A Note on Constant Geodesic Curvature Curves on Surfaces

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www.elsevier.com/locate/anihpc

A note on constant geodesic curvature curves on surfaces

Taoniu Sun

^1,2

University of Science and Technology of China, China and University Paris-Est, France Received 7 May 2008; accepted 30 June 2008

Available online 13 August 2008

Abstract

In this paper we are concerned with the structure of curves on surfaces whose geodesic curvature is a large constant. We first discuss the relation between closed curves with large constant geodesic curvature and the critical points of Gauss curvature. Then, we consider the case where a curve with large constant geodesic curvature is immersed in a domain which does not contain any critical point of the Gauss curvature.

Résumé

Dans cet article nous nous intéressons au comportement des courbes sur une surface, dont la courbure géodésique est une constante très grande. Dans un premier temps nous nous intéressons aux relations entre les courbes fermées dont la courbure géodésique est grande et les points critiques de la courbure de Gauss. Ensuite, nous nous intéressons au comportement asymptotique des courbes immergées dans un domaine de la surface qui ne contient aucun point critique de la courbure de Gauss.

MSC:53C21

Keywords:Constant geodesic curvature; Critical point; Gauss curvature

1. Introduction

Suppose(Mⁿ⁺¹, g) is a Riemannian manifold. We are interested in the structure of embedded spheres Sⁿ→ Mⁿ⁺¹ that have constant mean curvature. In the case where s, the scalar curvature function of (Mⁿ⁺¹, g), has a non-degenerate critical pointp, R. Ye has constructed constant mean curvature embedded spheres with high mean curvature which in fact form a local foliation of a neighborhood ofpin [9]. When the manifold is compact, F. Pacard and X. Xu have recently generalized this result relaxing the non-degeneracy assumption but loosing some control on the fact that the embedded spheres form a foliation in [3]. All these results point out the crucial role played by the critical points of the scalar curvature in the existence of embedded spheres with large enough mean curvature. More precisely, it is natural to ask the converse question:

E-mail address:tnsun@mail.ustc.edu.cn.

1 Present address: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China.

2 Partially supported by French Government Scholarship (2007/2304).

doi:10.1016/j.anihpc.2008.06.005

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Question 1.Assume thatp∈Mis fixed such that there exists a sequence of constant mean curvature hyper-surfaces H_i,i∈N, with mean curvaturem_i→ +∞which converges (for the Hausdorff distance) to the pointp. Is it true that pis a critical point of the scalar curvature function?

Recall that the Hausdorff distance between two setsAandBis defined to be DH(A, B):=inf

r>0

r >0:A⊂Tr(B)andB⊂Tr(A) , where

T_r(X):=

p∈M: dist(p, X) < r .

The result of O. Druet [2] tells us that the answer to this question is positive under the additional assumption that the constant mean curvature hyper-surfaces are solutions of the isoperimetric problem. In this note we give a positive answer to this question in the simplest case, that is when(M, g)is a 2-dimensional Riemannian manifold. In this case, constant mean curvature hypersurfaces are nothing but constant geodesic curvature curves and the scalar curvature function is nothing but the Gauss curvature.

Let(M, g)be an oriented 2-dimensional Riemannian manifold. Even though most of our results have straightforward generalization to the noncompact complete setting, we will always assume thatM is compact to simplify the statements. We first recall the results of R. Ye in our setting.

Theorem 1.(See [9].) Assume thatpis a non-degenerate critical point of the Gauss curvature functionK. Then, for allklarge enough, saykk_∗, the geodesic circle of radius1/k centered atpcan be perturbed intoΓ_k, a constant (=k) geodesic curvature embedded curve. More precisely, Γ_k is a normal graph for some function w_k over the geodesic circle of radius1/kcentered at a pointp_kwhere

w_k_C²ck⁻³ and dist(pk, p)ck⁻²

for some constantc >0which does not depend onk. Moreover, the curvesΓkform a local foliation of a neighborhood ofp.

We also have the following general property for curves on 2-dimensional Riemannian manifolds.

Theorem 2.(See [6].) Assume thatΓ is a closed embedded curve inMwith constant geodesic curvatureksatisfying k²>−minMK,

then,M\Γ has two disjoint connected components.

This result was obtained H. Rosenberg [6] for constant mean curvature surfaces in 3-manifolds but his proof extends to any dimension. A similar argument was used in [5] for constant mean curvature surfaces in flat 3-manifolds. For the sake of completeness, we give here a short proof of the result in the case of curves on surfaces.

IfΓ is a closed embedded curve inMwhich is the boundary of a compact domainΩ, the Gauss Bonnet theorem implies that

Γ

k=2π−

Ω

Kdvolg

and hence if the geodesic curvaturekis bounded from below by some positive constant, we conclude that the length ofΓ is bounded from above. more precisely, we have

|Γ|min|k|2π−

M

min(0, K)dvolg.

Hence if min|k|is large enough,|Γ|will be smaller than the injectivity radius ofM, thenΓ is topologically trivial.

Our main result gives, in dimension 2, a positive answer to the question raised above.

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Theorem 3.Assume thatp∈Mis fixed and assume that there exists a sequence of embedded closed curvesΓ_i,i∈N, with constant geodesic curvaturek_i→ +∞, which converges(for the Hausdorff distance)to the pointp. Thenpis a critical point of the Gauss curvature function.

If in addition the Gauss curvature functionKis a Morse function, we have:

Theorem 4.Assume that the Gauss curvature functionKis a Morse function. There existsc >0 (only depending on Mandg)such that ifΓ is a closed embedded curve inMwith constant geodesic curvaturek > c, then there existsp, a critical point of the Gauss curvature function, such that the Hausdorff distance betweenΓ and the geodesic circle of radius1/kcentered atpis bounded by ac/k².

The result of Theorem 2 together with the last result just says that, providedk >0 is large enough,Γ separatesM into two different connected components, one of which is close to a geodesic disc centered at a critical point of the Gauss curvature function. Unfortunately, we do not have an expression of the constantcwhich appears in this result.

Next we turn our attention to nonembedded curves. Since the equation which ensures that the geodesic curvature of a curve is a second order ordinary differential equation and since we are on a compact manifold, immersions ofR inMas a constant geodesic curvature curve (parameterized by arc length) exist in abundance. In fact once an initial pointpand an initial (unit) speedv∈T_pMhave been chosen there exists a unique curveΓ (p, v)passing throughp with speedv. We study the behavior of these curves as their geodesic curvature tends to∞. To be more precise, this curve is parameterized byγ=γ (p, v, k)such that

γ (0)=p and ∂_sγ (0)=v.

We chooser >0 smaller than the injectivity radius of the underlying manifold and defineI=I (p, v, k)⊂Rto be the largest interval containing 0 whose imageΓ (p, v, k)˜ byγ is included inB¯r(p). With these definitions, we have the following:

Theorem 5.Assume thatB¯r(p)∩ {q: K(q)=K(p)}does not contain any critical point ofK. Then asktends to∞, the sequence of constant geodesic curvature curves Γ (p, v, k)˜ converges in Hausdorff distance to the connected component ofB¯_r(p)∩ {q: K(q)=K(p)}which passes throughp.

Roughly speaking, ask tends to ∞, the curve Γ (p, v, k)˜ looks like the trajectory of a particle circling (at unit speed) at distance 1/k around a center which travels along the level curve of the functionK passing throughp at speeddKgk⁻³/8.

The above analysis leaves the possibility of having an immersed constant geodesic curvature curve circling around a critical point of the Gauss curvature function. To shed light over what is going on in this case, we restrict our attention to curves which are immersed in a simply connected domain ofMfor which it makes sense to define the degree of the curve. First of all, it is not surprising that the result of Theorem 4 holds, namely:

Theorem 6.LetΩ be a simply connected domain inMover whichKhas only non-degenerate critical points and let d∈Nbe fixed. There existsc >0such that, ifΓ is a closed curve of degreed immersed inΩwith constant geodesic curvaturek >0, then there existsp, a critical point of the Gauss curvature function, such that the Hausdorff distance betweenΓ and the geodesic circle of radius1/kcentered atpis bounded by ac/k².

More surprising is the following result we obtain.

Theorem 7.LetΩ be a simply connected domain inMover whichKhas only non-degenerate critical points and let d∈Nbe fixed. There existsk_∗>0such that, ifΓ is a closed curve of degreedimmersed inΩwith constant geodesic curvaturek > k_∗, thenΓ is ad-cover of an embedded constant geodesic curvature curve.

Now we outline the organization of this note briefly. In the beginning, we do some fundamental calculations about the metric and the geodesic curvature of the curves, these build Section 2. Then, we will prove Theorems 3 and 4 in Section 3 while Appendix A is devoted to Theorem 2. We study the immersed curves with large constant geodesic

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curvature in Section 4, the main result there is Theorem 5. The closed constant geodesic curvature curves immersed in a simple connected domain with degreed are considered in Section 5, where the main aim is to prove Theorem 7.

2. The geodesic curvature

We first give the expansion of the metric in polar geodesic coordinates. Next, we recall the expression of the geodesic curvature.

Given p∈M, we choose{e₁, e₂}an orthonormal basis of T_pM. To parameterize a neighborhood ofp, we use either geodesic normal coordinates(x₁, x₂)∈R²via the exponential map

Φ(x₁, x₂):=Exp_p(x₁e₁+x₂e₂), or polar coordinates(r, θ )∈ [0,∞)×S¹via

Ψ (r, θ ):=Exp_p

r(cosθ e1+sinθ e2) . It will be convenient to define

Θ(θ ):=cosθ e₁+sinθ e₂∈T_pM.

Gauss’s Lemma implies that, in polar geodesic coordinates, the metricgcan be written as Ψ^∗g=dr²+f²(r, θ ) dθ².

Recall [1] the expression ofK, the Gauss curvature, at the point of coordinates(r, θ )is given in terms off by K◦Ψ= −∂_r²f

f . (1)

We now recall the Taylor expansion of the functionf in powers ofr.

Proposition 8.(See [7,8].) The following expansion holds f (r, θ )=r−1

6K(p)r³− 1

12∇ΘK(p)r⁴+ 1 120

K(p)²−3∇Θ²K(p) r⁵+Op

r⁶

, (2)

where the subscriptpinOp(r⁶)is meant to remind the reader that this is a function ofp.

Proof. By definition of geodesic coordinates, we havef (0, θ )=0, and∂rf (0, θ )=1. Also the formula of the Gauss curvature tells us that

∂_r²f= −Kf,

where we write for shortK instead ofK◦Ψ. Hence∂_r²f (0, θ )=0. We take the derivative of (1) with respect tor and evaluate the result atr=0 to find

∂_r³f= −K∂rf− ∇ΘKf.

So∂_r³f (0, θ )= −K(p). Taking twice the derivative of (1) with respect torand evaluating the result atr=0, we get

∂_r⁴f= −K∂_r²f −2∇ΘK∂_rf− ∇Θ²K(p)f, therefore∂_r⁴f (0, θ )= −2∇ΘK(p). Similarly

∂_r⁵f (0)= −K∂_r³f−3∇ΘK∂_r²f−3∇_Θ²K∂_rf − ∇_Θ³Kf,

so that∂_r⁵f (0, θ )=K(p)²−3∇_Θ²K(p). Collecting these, we have completed the proof of the expansion. 2 We recall that formula for the geodesic curvature of a smooth curveΓ, which is parameterized in geodesic polar coordinates centered atpbyθ→(r(θ ), θ ).

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Fig. 1. Local graph ofΓ.

Lemma 9.The geodesic curvaturek_gofΓ at the pointΨ (r(θ ), θ ), is given by k_g= 1

(r²+f²)^3/2

r∂_θf +2r²∂_rf−rf +∂_rff²

, (3)

wherestands for∂θand wheref is computed at the point(r(θ ), θ ).

Proof. First we recall Liouville’s formula in [1,4] for the computation of the geodesic curvature: Suppose that(u, v) are isothermal coordinates on the surfaceMso that the metric can be written asg=E du²+G dv², whereEandG depend onuandv. Further assume thatC(s):=(u(s), v(s))is an immersed curve onMparameterized by arc-length.

Letαdenote the angle between the velocity vector∂_sC and∂_u. Then the geodesic curvature ofC is given by the formula

k_g=dα ds − 1

2√ G

∂lnE

∂v cosα+ 1 2√ E

∂lnG

∂u sinα.

In our case, we obtain k_g=dα

ds +∂rf f sinα,

whereαdenotes the angle betweenr-line and curveΓ. One can see Fig. 1.

It is easy to see that cosα= r

r²+f² and sinα= f

r²+f², (4) wheref is computed at the point(r(θ ), θ ). Differentiating the first formula with respect toθ and using the second formula, we get

dα

dθ =r(∂_θf+∂_rf r)−f r

r²+f² . (5)

Hence, we conclude that dα

ds = 1

(r²+f²)^3/2

r(∂_θf+∂_rf r)−f r . We can now use Liouville’s formula

k_g=dα ds +∂rf

f sinα= 1 (r²+f²)^3/2

r∂_θf+2r²∂_rf−rf+f²∂_rf . This completes the proof of the lemma. 2

We now specialize the previous general formula to curvesΓ_p,,wwhich, in polar coordinates centered at the pointp, are parameterized by

r(θ )=

1−w(θ )

, (6)

where >0 is a small parameter andwis small (smooth enough) function. We expand the geodesic curvature of this curve in powers ofandw. According to (3), the geodesic curvature ofΓp,,w reads:

kg(p, , w)=

1+²w² f²

−3/2

∂_rf f +w

f² −w∂_θf

f³ +2²w²∂_rf

f³ . (7)

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In order to make notations shorter, it will be convenient to use the following notations. An expression of the form L_p,(w)will denote a linear second order differential operator such that, there exists a constantc >0 independent of p∈Mand∈(0,1)such that

Lp,(w)

C⁰(S¹)cw_C²(S¹)

for allw∈C²(S¹). Similarly, givena∈N, any expression of the form Q^(a)_p,(w)denotes a nonlinear second order differential operator such that,Q^(a)_p,(0)=0 and there exists a constantc >0 independent ofp∈Mand∈(0,1)

Q^(a)_p,(w2)−Q^(a)_p,(w1)

C⁰(S¹)c

w2_C²(S¹)+ w1_C²(S¹)

a−1

w2−w1C²(S¹)

providedw_jC¹(S¹)1,j=1,2.

The following result gives the Taylor expansion ofk_g(p, , w)in powers ofwand: Proposition 10.The geodesic curvaturek_g(p, , w)of the curveΓ_p,,wcan be expanded as:

k_g(p, , w)=1−1

3K(p)²−1

4∇ΘK(p)³− 1

45K²(p)+ 1

10∇Θ²K(p) ⁴ +Op

⁵ +

1+1

3K(p)² ∂_θ²+1

w+³L_p,(w) +w²+1

2w²+2ww+Q⁽³⁾_p,(w)+²Q⁽²⁾_p,(w). (8)

The subscriptpinOp(⁵)is meant to remind the reader that this is a function ofpbounded by a constant times⁵. Proof. Using (2) withr=(1−w), we can write

²w²

f² =w²+Q⁽³⁾_p,(w)+²Q⁽²⁾_p,(w), w∂_θf

f³ =²Lp(w)+²Q⁽²⁾_p,(w), ³w²∂rf

f³ =w²+Q⁽³⁾_p,(w)+²Q⁽²⁾_p,(w), ²w

f² =w+2ww+1

3K(p)²w+Q⁽³⁾_p,(w)+²Q⁽²⁾_p,(w).

Using once more (2), we see that

∂_rf

f (r, θ )=1 r −1

3K(p)r−1

4∇ΘK(p)r²− 1

45K²(p)+ 1

10∇Θ²K(p) r³+O r⁴ at the point(r, θ ). Takingr=(1−w), we get

∂rfr

f =1−1

3K(p)²−1

4∇ΘK(p)³− 1

45K²(p)+ 1

10∇_Θ²K(p) ⁴+O ⁵ +

1+1

3K(p)² w+³L_,p(w)+w²+Q⁽³⁾_,p(w)+²Q⁽²⁾_,p(w).

Inserting these into (7), this completes the proof of the result. 2 3. Constant geodesic curvature curves

In this section, we assume thatΓ is an embedded closed curve inMwith constant geodesic curvaturek=1/. We assume thatkis large enough so that the result of Theorem 2 holds true.

We now show thatΓ is in fact a normal graph over a geodesic circle of radius 1/k. For the sake of simplicity, let us assume that(M, g)is compact.

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Proposition 11.There existsk_∗>0andc >0such that, ifΓ is an embedded closed curve with constant geodesic curvaturek=1/k_∗, then there exist a pointp∈M such that Γ can be parameterized in polar coordinates centered atpbyr(θ )=(1−w(θ ))where the functionw∈C²(S¹)satisfies

w_C²c², and

2π 0

w(θ )cosθ dθ= 2π 0

w(θ )sinθ dθ=0. (9)

Proof. The proof goes as follows. We first show that, there exists p˜∈M such that Γ can be written as a normal graph over the geodesic circle of radiuscentered atp, for some function which is bounded by a constant times˜ ³. Obviously, there is no uniqueness in the choice ofp˜ and next, we show that, moving the pointp˜ if this is necessary, one can arrange in such a way that the function satisfies the orthogonality condition (9).

We pick a pointq∈Γ and consider the pointp˜defined as follows: The pointp˜is at distance=1/kfromqalong the geodesic starting atq with velocity the normal vector aboutΓ (see Fig. 1). We assume thatkis large enough so thatis less than the cut locus ofp˜and we denote byΓ˜ the geodesic circle of radius=1/kcentered atp. Clearly,˜ nearq, the curveΓ can be written as a normal graph overΓ˜ and hence we can parameterizeΓ nearqusing geodesic polar coordinates centered atp, namely˜

θ→ r(θ ), θ

withr(0)=andr(0)=0. Letθ˜∈(0, π]be the largest value such that r(θ )−² and r(θ )²,

for allθ∈ [− ˜θ ,θ˜]. Obviouslyθ >˜ 0.

Sincekg(Γ )=k, by Lemma 9, we know thatr is a solution of the following second order ordinary differential equation

r=f ∂_rf −1 f²+r

f∂_θf+2 r

f

2

f ∂_rf−f²

1+

r f

2 3/2

−1 .

Thanks to the expansion of the functionf given in (2), we obtain the following estimates forθ∈ [− ˜θ ,θ˜].

r

f∂θf =O ⁵

, f ∂rf−1

f²=r−r² +O

³ ,

2 r

f

2

f ∂_rf =O ³

, −f²

1+

r f

2 3/2

−1 =O ³

. Therefore, we conclude thatθ→r(θ )is a solution of the equation

r=r−r² +O

³ ,

withr(0)=andr(0)=0. We setr=(1−w), so that w= −w+w²+O

² ,

withw(0)=w(0)=0. It is easy to seew=O(²),w=O(²)and hencew=O(²). Going back to the original function, we see that

r=+O ³

, r=O ³

.

This implies thatθ˜=π, forsmall enough. In addition, the curveΓ being an embedded curve, we conclude easily that

r(−π )=r(π )

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providedis small enough and this completes the proof of the first part of the result. It remains to prove that, modulo some small change in the position ofp, one can ensure that (9) holds.˜

As already mentioned, the pointp˜is not unique and in fact once that we know thatΓ is a normal graph over the geodesic circle of radiuscentered atp, for some function˜ w_p_˜, we conclude that the same is true if instead ofp, we˜ choose any pointpˆclose enough top. We claim that it is possible to choose˜ pˆin such a way that (9) is fulfilled. This follows at once from the following argument.

It is easy to check that there existsc >0 small enough such that, for allv˜∈T_p_˜Mand allw˜ ∈C²(S¹)satisfying

˜vgc and ˜w_C²c

the curve Γ (p, ,˜ w)˜ can also be written as the normal graph over the geodesic circle of radiuscentered at p= Exp_p_˜(v)˜ for some functionw=w_p,_˜_v,_˜_w_˜. In other words, we can write

Γ (p, ,˜ w)˜ =Γ (p, , w_p,_˜_v,_˜_w_˜).

We define P (,v,˜ w)˜ = 1

π 2π

0

w_p,_˜_v,_˜_w_˜Θ dθ∈T_p_˜M.

It is easy to check thatP is depends smoothly onvand(at least when >0 is small enough) and extends smoothly to=0. Moreover,P (0,0,0)=0 and

D_v_˜P_(0,0,0)(v)˜ = ˜v.

The implicit function theorem implies that, for all >0 and ˜w_C² small enough, there exists a vectorv˜∈T_p_˜Msuch thatP (,v˜,w)˜ =0. In addition dist(p,p)˜ c ˜w_C²ifp=Exp_p_˜(v). This completes the proof of the result.˜ 2

We keep the notations, assumptions and conclusions of Proposition 11. Making use of Proposition 10, we get:

Proposition 12.There exists a constantc >0such that dK(p)

gc² providedk=1/k_∗.

Proof. By Proposition 11,Γ can be parameterized by r(θ )=(1−w(θ )) in polar coordinates centered at p. In addition, we know thatw_C²=O(²). Using this information in (8), we conclude that the functionwis a solution of

1:=k_g(, w)=1−1

3K(p)²+

∂_θ²+1 w+O

³ . In particular, we get

∂_θ²+1 w=1

3K(p)²+O ³

,

moreover we know that, by construction,wisL²(S¹)-orthogonal to the functions cosθand sinθ. Hence we conclude that

w=1

3K(p)²+O ³

. Therefore, we get

2π 0

w²+1

2w²+2ww cosθ dθ=O ⁵

.

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Obviously, 2π 0

cosθ 1

3K(p)²+ 1

45K²(p)+ 1

10∇_Θ²K(p) ⁴ dθ=0, and, thanks to (9)

1+1

3K(p)² 2π 0

cosθ

∂_θ²+1

w dθ=0.

Multiplying (8) by cosθ, using the fact thatk_g(, w)=1, and integrating the result over(0,2π ), we conclude that 1

3³ 2π 0

∇ΘK(p)cosθ dθ=O ⁵

.

Similarly, we get 1

3³ 2π 0

∇ΘK(p)sinθ dθ=O ⁵

which implies that dK(p)

gc²

as claimed. This completes the proof of the result. 2 We are now in a position to prove both Theorems 3 and 4.

Proof of Theorem 3. By assumption, we have a sequence of closed embedded curvesΓ_i with constant geodesic curvaturek_i→ +∞. According to the result of Proposition 12, wheniis large enough, we can writeΓ_i as a normal graph, for some function bounded by a constant times 1/k³_i, over a geodesic circle of radius 1/kicentered at a pointp_i with dK(pi)

gc/k²_i.

The fact that Γ_i converges to p forces p_i to converge to p. Passing to the limit, asi→ +∞, we conclude that dK(p)=0. This completes the proof of Theorem 3. 2

Proof of Theorem 4. We keep the notations of the previous paragraph. The novelty being thatKis now assumed to be a Morse function onM. In particular it has finite number of critical points which are all isolated. The curveΓ is known to be a normal graph (for some function bounded by a constant times 1/k³) over a geodesic circle of radius 1/kcentered at a pointp˜such thatdK(p)˜ g=O(1/k²). SinceKis a Morse function, we conclude thatp˜is at most at distance a constant times 1/k²from one of the critical points ofK. This completes the proof of the result. 2 4. Limit of constant geodesic curvature curves as their curvature tends to infinity

In this section we consider curves with large constant geodesic curvature which are immersed in some open domain Ωwhich does not contain any critical point of the Gauss curvature. Without loss of generality, we can assume that this curve is parameterized bys∈I →γ (s)∈M wheresis the arc length andI the maximal interval for whose image byγ lies inΩ.

As usual we set =1/k.

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Fig. 2. Local description ofΓ.

We pick a pointq∈Γ and consider the pointpwhich is at distancefromq along the geodesic starting atq with velocity the normal vector about Γ. By Proposition 11, Γ can be parameterized byr(θ )=(1−w(θ )) in polar coordinates centered at p. (The point q corresponds toθ=0.) In addition, we know that w_C² =O(²)on any interval of fixed length and also thatw(0)=w(0)=0 (observe that here we do not assume thatwsatisfies (9)).

Using this information in (8), we conclude that the functionwis a solution of ∂_θ²+1

w=1

3K(p)²+1 4

∂x₁K(p)cosθ+∂x₂K(p)sinθ ³+O

⁴ . It is easy to check that

w(θ )=1

3K(p)²(1−cosθ ) +1

8³ θ

∂x₁K(p)sinθ−∂x₂K(p)cosθ

+∂x₂K(p)sinθ +O

⁴ and in particular, we conclude that

w(θ+2π )=w(θ )+π 4³

∂x₁K(p)sinθ−∂x₂K(p)cosθ +O

⁴

. (10)

Since the metric can be written as Ψ^∗g=dr²+f²dθ²,

we get

gradK=∂_rK∂_r+ 1

f²∂_θK∂_θ.

Ifvis a tangent vector toM, we denotev^⊥the vector obtained fromvby rotation of angleπ/2. We have the formula gradK^⊥= 1

f(∂rK∂θ−∂θK∂r).

IfΓ is parameterized in geodesic polar coordinates byγ:θ→(θ, r(θ ))withr(θ )=(1−w(θ )), we have

∂θγ=∂θ−w∂r. Therefore,

g

gradK^⊥, ∂_θγ

=f ∂_rK+ 1

fw∂_θK.

The first lemma ensures that, inI, the set ofssuch that∂_sγ is colinear to gradK(γ ), with opposite orientation, is a sequence of isolated points whose mutual distance are roughly multiple of 2π .

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Proposition 13.There existsk_∗>0such that, if the geodesic curvature ofΓ is constant and larger thank_∗then the set of parametersθfor which∂_θγ andgradK(γ )are co-linear and have opposite orientations is a finite sequence of pointsθ₀< θ₁<· · ·< θ_m(which depend onΓ). In addition

|θ_j₊₁−θ_j−2π|c⁴,

for some constantc >0only depending onΩ.

Proof. The proof of this result follows from the implicit function theorem which is applied to the function φ:=g

gradK^⊥(γ ), ∂_θγ . We find

f⁻¹φ=

∂_x₁K(γ )+ 1

f²²w(1−w)∂_x₂K(γ ) cosθ +

∂_x₂K(γ )− 1

f²²w(1−w)∂_x₁K(γ ) sinθ.

One has to be careful that the set of zeros ofφis not exactly the set we are looking but the set of points where∂_sγ and gradK(γ )are colinear, independently of the choice of orientation. It is easy to check that

1

f²²w(1−w)=O ²

and ∂_θ 1

f²²w(1−w) =O ²

. Moreover

∂_θ

∂_x₁K(γ )

=O() and ∂_θ

∂_x₂K(γ )

=O().

Therefore,

f⁻¹φ=∂x₁K(γ )cosθ+∂x₂K(γ )sinθ+O() and

∂_θ f⁻¹φ

= −∂_x₁K(γ )sinθ+∂_x₂K(γ )cosθ+O(). (11) Letθ0∈(−π, π]such that

∂_x₁K(p)cosθ₀+∂_x₂K(p)sinθ₀=0.

Since the distance betweenpandγ can be estimated by a constant times, we see that the zeros ofφare given by θ_n=θ₀+nπ+O(),

wheren∈Z. But using (11) we can estimate∂_θ(f⁻²φ)at any zero ofφand show that the zeros of φare isolated and the distance between two consecutive zeros is aboutπ. Taking into account the fact that we are only interested in points where∂sγ and gradK(γ )have opposite orientation, we conclude that the distance between the points we are interested in is about 2π. We can assume thatθ0is chosen so that these points correspond toθ2nforn∈Z.

To get a better estimate for the distance between two such zeros ofφwe use (10) which implies that w(θ+2π )=w(θ )+π

4³

∂x₁K γ (θ )

sinθ−∂x₂K γ (θ )

cosθ +O

⁴ since the distance betweenγ (θ )andpis estimated by a constant times. We also get

w(θ+2π )=w(θ )+π 4³

∂x₁K γ (θ )

cosθ+∂x₂K γ (θ )

sinθ +O

⁴ since∂_θ∂_x_jK(γ (θ ))=O(). Using these informations we estimate

f⁻¹φ

(θ+2π )− f⁻¹φ

(θ )

=π 4³

∂_x₁K γ (θ )

cosθ+∂_x₂K γ (θ )

sinθ

∂_x₂K γ (θ )

cosθ−∂_x₁K γ (θ )

sinθ +O

⁴

(12)

Fig. 3. “Spring”.

which can also be written as f⁻¹φ

(θ+2π )=

1+π

4³∂θ(f⁻¹φ)(θ ) f⁻¹φ

(θ )+O ⁴

. This together with the estimate of the derivative off⁻¹φimplies that

θ2n+2=θ2n+2π+O ⁴

.

This completes the proof of the result by dividing the subscript ofθby two. 2 We set

p_j=γ (θ_j).

Since

w(θ+2π )=w(θ )+π 4³

∂_x₁K(p)sinθ−∂_x₂K(p)cosθ +O

⁴ we conclude that

p_j₊₁=Exp_p_j(V_j),

whereVj=^π₄⁴gradK(pj)^⊥+O(⁵). This completes the proof of Theorem 5.

In Fig. 3 is the global picture of this process, the curve goes along the level curve ofKlike a spring.

5. Constant geodesic curvatured-circles: the proof of Theorem 7

In this section we study the case of immersed closed curves with constant geodesic curvature. We start with:

Definition 1.Givend∈N⁺, a closed curve immersed inMis called ad-circleif it is a degreedcurve immersed in a simple connected domain ofM.

The proof of Theorem 7 is based on the following idea: It is easy to check that the results of Theorems 6 and 4 hold when “embedded curves” are replaced by “d-circles”, withd fixed. Therefore, ifΓ is a constant geodesic curvature d-circle, it is a graph over the geodesic circle of radius=1/kcentered at a pointq which is at distancec/k²from pa critical point of the Gauss curvature function, for some 2π d-periodic function which can be estimated byc³in C²topology.

We now prove that, for allsmall enough, there exists a unique normal graph over the geodesic circle of radius =1/k centered at a pointq at distancec/k²fromp for some 2π d-periodic function which can be estimated by c³inC²topology. Since thed-cover of the embedded curve obtained by R. Ye in Theorem 1 has constant geodesic curvature equal tok,Γ has to be thed-cover of this embedded curve.

From now on, we focus our attention on proving Theorem 7. We will use the fixed point argument to derive an uniqueness property which is enough to obtain the theorem. Assume that we have Γ₁ andΓ₂ two d-circles with constant geodesic curvature. It follows from the result of Theorem 4 thatΓ_j a normal multi-graph for some 2π d- periodic function wj over the geodesic circle of radius =1/k centered at the point pj. Furthermore, we can assume that

2dπ 0

w_jcosθ dθ= 2dπ 0

w_jsinθ dθ=0

(13)

Fig. 4. A 2-circle.

and that

dist(pj, p)+ w_j_C²(S¹)c², (12)

wherepis a non-degenerate critical point ofK.

Letkg(q, , w)denote the geodesic curvature of the curve parameterized by((1−w(θ )), θ )in geodesic polar coordinates centered atq. We use the result of Proposition 10 to get the expansion

kg(q, , w)=1−1

3K(q)²−1

4∇ΘK(q)³+Oq

⁴

+

∂_θ²+1

w+²L_q,(w)+Q⁽²⁾_q,(w), (13)

where the subscriptq inOq(⁴)means that this is a function ofq. We denote F (, q):= −1

3K(q)−1

4∇ΘK(q)+Oq

² . Sincekg(pj, , wj)=1 we get by substraction

∂_θ²+1

(w₂−w₁)=

F (, p₁)−F (, p₂)

²+²(L_p₁_,w₁−L_p₂_,w₂) +

Q⁽²⁾_p

1,(w1)−Q⁽²⁾_p

2,(w2) .

Sincew₂−w₁isL²-orthogonal to cosθand sinθ, we conclude easily that w₂−w₁_C²_(S¹₎c²

dist(p2, p₁)+ w₂−w₁_C²_(S¹₎ ,

where we have implicitly used (12). Hence, forsmall enough, we conclude that

w2−w1_C²(S¹)c²dist(p2, p1). (14)

Now, we project onto cosθ, the identityk_g(p₂, , w₂)−k_g(p₁, , w₁)=0. Using the arguments already used in the proof of Proposition 12 we get

2dπ

0

∂_θ²+1

(w₂−w₁)cosθ dθ= 2dπ

0

K(p₂)−K(p₁)

cosθ dθ=0.

Moreover, using (12), we easily conclude that ²

2dπ 0

(L_p₂_,w2−L_p₁_,w1)cosθ dθ c

⁴dist(p2, p1)+²w2−w1_C²(S¹)

and

2dπ 0

Q⁽²⁾_p

2,(w2)−Q⁽²⁾_p

1,(w1) cosθ dθ

c

⁴dist(p2, p1)+²w2−w1_C²(S¹)

.

(14)

Fig. 5. Graph ofM\Γ.

Therefore, we conclude that ³

2dπ

0

∇ΘK(p₂)− ∇ΘK(p₁) cosθ dθ

c

⁴dist(p2, p₁)+²w₂−w₁_C²(S¹)

.

Similarly, we have ³

2dπ

0

∇ΘK(p₂)− ∇ΘK(p₁) sinθ dθ

c

⁴dist(p2, p₁)+²w₂−w₁_C²(S¹)

.

This implies that

dist(p2, p₁)cw₂−w₁_C²(S¹), (15)

providedis small enough.

Using (14) and (15), we conclude thatw2=w1andp1=p2. This completes the proof of Theorem 7.

Acknowledgements

The author would like to thank Professor F. Pacard, his thesis advisor, for his helpful advice and warm encourage- ment. Thanks also go to University Paris-Est where this work was done. Moreover, he is grateful for the continued help and support of Professors Q. Chen and X. Chen.

Appendix A. The proof of Theorem 2

We recall the proof of Theorem 2 following [6]. The proof is by contradiction. Assume that the result is not correct, so that we have an embedded curveΓ with constant geodesic curvatureksuch thatM\Γ has only one connected component. We can consider thatM0:=M\Γ is a manifold with two boundariesΓandΓ.

We consider the curveγ⊂M0which minimizes the distance between a point ofΓand a point ofΓ. Namely, γ is a solution of the problem variational problem

Inf

Length(γ ): γ:[0, ] →M₀, γ (0)∈Γandγ ()∈Γ .

We parameterizeγ by arc-length and definep:=γ (0)andq:=γ ()where 0< :=Length(γ ).

We parameterize a small neighborhood ofpinΓby arc lengths∈(−, )→φ(s)where >0 andφ(0)=p.

We denote by n(s)the normal vector aboutΓ at the pointφ(s)and assume that the orientation is chosen so that k >0. We further assume that the orientation ofγ is chosen so that so thatγ (0)˙ =natp(if not, just changepintoq andΓwithΓ).

We define the map

A:(−, )× [0, ] →M, (s, t )→Exp_{φ (s)}

t n(s) . We claim that the following is true:

(15)

Fig. 6. MovingΓalongγ.

Lemma 14.We have

∂A

∂s(0, t )=0, for allt∈ [0, ].

A straightforward application of the implicit function theorem implies that, provided >0 is chosen sufficiently small, the mappingAdefined is a diffeomorphism from(−, )× [0, ]onto its image.

Let us assume that we have already proven the claim and let us complete the proof of Theorem 2.

In the image ofA, we can decompose the metricgas A^∗g=dt²+f²(s, t ) ds²,

wheref (s,0)=1 and, because of the chosen orientation,

∂_tf

f (s,0)= −k,

the (constant) geodesic curvature ofΓ.

For allt∈ [0, ], we denote bykg(s, t )the geodesic curvature of the curves→A(s, t ), at the pointA(s, t )and we denote byK(s, t )the Gauss curvature at the pointA(s, t ). We have

K(s, t )= −∂_t²f f (s, t )

and, again because of the chosen orientation, k_g(s, t )= −∂_tf

f (s, t ).

We compute

∂_tk_g(s, t )= −∂_t ∂_tf

f (s, t )=k_g²(s, t )+K(s, t ). (A.1)

By assumption, we have k²>−minMK,

so that∂_tk_g(s,0) >0, then (A.1) implies that∂_tk_g(s, t ) >0 for allt∈ [0, ]. In particular

k_g(0, ) > kg(0,0):=k. (A.2)

However, because the minimizing property ofγ, the curves→A(s, )is on one side ofΓand tangent toΓatq. Therefore, we can compare the geodesic curvature of these two curves and given the chosen orientation we necessarily have

kkg(0, )

sinceΓhas constant geodesic curvature equal tok. But this clearly contradicts (A.2). This completes the proof of Theorem 2.