Prescribing the penetration

In Section4, we will use the following operation repeatedly: a geodesic ray or line γ starting from a given pointξ₀ is given that penetrates twoǫ-convex sets C and C^′ with penetration maps f and f^′, first entering C with f (γ)=h, and then C^′ with f^′(γ)≥h^′. We will need to pick a new geodesic ray or lineγ^′ starting fromξ₀which intersects C before C^′, for which we still have f (γ^′) =h, and for which we now have the equality f^′(γ^′)=h^′. In the following result, we show that this operation is possible in a number of geometric cases. These cases will be used in Section5for various applications.

Proposition 3.7 Let X be a complete, simply connected Riemannian manifold with sectional curvature at most−1 and dimension at least 3. Letǫ >0 and δ,h,h^′ ≥0.

Let C and C^′ be ǫ-convex subsets of X , and ξ0 ∈ (X∪∂_∞X)−(C∪∂_∞C). Let f and f^′ be maps T_ξ¹

0X → [0,+∞], with f^′ continuous andκ^′ =kf^′−ℓ_C^′k∞ <+∞. Consider the following cases:

(1) C is a horoball with diam(C∩C^′) ≤δ; f is either the penetration height map ph_C or the inner-projection penetration mapipp_C;

h≥h^min=2c^′₁(ǫ)+2δ+kf −ph_Ck∞

and h^′ ≥h^min₀ =κ^′+2δ; if C^′ is also a horoball we may take ǫ= +∞ in the definition of h^min.

(2) C is a ball of radius R (≥ǫ)with diam(C∩C^′) ≤δ; f is either the penetration height map ph_C or the inner-projection penetration mapipp_C;

h>h^min=2c^′₁(ǫ)+2δ+kf−ph_Ck^∞ ≤ h ≤ 2R−2c^′₁(ǫ)− kf −ph_Ck^∞=h^max and h^′ ≥h^min₀ =κ^′+2δ;

(3) C is the ǫ-neighbourhood of a complete totally geodesic subspace L of dimen-sion at least 2, with diam(C ∩C^′) ≤ δ; either f = ℓC and X has constant curvature, or f is the fellow-traveller penetration mapftp_L;

h≥h^min=4c^′₁(ǫ)+2ǫ+δ+2kf −ftp_Lk∞

and h^′ >h^min₀ =κ^′+δ;

(4) _• C is theǫ-neighbourhood of a geodesic line L;

• h≥h^min=4c^′₁(ǫ)+2ǫ+δ+kf −ftp_Lk∞ ;

• either f = ℓ_C and X has constant curvature, or f is the fellow-traveller penetration map ftp_L, or ξ0 ∈∂_∞X, f =crp_L, and the metric spheres of the Hamenstädt distance on ∂_∞X− {ξ₀}are topological spheres;

• either C^′ is anyǫ-convex subset that does not meet C (in which case δ = 0) and h^′ >h^min₀ =κ^′, or C^′ is the ǫ-neighbourhood of a totally geodesic subspace with codimension at least two such that diam(C∩C^′)≤δ and

h^′ ≥h^min₀ =3c^′₁(ǫ)+3ǫ+δ+kf^′−ftp_L^′k∞.

Assume that one of the above cases holds. If there exists a geodesic ray or line γ starting from ξ0 which meets first C and then C^′ with f (γ)=h and f^′(γ) ≥h^′, then there exists a geodesic ray or line γ starting from ξ₀ which meets first C and then C^′ with f (γ)=h and f^′(γ)=h^′.

Proof. Let γ be as in the statement, and x (resp. y) be the point where γ enters (resp. exits) C (with y in X since f (γ) = h < +∞). Let x^′ (resp. y^′) be the point where γ enters (resp. exits) C^′ (with x^′ ∈ X but possibly with y^′ at infinity). By convexity,ξ0∈/C^′∪∂_∞C^′. For every h≥0, we define A as the set of points α(+∞) where α ∈ T_ξ¹

0X satisfies f (α) = h. Let A₀ be the arcwise connected component of A containing γ(+∞). By considering the various cases, we will prove below the following two claims :

a) every geodesic ray or line, starting from ξ0 and meeting C^′, first meets C and then C^′;

b) there exists a geodesic ray or line γ₀ starting from ξ₀ with γ₀(+∞) belonging to A0, and f^′(γ₀)≤h^min₀ .

As f^′ is continuous and A0 is arcwise connected, the intermediate value theorem im-plies the existence of a geodesic γ with the desired properties, and Proposition3.7is proven.

Case (1). Letκ=kf −ph_Ck∞. Letξ be the point at infinity of C , which is different from ξ0, and let p_ξ be the closest point to ξ on γ. As f (γ) = h > 0, the point p_ξ belongs to the interior of the horoball C . Let γ_ξ be the geodesic ray or line starting fromξ0 withγ_ξ(+∞)=ξ.

x^′

ξ0

p_ξ

u^′ C[δ]

y

ξ γξ

x

C C^′

y^′ γ

We start by proving the (stronger) first claim that every geodesic ray or line starting from ξ0 and meeting C^′ meets C[δ] first (hence it meets C before C^′ and Assertion a) holds). Note that

d(y,pξ)≥d(pξ, ∂C)= ph_C(γ)

2 ≥ f (γ)−κ

2 = h−κ

2 ≥ h^min−κ

2 =c^′₁(ǫ)+δ > δ . As f^′(γ)≥ h^′ ≥ h^min₀ =κ^′+2δ, we have ℓ_C^′(γ) ≥f^′(γ)−κ^′ > δ, unless ℓ_C^′(γ) = δ = 0. Note that γ∩C^′ is not contained in the geodesic segment [x,y]. Otherwise, this would contradict the assumption that diam(C∩C^′)≤δ when ℓ_C^′(γ) > δ. When ℓ_C^′(γ) =δ =0, as γ meets C^′, the segment γ∩C^′ would be reduced to a point by the convexity of C^′. This point would be {x} or{y} (as C^′ is not a singleton). But then the tangent vector of γ at x or its opposite at y would both enter strictly C and be tangent to C^′, which contradicts the fact that δ=0.

Asγ meets C before C^′, this implies, in particular, that the geodesic ray [y, γ(+∞)[

meets C^′, and that the point pξ belongs to ]x^′, ξ₀] : otherwise C∩C^′ would contain a segment of length at least d(p_ξ,y)> δ, which is impossible. Hence, by convexity, any geodesic ray or line starting from ξ₀ and meeting B(x^′,c^′₁(ǫ)) first meets B(p_ξ,c^′₁(ǫ)).

By Lemma2.3, every geodesic ray or line, starting fromξ0and meeting C^′, meets the ball B(x^′,c^′₁(ǫ)) before entering C^′ (and we may takeǫ= +∞ if C^′ is also a horoball, by Lemma2.9). This proves the first claim, as the ball B(p_ξ,c^′₁(ǫ)) is contained in C[δ], since d(p_ξ, ∂C)≥c^′₁(ǫ)+δ, as seen above.

Let us prove now the (stronger) second claim that there exists a geodesic ray or lineγ₀ starting fromξ₀ withγ₀(+∞) belonging to A₀, and avoiding the interior of C^′, which implies assertion b), as then f^′(γ₀)≤ℓC^′(γ₀)+κ^′ =κ^′ ≤h^min₀ .

The subspace A of ∂_∞X is a codimension 1 topological submanifold of the topo-logical sphere ∂_∞X , which is homeomorphic to the sphere Sⁿ⁻², hence it is arcwise connected. Indeed, if f =ph_C, then A is the subset of endpoints of the geodesic rays or lines starting from ξ₀ that are tangent to ∂ C[h/2]

. If f = ipp_C, the subset A is the preimage of a point in ]ξ₀, ξ[ by the closest point map from∂_∞X to [ξ₀, ξ], which is, over ]ξ0, ξ[, a trivial topological bundle with fibers homeomorphic to Sⁿ⁻². Note that f is continuous, f (γ_ξ)=∞>h, and f (α)=0 ifα is a geodesic ray or line starting fromξ₀ withα(+∞) close enough toγ(−∞). Therefore A separates γ(−∞) and γ_ξ(+∞), as the connected components of ∂_∞X−A are arcwise connected.

If the (stronger) second claim is not true, then the topological sphere A₀ = A of dimension n−2 is contained in the interior of the shadow O_ξ

0C^′. Asξ0∈/C^′∪∂_∞C^′, this shadow is homeomorphic to a ball of dimension n−1. Thus, by Jordan’s theorem, one of the two connected components of ∂_∞X −A is contained in the interior of O_ξ

0C^′. As γ(−∞) does not belong to O_ξ

0C^′ and A separates γ(−∞) and γ_ξ(+∞), this implies thatγ_ξ(+∞) belongs to the interior ofO_ξ

0C^′. Henceγ_ξ meets the interior of C^′.

Therefore, by the first claim, the geodesic ray or line γ_ξ meets C[δ] before meeting C^′. Let u^′ be the entering point of γ_ξ in C^′. Asξ is the point at infinity of C[δ], the pointsξ₀,u^′, ξ are in this order onγ_ξ. Hence by convexity, this implies that u^′ belongs to C[δ]. As f^′(γ)≥h^′ ≥h^min₀ =κ^′+2δ, we have

d(x^′,y^′)=ℓ_C^′(γ)≥f^′(γ)−κ^′ ≥2δ .

Hence by the triangular inequality, one of the two distances d(u^′,x^′),d(u^′,y^′) is at least δ, and by the strict convexity of the distance, it is strictly bigger than δ (as u^′ does not belong toγ (asγ 6=γ_ξ). Hence, if u^′′is a point close enough to u^′ in ]u^′, ξ[, then u^′′ belongs to the interior of C^′ and to the interior of C[δ], and is at distance strictly greater than δ from either z^′ =x^′ or z^′ =y^′. Therefore, the geodesic segment [u^′′,z^′]∩C has length strictly bigger thanδ, and is contained in the intersection C∩C^′. This contradicts the assumption that diam(C∩C^′)≤δ.

Case (2). The proof is completely similar to Case (1). Let now z be the center of the ball C , let pz be the point of γ the closest to z, let γz be the geodesic ray or line starting from ξ₀ and passing through z, and let κ = kf −ph_Ck∞. Note that R>h^max/2≥h^min/2≥δ, so that C[δ] is non empty. We only have to replace ξ by z, p_ξ by pz and γ_ξ by γz, and to replace two arguments in the above proof, the one in order to show that A separatesγ(−∞) from γ_z(+∞), and the one in order to show that ξ0,u^′,z are in this order on γz, where u^′ is the entering point ofγz in C^′.

To prove that A separates γ(−∞) from γ_z(+∞), we simply use now that f (γ_z) = 2R>h^max≥h instead of f (γ_ξ) =∞>h. Let us prove that ξ₀,u^′,z are in this order onγz. We have

d(z,p_z)=R−ph_C(γ)

2 ≥R−f (γ)+κ

2 ≥R−h^max+κ

2 =c^′₁(ǫ).

By Lemma2.3, we have d(x^′,u^′)≤c^′₁(ǫ). Asγ_z meets the interior of C^′, by the same argument as in Case 1, we even have d(u^′, γ) <c^′₁(ǫ). Hence by strict convexity, we do have u^′ ∈]z, ξ₀[. The rest of the argument in the proof of Case (1) is unchanged.

Before studying the last two cases, we start by proving two lemmas. The first one implies the first of the two claims we need to prove in Cases (3), (4), and the second one gives the topological information on A that we will need in these last two cases.

Lemma 3.8 Let L be a complete totally geodesic subspace with dimension at least 1, ǫ > 0, C = N_ǫL, ξ₀ ∈ (X∪∂_∞X)−(C∪∂_∞L), and C^′ be an ǫ-convex subset of X such that diam(C∩C^′) ≤ δ. Let f,f^′ : T_ξ¹

0X → [0,+∞] be maps such that κ = kf −ftp_Lk∞ < +∞, κ^′ = kf^′ −ℓ_C^′k∞ < +∞. Let γ be a geodesic ray or line starting fromξ0, entering C before entering C^′, such that 4c^′₁(ǫ)+2ǫ+δ+κ ≤ f (γ) <+∞ and f^′(γ)> δ+κ^′. Ifeγ is a geodesic ray or line starting from ξ0 which meets C^′, thenγemeets the interior of C before meeting C^′.

Proof. Note that ξ0 ∈/ C^′∪∂_∞C^′, by convexity and the assumptions on γ, as ξ0 ∈/ C∪∂_∞C . Let L0 be the geodesic line passing through the closest points p_ξ0,p_γ(+∞)

on L ofξ₀, γ(+∞), respectively. Note that

(- 13 -) d(p_ξ0,pγ(+∞))=ftp_L(γ)≥f (γ)−κ≥4c^′₁(ǫ)+2ǫ+δ >0. Hence, by Lemma3.4, and asftp_L0(γ)=ftp_L(γ), we have

ℓN_ǫL0(γ) ≥ftp_L0(γ)−2c^′₁(ǫ)−2ǫ >0.

In particular, γ entersN_ǫL0 at a point x0 and exits it at a point y0 in X (asγ(+∞)∈/

∂_∞L since f (γ)<+∞). Let u7→pube the closest point map from X∪∂_∞X onto L0∪

∂_∞L₀. Recall that this map does not increase the distances (and even decreases them, unless the two points under consideration are on L0), and that it preserves betweenness, that is, if u^′′ ∈ [u,u^′], then p_u^′′ ∈ [pu,p_u^′]. Let x^′ (resp. ex^′) be the point where γ (resp. eγ) enters C^′, and q_ξ0 and q_γ(+∞) be the closest point to ξ₀ and γ(+∞) respectively onN_ǫL0.

γ(+∞)

p_ξ0 ξ0

p_γ(+∞)

eγ(+∞)

p_eγ(+∞)

p_ex^′ p_x^′ p_y0 y0

q_ξ0 q_γ(+∞)

L0⊂L eγ

x^′ x0

ex^′

γ

C^′

Recall that by Lemma2.3, the distances d(ex^′,x^′),d(x₀,q_ξ0),d(y₀,q_γ(+∞)) are at most c^′₁(ǫ). Note thatex^′∈[ξ0,eγ(+∞)]. Hence, as betweenness is preserved,

ftp_L0(eγ)=d(p_ξ0,p_γ(e+∞))≥d(p_ξ0,p_ex^′)≥d(p_ξ0,p_x^′)−d(p_x^′,p_ex^′)

≥d(p_ξ0,p_x^′)−d(x^′,ex^′)≥d(p_ξ0,p_x^′)−c^′₁(ǫ).

Note that d(p_ξ0,p_x^′)≥d(p_ξ0,p_y0) whenξ₀,y₀,x^′ are in this order onγ. Whenξ₀,y₀,x^′ are not in this order onγ, as γ enters in C before C^′, asℓC^′(γ)≥f^′(γ)−κ^′ > δ and as diam(C∩C^′)≤δ, we have d(x^′,y0)≤δ; hence

d(p_ξ0,p_x^′)≥d(p_ξ0,p_y0)−d(p_y0,p_x^′)≥d(p_ξ0,p_y0)−d(y₀,x^′)≥d(p_ξ0,p_y0)−δ . Therefore, in both cases, as y0∈[ξ0, γ(+∞)] and u7→pupreserves the betweenness, and since p_γ(+∞) =p_qγ(+∞), we have, using (- 13 -),

ftp_L0(γe)≥d(p_ξ0,p_ex^′)≥d(p_ξ0,p_x^′)−c^′₁(ǫ)≥d(p_ξ0,p_y0)−δ−c^′₁(ǫ)

≥d(p_ξ0,p_γ(+∞))−d(p_qγ(+∞),p_y0)−c^′₁(ǫ)−δ

>ftp_L(γ)−d(q_γ(+∞),y₀)−c^′₁(ǫ)−δ≥ ftp_L(γ)−2c^′₁(ǫ)−δ≥2c^′₁(ǫ)+2ǫ . By Lemma3.4, we hence have

ℓN_ǫL(eγ)≥ℓN_ǫL0(eγ)≥ftp_L0(eγ)−2c^′₁(ǫ)−2ǫ >0.

In particular, eγ does enter the interior of C , at a pointex. Note that the geodesic from ξ₀ through p_ξ0 enters C at q_ξ0. Now by absurd, if γe enters the interior of C after it enters C^′, thenex^′∈[ξ₀,ex], so that

c^′₁(ǫ)≥d(q_ξ0,ex)≥d(p_ξ0,p_ex)≥d(p_ξ0,p_ex^′)>2c^′₁(ǫ)+2ǫ ,

as seen above, a contradiction.

Lemma 3.9 Let X be a complete, simply connected Riemannian manifold with sec-tional curvature at most−1 and dimension at least 3. Letǫ,h >0. Let L be a com-plete totally geodesic submanifold with dimension at least 1 andξ₀ ∈ (X∪∂_∞X)− (N_ǫL∪∂_∞L). Assume either that

(1) f =ℓN_ǫL, X has constant curvature and h∈[4c^′₁(ǫ)+2ǫ,+∞[, or (2) ξ0 ∈∂_∞X, dim L =1, f =crp_L, h ∈

log 2,+∞

, and the metric spheres of the Hamenstädt distance on ∂_∞X− {ξ₀}are topological spheres, or

(3) f =ftp_L. Then

A={α(+∞) : α∈T_ξ¹₀X, f (α) =h}

is a codimension 1 topological submanifold of the topological sphere ∂_∞X, which is homeomorphic to the torus S^{dim L−1}×S^{codim L−1}. Furthermore,

(a) if dim L = 1, then A has two arcwise connected components, homeomorphic to a sphere of dimension n−2. If f =crp_L or if h>c^′₁(ǫ), then each of them separates γ(−∞) and exactly one of the two points at infinity of L, for every geodesic ray or line γ starting from ξ₀ if f =crp_L, and for those meetingN_ǫL if f 6=crp_L.

(b) if codim L=1, then A has two arcwise connected components, homeomorphic to a sphere of dimension n−2, separated by ∂_∞L.

(c) if dim L≥2and codim L≥2, then A is arcwise connected.

In cases (b) and (c), for every component A0 of A, for every geodesic rayρ in L with ρ(0) the closest point to ξ₀ on L, there exists η ∈A₀ such that ρ(h) is at distance at mostkf −ftp_Lk∞ from the closest point to η onρ.

Proof. Let πL : X∪∂_∞X → L∪∂_∞L be the closest point map, and p0 = πL(ξ0).

Note that L has codimension at least one, by the existence ofξ0.

Assume first that f = ftp_L. As h > 0, the subspace A of ∂_∞X is the preimage of the sphere (of dimension dim L−1) of center p0 and radius h in L, by π_L. As πL : ∂_∞X \∂_∞L → L is a trivial topological bundle whose fibers are spheres of dimension codim L−1, the topological structure (including the assertions (b) and (c) ) of A is immediate. The final statement on (b) and (c) is trivial as, by definition, ρ(h) is the closest point to some point in A0.

If dim L = 1 and h > c^′₁(ǫ), if γ ∈ T_ξ¹

0X meets N_ǫL, then by Lemma2.3 and by convexity, d(π_L(γ(−∞)),p₀) ≤ c^′₁(ǫ) < h. Hence, the separation statement in (a) follows.

Assume then that f =crp_L, and that the hypotheses of (2) are satisfied. The result in this case (only assertion (a) needs to be checked) follows from the discussion before Lemma3.5.

Assume now that f =ℓN_ǫL, X has constant curvature, and h∈[4c^′₁(ǫ)+2ǫ,+∞[ . Let S₀=π⁻_L¹(p₀)∩∂_∞X . Using normal coordinates along L, the topological sphere∂_∞X is homeomorphic to the topological join of the spheres ∂_∞L of dimension dim L−1 and S₀ of dimension codim L−1

S₀∨∂_∞L= S₀×[0,+∞]×∂_∞L /∼,

where ∼ is the equivalence relation generated by (a,0,b) ∼ (a,0,b^′) as well as (a,+∞,b) ∼ (a^′,+∞,b), for every a,a^′ in S0 and b,b^′ in ∂_∞L. We denote by [a,t,b] the equivalence class of (a,t,b). We choose the parametrization of ∂_∞X by S₀∨∂_∞L such that [a,0,b] = a, [a,+∞,b] = b, d(π_L([a,t,b]),p₀) = t, and the geodesic rays

π_L([a,t,b]),[a,t,b]

are parallel transports of [p₀,a[ along the geodesic ray [p0,b[ , for 0<t<+∞.

For every t in ]0,+∞[ and every (a,b) in S0×∂_∞L, letγ_[a,t,b] be the geodesic ray or line starting fromξ₀ and ending at [a,t,b]. By the proof of Lemma3.4and by Lemma 3.2, we have

−2c^′₁(ǫ)≤ℓN_ǫL(α)−ftp_L(α) ≤2c^′₁(ǫ)+2ǫ,

for everyα∈T_ξ¹₀X . In particular, if t=ftp_L(γ_[a,t,b])>2c^′₁(ǫ), thenℓN_ǫL(γ_[a,t,b])>0, that is γ[a,t,b] meets the interior of N_ǫL. The points p0,a,b, ξ0 are contained in an isometrically embedded copy of H³_R in X , and therefore we can restrict to the case when X is the upper halfspace model ofH³_R and L has dimension 1. If we normalize so that ξ0=∞ and the endpoints of L are b=1 and −b, then the level sets of ℓN_ǫL

for positive values are drawn in the following picture. The level set ℓ⁻_N¹

ǫL(2ǫ) is the unique figure-8 curve. Each level set ℓ⁻_N¹

ǫL(t) for t>2ǫ has exacty two components, one in each bounded component of the complement ofℓ⁻_N¹_ǫL(2ǫ).

The curve t 7→ [a,t,b], t ≥ 0, is a segment of a circle through b and −b that con-nects the point a on the imaginary axis to b. Thus, it is easy to see that the map

from [2c^′₁(ǫ),+∞[ to [0,+∞[ defined by t7→ℓN_ǫL(γ_[a,t,b]) is continuous and strictly increasing, for every fixed (a,b) in S₀×∂_∞L.

Hence, as Lemma3.4gives ℓN_ǫL(γ_[a,2c^′

1(ǫ),b])≤4c^′₁(ǫ)+2ǫ≤h<+∞,

there exists a unique ta,b∈[2c^′₁(ǫ),+∞[, depending continuously on (a,b), such that ℓN_ǫL(γ_[a,ta,b,b]) = h. In particular, the subset of points of ∂_∞X of the form [a,t_a,b,b]

for some (a,b) in S₀ ×∂_∞L is indeed a codimension 1 topological submanifold of

∂_∞X , which is homeomorphic to the torus S^{dim L−1}×S^{codim L−1}. The statements (b) and (c) follow.

If L has dimension 1, and if γ ∈ T_ξ¹

0X meets N_ǫL, then by Lemma 2.3 and by convexity, d(π_L(γ(−∞)),p₀) ≤ c^′₁(ǫ). For every ξ in a component A₀ of A, if as above ξ =[a,t_a,b,b], then we have d(π_L(ξ),p₀) =t_a,b ≥2c^′₁(ǫ), hence A₀ separates γ(−∞) and b. This proves (a).

Let us prove the last assertion of the lemma. Let κ = kf −ftp_Lk∞, and let A₀ be a connected component of A. For every u in L such that d(u,p0)=h, letη0 =[a,h,b], on the same side of ∂_∞L as A₀ if codim L = 1, be such that π_L(η₀) = u. Let η_t =[a,h+t,b], which is on the same side of ∂_∞L as A₀ if codim L=1. Note that

f (γ_[a,h+κ,b])≥ftp_L(γ_[a,h+κ,b])−κ=h,

and similarly, f (γ_[a,h−κ,b]) ≤ h. By the intermediate value theorem, there exists t ∈ [−κ,+κ] such thatη_t ∈A₀. Hence d(u, π_L(η_t))=|t| ≤κ.

Now we proceed with the proof of the remaining parts of Proposition3.7.

Case (3). By Lemma3.8, we only have to prove the second claim, that there exists a geodesic ray or line γ0 starting from ξ0 with γ0(+∞) belonging to A0, such that f^′(γ₀)≤h^min₀ .

Let κ = kf−ftp_Lk_∞. Let p0 (respectively p_γ) be the point of L the closest to ξ0

(respectively γ(+∞)), so that, in particular,

d(p0,p_γ) =ftp_L(γ)≥f (γ)−κ=h−κ >0.

Let p^′_γ be the point on the geodesic line L0(contained in L) passing through p0 and pγ

on the opposite side of p_γ with respect to p0, and at distance h from p0. By Lemma 3.9, there exists a geodesic line γ₀ starting fromξ₀ and ending at a point in A₀ whose closest point pγ0 on L is at distance at most κ from p^′_γ.

y^′ y

p_γ p0

y0

y^′₀

ξ0 γ(+∞)

p^′_γ p_γ0 γ0(+∞)

x0 x

γ1

L0⊂L

≥h−κ ≥h−κ γ

γ0

Assume by absurd that f^′(γ₀)>h^min₀ . We have

ℓ_C^′(γ)≥f^′(γ)−κ^′ ≥h^′−κ^′ ≥h^min₀ −κ^′ > δ.

Similarly,ℓ_C^′(γ0)> δ, and, in particular, γ0 enters C^′. Let y^′ (resp. y^′₀) be the point, possibly at infinity, whereγ (resp.γ₀) exits C^′. By Lemma3.8,γ₀meets C before C^′. Let x0(resp. y0) be the point whereγ₀ enters in (resp. exits) C . As diam(C∩C^′)≤δ, we have y^′₀ ∈ ]y0, γ0(+∞)[ and y^′ ∈ ]y, γ(+∞)[, so that in particular d(y^′₀,L) > ǫ and d(y^′,L)> ǫ.

Let γ₁ be the geodesic line through y^′ and y^′₀. The points at infinity of γ₁ do not belong to∂_∞L0, so that ftp_L0(γ₁) and ℓ_C(γ₁) are finite. Note that by strict convexity and by Lemma2.3, we have

d(y^′,[p_γ, γ(+∞)[)<d(y,[p_γ, γ(+∞)[)≤c^′₁(ǫ),

and, similarly, d(y^′₀,[p_γ0, γ0(+∞)[)<c^′₁(ǫ). Hence, withπL0 the closest point map to L₀, which preserves betweenness and does not increase distances,

ftp_L0(γ₁)≥d(π_L0(y^′₀), π_L0(y^′))

>d(p_γ0,p_γ)−2c^′₁(ǫ)=d(p_γ0,p0)+d(p0,p_γ)−2c^′₁(ǫ)

≥h−κ+h−κ−2c^′₁(ǫ)=2h−2κ−2c^′₁(ǫ). In particular, by Lemma3.4,

ℓC(γ1)≥ℓN_ǫL0(γ1)≥ftp_L0(γ1)−2c^′₁(ǫ)−2ǫ >2h−2κ−4c^′₁(ǫ)−2ǫ≥δ , by the definition of h^min. Hence γ1 meets C in a segment I of length > δ. But as y^′₀ and y^′ are at a distance strictly bigger than ǫ of L, the segment I is contained in [y^′,y^′₀], which is contained in C^′, by convexity. This contradicts the assumption that diam(C∩C^′)≤δ.

Case (4). Letκ^′′ =kf^′−ftp_L^′k∞. Note that f^′(γ)≥h^′ ≥h^min₀ > δ+κ^′. This is true under both assumptions on the value of h^min₀ , as when h^min₀ =3c^′₁(ǫ)+3ǫ+δ+κ^′′, we have, by Lemma3.4,

δ+κ^′ ≤δ+κ^′′+2c^′₁(ǫ)+2ǫ <h^min₀ .

By Lemma3.8, we only have to prove the second claim that there exists a geodesic line γ₀ starting fromξ0 with γ₀(+∞) belonging to A0, such that f^′(γ₀)≤h^min₀ . We first consider the case C^′ = N_ǫL^′ where L^′ is a totally geodesic subspace of codimension at least 2, with diam(C∩C^′)≤δ, and h^′≥h^min₀ =3c^′₁(ǫ)+3ǫ+δ+κ^′′. Assume by absurd that every geodesic ray or lineα starting fromξ₀ with α(∞)∈A₀ satisfies f^′(α) > h^min₀ . In particular, every such α satisfies ℓ_C^′(α) ≥ f^′(α)−κ^′ ≥ h^min₀ −κ^′> δ≥0,henceα meets the interior of C^′. Let

B^′={β(∞) :β∈T_ξ¹₀X, ftp_L^′(β)>h^min₀ −κ^′′}.

By the absurdity hypothesis and the definition ofκ^′′, we have A0 ⊂B^′. Let p^′₀ be the closest point toξ0 on L^′. Note that B^′ is a (topological) open tubular neighbourhood of∂_∞L^′, whose fiber over a point ξ in∂_∞L^′ is the preimage ofρ_ξ(]h^min₀ −κ^′′,+∞]) by the closest point map from ∂_∞X to L^′∪∂_∞L^′, where ρ_ξ is the geodesic ray with ρ(0)=p^′₀ and ρ(+∞)=ξ.

By Lemma 3.9 (a), let ξ₁ be the point at infinity of L separated from γ(−∞) by A₀. Let p^′_γ(−∞) be the closest point to γ(−∞) on L^′. Recall that γ enters C^′ at x^′, and ξ0 ∈/ C^′, so that ξ0 ∈ [γ(−∞),x^′[ . Hence, by Lemma2.3and the fact that closest point maps preserve betweenness and do not increase the distances, we have d(p^′_γ(−∞),p^′₀)≤c^′₁(ǫ)<h^min₀ −κ^′′by the definition of h^min₀ . Hence the complement of B^′ in ∂_∞X , which is connected as codim L^′ ≥ 2, contains γ(−∞). As A0 separates γ(−∞) from ξ₁ and is contained in B^′, it follows that B^′ contains ξ₁.

x^′₀

ξ1

L^′₀⊂L^′ L

p^′_ξ1 p^′_u

ξ0

u

p^′₀

Let x^′₀ be the intersection point of ]ξ₀,p^′₀] with ∂C^′. Lemma2.3 then implies that d(x^′,x^′₀) ≤ c^′₁(ǫ). Hence, by convexity and as γ first meets C and then C^′, we have

d x,]ξ₀,p^′₀]

≤ c^′₁(ǫ). If u ∈ L is the closest point to x on L, we therefore have d(u,]ξ₀,p^′₀])≤c^′₁(ǫ)+ǫ. Let p^′_ξ

1 be the closest point toξ₁ on L^′, and L^′₀the geodesic line (contained in L^′) through p^′₀ and p^′_ξ

1. As the closest point map does not increase distances, the closest point p^′_u to u on L^′₀ satisfies d(p^′₀,p^′_u) ≤c^′₁(ǫ)+ǫ. Then, since the closest point map to L^′₀ preserves betweenness, by the triangle inequality, and as ξ1 belongs to B^′, we have

ftp_L^′

0(L)≥d(p^′_u,p^′_ξ1)≥d(p^′_ξ1,p^′₀)−d(p^′₀,p^′_u)>h^min₀ −κ^′′−c^′₁(ǫ)−ǫ . Therefore, using Lemma3.4,

diam(C∩C^′)≥diam(C∩N_ǫL^′₀)≥ℓN_ǫL^′₀(L)≥ftp_L^′

0(L)−2c^′₁(ǫ)−2ǫ

>h^min₀ −κ^′′−3c^′₁(ǫ)−3ǫ=δ , a contradiction.

Assume now that C^′ is any ǫ-convex subset such that C∩C^′ = ∅, and that h^′ >

h^min₀ = κ^′′. Let us prove that there exists a geodesic ray or line γ₀ starting from ξ₀ withγ₀(+∞) in A₀, and avoiding the interior of C^′. This implies the result as in Case (1).

By absurd, suppose that for every ξ in A₀, the geodesic ray or line γ_ξ starting from ξ0 and ending at ξ meets the interior of C^′. By Lemma3.8(applied with δ =0), γ_ξ meets the interior of C before meeting C^′. Let x^′_ξ be the entering point of γ_ξ in C^′ and y_ξ be its exiting point out of C . As C and C^′ are disjoint, note that ξ₀,y_ξ,x^′_ξ, ξ are in this order along γ_ξ. The maps ξ7→y_ξ and ξ 7→x^′_ξ are injective and continuous on A₀ (by the strict convexity of C and C^′, asγ_ξ meets the interior of C and C^′). We know that A₀ is a topological sphere, by Lemma3.9 (a), separating the endpoints of L. Hence the subsets A0 and S^′ ={x^′_ξ :ξ ∈A0}are spheres, that are homotopic (by the homotopy along the geodesic ray or line γ_ξ that does not meet L between x^′_ξ and ξ) in the complement of L in X∪∂_∞X . By a homology argument, every disc with boundary S^′ in X∪∂_∞X has to meet L. But by convexity of C^′, there exists a disc contained in C^′ with boundary S^′ (fix a point of S^′ and take the union of the geodesic arcs from this point to the other points of S^′). This contradicts the fact that C∩C^′ =∅.

Remarks. (1) In Case (2), we have h^max ≥ h^min if R is big enough, as c^′₁(ǫ) has a finite limit asǫ→ ∞.

(2) In Case (3), if the codimension of L is 1, then we may assume that γ meets L ifγ meets L. Indeed, as we have seen in Lemma3.9(b), L∪∂_∞L separates X∪∂_∞X into

two connected components, and A (defined in the beginning of the proof) has exactly two components separated by L∪∂_∞L. If A⁺₀ is the component of A on the same side of L∪∂_∞L asξ0, and A⁻₀ the component of A on the other side, then a geodesic ray or line starting fromξ₀ and ending in A⁺₀ does not meet L (as L is totally geodesic), and any geodesic line starting from ξ₀ and ending in A⁻₀ meets L, by separation. This observation on the crossing property will be used in the proof of Corollary 5.12 to make sure that the locally geodesic ray or line constructed in the course of the proof stays in the convex core.

(3) Case (4) is not true if C^′ is assumed to be anyǫ-convex subset, as shown by taking X the real hyperbolic 3-space, and C^′ the ǫ-neighbourhood of the (totally geodesic) hyperbolic plane perpendicular to L at a point at distance h from the closest point to ξ₀ on L: any geodesic ray or lineα starting from ξ₀, with ftp_L(α) =h and meeting C^′ satisfies f^′(α) = +∞for every f^′ which is a κ^′-penetration map in C^′.

Dans le document 3 Properties of penetration in ǫ-convex sets (Page 39-51)