The Legendrian foliation near isolated points

∂Ω_r

(∇0U )^⊥· nrds (4.40)

=

Γ_r

(∇0U )^⊥· nrds =−

Γ_r

|∇0U|νH^⊥(p⁺)· nrds = (−r + o(r))|Γr|.

On the other hand, the divergence theorem implies g(r) =

Ω_r

div(∇0U )^⊥dx₁dx₂=−

r 0

|Γs| ds. (4.41)

From (4.40) and (4.41) we obtain the ODE (g^/g)(r) = (1/r) + o(1/r) whose solution g(r) = cr²+ o(r²) contradicts (4.39). Thus ˜γ₁⁺= ˜γ₂⁺. A similar argument yields the uniqueness and non-transversality of ˜γ⁻.

4.4.2 The Legendrian foliation near isolated points of the characteristic locus

Next, we turn our attention to isolated characteristic points of t-graphs. We will show that every characteristic curve intersecting a small neighborhood of the pro-jection of such a point p will reach p in finite time. Moreover, to every tangent direction a∈ T(p,u(p))Gu there corresponds one and only one curve of the Legen-drian foliation tangent to a at p.

For u∈ C²(Ω) we define the vector field T (x1, x2) = ∂_x2u−x1

2 ,−∂x₁u−x1

2

!

= (∇0U )^⊥(x1, x2,·).

4.4. Analysis at the characteristic set and fine regularity of surfaces 85

Note that T = 0 on Su and that the projections of curves in the Legendrian foliation are tangent toT in Ω \ Su. We will also consider the differential ofT :

d_pT =

∂_x²_1,x2u(p)−¹₂ ∂_x²₂u(p)

−∂x²₁u(p) −∂x²₁,x₂u(p)−¹₂



. (4.42)

Lemma 4.36. Assume p∈ Su is an isolated (projection of a) characteristic point, and|H0(z, u(z))| = o(dist(p, z)⁻¹). Then

Next, we recall from (4.12) that H0= 1

Substituting (4.37) in (4.43) we obtain

H0=

which contradicts the hypothesis|H0| = o(dist(p, x)⁻¹). Setting ∆x = and arguing in the same fashion gives b = 0. Next, set b = c = 0 in (4.44) to obtain

H0=−ad (a + d)∆x∆y + o((∆s)²) (a²(∆x)²+ d²(∆y)²)^3/2+ o((∆s)³),

from which we deduce that 2∂_x²_1,x2u(p) = a + d = 0. Thus a =−d and b = c = 0 which gives (i). To conclude the proof we observe that det d_pT = 1/4 > 0, whence

Index_pT = 1.

86 Chapter 4. Horizontal Geometry of Submanifolds

Lemma 4.37. With the hypotheses of Lemma 4.36 in force, there exists a neigh-borhood U_p of p such that every characteristic curve γ which intersects U_p\ {p}

reaches p in finite time.

Proof. Let q ∈ Ω and use polar coordinates to represent q − p = (∆x, ∆y) = (∆s)e^{i φ}. Let α, β ∈ R satisfy

ν_H^⊥ = αe^{i φ}+ βi e^{i φ}. In view of (4.37) and Lemma 4.36(i),

∂_x1u +1 2x₂=1

2∆y + o(∆s),

∂_x2u−1

2x₁=−1

2∆x + o(∆s), ν_H^⊥ =− ∆x

∆s + o(1),∆x

∆s + o(1)

!

(4.45)

as ∆s→ 0. Consequently, α = νH^⊥· e^{i φ}=−1 + o(1) and β = o(1).

Next, consider a curve γ as in the statement of the lemma, and represent it in polar coordinates: γ(t) = ∆s(t)e^{i φ(t)}. Recalling from Section 4.3.2 the identity γ^=−νH^⊥, we obtain

(∆s)^e^{i φ}+ (∆s)φ^i e^{i φ}=−νH^⊥ =−(1 + o(1))e^{i φ}+ o(1)i e^{i φ}. Thus

(∆s)^ =−1 + o(1) (4.46)

so, in sufficiently small neighborhoods of p, ∆s(t) reaches zero at a finite time

t = T .

We now combine all of the preceding work to deduce a structure theorem for the projection of isolated points in the characteristic locus.

Theorem 4.38 (Cheng–Hwang–Malchiodi–Yang). Assume:

(H1) p∈ Su is an isolated (projection of a) characteristic point;

(H2) the bound|H0(z, u(z))| = o(dist(p, z)⁻¹) holds for z∈ Su near p;

(H3) for some r₀> 0,

r₀ 0

sup

z∈∂B(p,r)|H0(z, u(z))| dr < ∞.

Then for all a∈ S¹ there exists a unique C¹ curve γa such that

(i) γ_a is characteristic, i.e., the projection of a curve in the Legendrian foliation of Gu,

(ii) p lies in the closure of γa,

(iii) lim_{q∈γa,q→p}ν_H(q) exists and is orthogonal to a.

Moreover, as a ranges over all of S¹, such curves γa cover Up\ {p} for some neighborhood Up of p.

4.4. Analysis at the characteristic set and fine regularity of surfaces 87

Proof. Let Up be as in Lemma 4.37. Choose δ > 0 sufficiently small so that B(p, δ) ⊂ Up. For each q ∈ ∂B(p, δ) denote by γ the unique characteristic curve through q. In view of Lemma 4.37, the curve γ will reach p at a finite time T . Con-sider a sequence of parameter values t_j T and set qj= γ(t_j) so lim_j→∞q_j= p.

Recall from Remark 4.25 that if ν_H(γ(t)) = exp(i θ(t)) then

H0(γ(t), u(γ(t)) =−θ^(t). (4.47) Denote by θ_j the angle corresponding to q_j, i.e., ν_H(q_j, u(q_j)) = (cos θ_j, sin θ_j).

Using hypothesis (H3) we will show that{θj} is a Cauchy sequence. First, observe that

θ_j− θk =

t_j t_k

θ^(t) dt =

t_j t_k

H0dt. (4.48)

Recall from (4.46) that (∆s)^(t) = −1 + o(1) for t near T . Consequently, we can express the parameter t in terms of r = ∆s and estimate t^(r) ≈ 1 in a neighborhood of p. Using this observation, letting r_j = r(t_j), r_k = r(t_k) and in view of (4.48) we obtain

|θj− θk| ≤

r_j r_k

sup

z∈∂B(p,r)|H0(z, u(z))| |t^(r)| dr → 0 as j, k→ ∞, (4.49) thus proving that (θj) is Cauchy. Let us denote by θ_(p,q) its limit as j→ ∞. Now we can define a map ψ : ∂B(p, δ)→ S¹as follows:

ψ(q) = e^{i θ(p,q)}.

To conclude the proof of the theorem it suffices to show that ψ is a homeomor-phism.

Step 1 (ψ is continuous): Essentially we need to prove a result of C¹ continuity of solutions of a certain ODE with respect to initial data. Consider a sequence of points q_j ∈ ∂B(p, δ) converging to q ∈ ∂B(p, δ). Denote by θj = θ_(p,qj) (resp.

θ = θˆ _(p,q)) and by γ_j (resp. ˆγ) the corresponding characteristic curves joining q_j to p (resp. q to p). We must have that θ_j→ θ(p,q). Let φ_j be the angle in the polar coordinate representation of qj−p. Without loss of generality we may assume that φj is strictly decreasing. Since two curves in the Legendrian foliation cannot cross in B(p, δ)\ {p}, we also have θj ≥ θj+1 for all j. As a monotone and bounded from below sequence, (θ_j) has a limit θ.

We argue by contradiction. If θ = θ(p,q) then necessarily θ > θ_(p,q). In this case we find two rays emanating from p and forming an angle less than θ− θ(p,q)

such that for j sufficiently large, both γ_j and γ avoid a “fan-shaped” region ˜Ω surrounded by these two rays and ∂B(p, R) for some R > 0.

For any point ˜p∈ ˜Ω we consider the unique characteristic curve ˜γ joining ˜p to p. This curve will intersect ∂B(p, δ) at a point ˜q. Since ˜γ∩ ˜Ω does not intersect any γj, θj> ˜θ_{(p, ˜q)}and ˜q must lie in the arc between qj and q for all j. Thus ˜q = q and ˜γ = ˆγ which is a contradiction.

88 Chapter 4. Horizontal Geometry of Submanifolds

Step 2 (ψ is surjective): If ψ is not surjective then there exists ˜θ such that e^{i ˜θ}∈ S¹\ ψ(∂B(p, δ)). Since the latter set is open, we can find a neighborhood Iθ˜ of θ such that e˜ ^{i Iθ˜} is disjoint from the range of ψ. Arguing as in the proof of the continuity of ψ, we deduce the existence of a fan-shaped region ˜Ω, surrounded by two rays and a portion of a circle ∂B(p, R), which avoids all characteristic curves connecting points q∈ ∂B(p, δ) to p. For any ˜p ∈ ˜Ω we consider the unique characteristic curve ˜γ through ˜p and let ˜q be its intersection with ∂B(p, δ). Then clearly we must have e^{i θ(p,˜q)} ∈ e^{i Iθ˜}⊂ S¹\ ψ(∂B(p, δ)) which is a contradiction.

Step 3 (ψ is injective): Consider q1, q2 ∈ ∂Bδ distinct such that θ_(p,q1) = θ_(p,q2). Denote by γ_i a characteristic curve joining p to q_i. Then the angle between the tangent vectors of γ1 and γ2 at p is zero. Consider regions Ωrsurrounded by γ1, γ2and ∂B(p, r). Clearly Ωr is contained in a fan-shaped region with vertex p and aperture θr→ 0. Set Γr= ∂Ωr∩ ∂B(p, r), then

|Γr| ≤ rθr (4.50)

as before. Let n_rdenote the outer normal to ∂Ω_rand observe that n_r⊥ νH^⊥ along γ₁ and γ₂. Consequently

g(r) :=

∂Ω_r

(∇0U )^⊥· nrds =

Γ_r

(∇0U )^⊥· nrds

=−

Γ_r

(q(s)− p + o(r)) ·q(s)− p

r ds = (−r + o(r))|Γr|.

On the other hand, the divergence theorem implies g(r) =

Ω_r

div(∇0U )^⊥dx₁dx₂=−

r 0

|Γs| ds. (4.51)

From (4.4.2) and (4.51) we obtain the ODE g^

g =1 r + o 1

r

! ,

which yields g(r) = cr²+ o(r²) for some c > 0, in contradiction with (4.50).

Step 4 (ψ⁻¹is continuous): We argue by contradiction. Assume there is a sequence (q_j)⊂ ∂B(p, δ) converging to ˜q so that

θ_j := θ_(p,qj)→ θ := θ(p,q) (4.52) with q= ˜q ∈ ∂B(p, δ). Without loss of generality we may assume θj ≥ θj+1≥ θ.

Since q = ˜q we can find ¯q ∈ ∂B(p, δ) such that q = ¯q, ˜q = ¯q and θj ≥ θ(p, ¯q)≥ θ.

But then, by (4.52) and the injectivity of ψ we must have ¯q = q. With this contradiction we complete the proof of the theorem.

Nel documento Progress in Mathematics Volume 259 (pagine 98-103)