, we end up with a
1
curve γ(cid:48) ∈M0(1,k) such that S0(γ(cid:48))<0.
k
The group of Deck transformations of the abelian covering M is, by definition,
0
the first homology group H1(M;Z) (cid:39) Zb1. Let e1,...,eb1 be the generators of
HH1((vM):;=Z(cid:104))e.v1F,.o..r,eevabc1h(cid:105) ovfeHcto(rMv).=Th(ve1q,u..o.,tviebn1)t H∈ (ZMb1),/Hwe(vc)oinssaidfiernittheegrsouubpg,roaunpd
1 b1 1 1
therefore M admits a finite covering M(cid:48) whose fundamental group is this quotient.
The abelian covering M is a covering of M(cid:48). If all the entries of the vector v are
0
largeenough,γ(cid:48) isprojectedtoanull-homologouscurveγ inM(cid:48) thatisapiecewise
solution of the Euler-Lagrange equation with energy k of the lift of L to TM(cid:48), has
no self-intersections (in particular γ ∈M(cid:48)(1,n)), and satisfies S(cid:48)(γ)<0. (cid:4)
k
From now on, in order to simplify the notation, we will assume that M = M(cid:48),
and we will denote by n the minimal positive integer so that the conclusion of
neg
Lemma 2.7 holds for all n≥n .
neg
Let n ≥ n . Since M(n) is compact and the free-period action functional
neg
S :M(n)→R is continuous, there exists γ =(γ ,...,γ )∈M(n) such that
k 1 m
S (γ)= min S <0. (2.3)
k k
M(n)
Noticethatsomeconnectedcomponentofγ maybeacollapsedcurveγ :{0}→M.
i
However, in this case S (γ ) = 0. Since S (γ) < 0, there exists a connected
k i k
component γ of γ such that S (γ ) < 0. Such a γ is not a collapsed curve.
j k j j
Therefore, after removing all the collapsed connected components of γ we are left
with a multicurve in M(n), which we still denote by γ, that satisfies (2.3).
We say that γ :R/τ Z→M is non-iterated when τ is its minimal period.
i i i
Lemma 2.8. Assume that γ = (γ ,...,γ ) ∈ M(n) is a multicurve that satis-
1 m
fies (2.3) and has no connected component that is a collapsed curve. If in addition
γ ∈ M(n−1), then each of its connected components is a non-iterated periodic
orbit of the Lagrangian system of L with energy k.
Proof. Every connected component of γ has the form γ : R/τ Z → M, and there
i i
exists a sequence of time parameters 0 = τ < τ < ... < τ = τ such that
i,0 i,1 i,ni i