**4.4 Analysis at the characteristic set and ﬁne regularity of surfaces**

**4.4.2 The Legendrian foliation near isolated points**

*∂Ω*_{r}

(*∇*0*U )*^{⊥}*· n**r**ds* (4.40)

=

Γ_{r}

(*∇*0*U )*^{⊥}*· n**r**ds =−*

Γ_{r}

*|∇*0*U|ν**H*^{⊥}*(p*^{+})*· n**r**ds = (−r + o(r))|Γ**r**|.*

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

Ω_{r}

div(*∇*0*U )*^{⊥}*dx*_{1}*dx*_{2}=*−*

*r*
0

*|Γ**s**| 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) = cr*^{2}*+ o(r*^{2}) contradicts (4.39). Thus ˜*γ*_{1}^{+}= ˜*γ*_{2}^{+}. 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))**G**u* there corresponds one and only one curve of the
*Legen-drian foliation tangent to a at p.*

*For u∈ C*^{2}(Ω) we deﬁne the vector ﬁeld
*T (x*1*, x*2) = *∂*_{x}_{2}*u−x*1

2 *,−∂**x*_{1}*u−x*1

2

!

= (*∇*0*U )*^{⊥}*(x*1*, x*2*,·).*

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

Note that *T = 0 on S**u* and that the projections of curves in the Legendrian
foliation are tangent to*T in Ω \ S**u*. We will also consider the diﬀerential of*T :*

*d*_{p}*T =*

*∂*_{x}^{2}_{1}_{,x}_{2}*u(p)−*^{1}_{2} *∂*_{x}^{2}_{2}*u(p)*

*−∂**x*^{2}_{1}*u(p)* *−∂**x*^{2}_{1}*,x*_{2}*u(p)−*^{1}_{2}

*.* (4.42)

**Lemma 4.36. Assume p**∈ S*u* *is an isolated (projection of a) characteristic point,*
*and|H*0*(z, u(z))| = o(dist(p, z)*^{−1}*). Then*

Next, we recall from (4.12) that
*H*0= 1

Substituting (4.37) in (4.43) we obtain

*H*0=

which contradicts the hypothesis*|H*0*| = 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*

*H*0=*−ad* *(a + d)∆x∆y + o((∆s)*^{2})
*(a*^{2}*(∆x)*^{2}*+ d*^{2}*(∆y)*^{2})^{3/2}*+ o((∆s)*^{3})*,*

*from which we deduce that 2∂*_{x}^{2}_{1}_{,x}_{2}*u(p) = a + d = 0. Thus a =−d and b = c = 0*
*which gives (i). To conclude the proof we observe that det d*_{p}*T = 1/4 > 0, whence*

Index_{p}*T = 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}*+ β i e*

^{i φ}*.*In view of (4.37) and Lemma 4.36(i),

*∂*_{x}_{1}*u +*1
2*x*_{2}=1

2*∆y + o(∆s),*

*∂*_{x}_{2}*u−*1

2*x*_{1}=*−*1

2*∆x + o(∆s),*
*ν*_{H}* ^{⊥}* =

*−*

*∆x*

*∆s* *+ o(1),∆x*

*∆s* *+ o(1)*

!

(4.45)

*as ∆s→ 0. Consequently, α = ν**H*^{⊥}*· 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*

*=*

^{}*−ν*

*H*

*, we obtain*

^{⊥}*(∆s)*^{}*e*^{i φ}*+ (∆s)φ*^{}**i e***^{i φ}*=

*−ν*

*H*

*=*

^{⊥}*−(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∈ S**u* *is an isolated (projection of a) characteristic point;*

(H2) *the bound|H*0*(z, u(z))| = o(dist(p, z)*^{−1}*) holds for z∈ S**u* *near p;*

(H3) *for some r*_{0}*> 0,*

*r*_{0}
0

sup

*z**∈∂B(p,r)**|H*0*(z, u(z))| dr < ∞.*

*Then for all a∈ S*^{1} *there exists a unique C*^{1} *curve γ**a* *such that*

*(i) γ*_{a}*is characteristic, i.e., the projection of a curve in the Legendrian*
*foliation of* *G**u**,*

*(ii) p lies in the closure of γ**a**,*

(iii) lim_{q}_{∈γ}_{a}_{,q}_{→p}*ν*_{H}*(q) exists and is orthogonal to a.*

*Moreover, as a ranges over all of* S^{1}*, such curves γ**a* *cover U**p**\ {p} for some*
*neighborhood U**p* *of p.*

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

*Proof. Let U**p* *be as in Lemma 4.37. Choose δ > 0 suﬃciently small so that*
*B(p, δ)* *⊂ U**p**. 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 q**j**= 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*

*H*0*(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*{θ**j**} is a Cauchy sequence. First, observe*
that

*θ*_{j}*− θ**k* =

*t*_{j}*t*_{k}

*θ*^{}*(t) dt =*

*t*_{j}*t*_{k}

*H*0*dt.* (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

*|θ**j**− θ**k**| ≤*

*r*_{j}*r*_{k}

sup

*z**∈∂B(p,r)**|H*0*(z, u(z))| |t*^{}*(r)| dr → 0* *as j, k→ ∞,* (4.49)
*thus proving that (θ**j**) is Cauchy. Let us denote by θ*_{(p,q)}*its limit as j→ ∞. Now*
*we can deﬁne a map ψ : ∂B(p, δ)→ S*^{1}as 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 C**^{1} 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 θ**j* *= θ*_{(p,q}_{j}_{)} (resp.

*θ = θ*ˆ _{(p,q)}*) and by γ** _{j}* (resp. ˆ

*γ) the corresponding characteristic curves joining q*

_{j}*to p (resp. q to p). We must have that θ*

_{j}*→ θ*

*(p,q)*

*. Let φ*

*be the angle in the polar*

_{j}*coordinate representation of q*

*j*

*−p. Without loss of generality we may assume that*

*φ*

*j*is strictly decreasing. Since two curves in the Legendrian foliation cannot cross

*in B(p, δ)\ {p}, we also have θ*

*j*

*≥ θ*

*j+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 γ**j**, θ**j**> ˜θ*_{(p, ˜}* _{q)}*and ˜

*q must lie in the arc between q*

*j*

*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}*∈*
S^{1}*\ ψ(∂B(p, δ)). Since the latter set is open, we can ﬁnd a neighborhood I**θ*˜ of
*θ such that e*˜ ^{i I}^{θ}^{˜} *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 I}^{θ}^{˜}*⊂ S*^{1}*\ ψ(∂B(p, δ)) which is a contradiction.*

* Step 3 (ψ is injective): Consider q*1

*, q*2

*∈ ∂B*

*δ*

*distinct such that θ*

_{(p,q}_{1}

_{)}

*= θ*

_{(p,q}_{2}

_{)}.

*Denote by γ*

_{i}*a characteristic curve joining p to q*

*. Then the angle between the*

_{i}*tangent vectors of γ*1

*and γ*2

*at p is zero. Consider regions Ω*

*r*

*surrounded by γ*1,

*γ*2

*and ∂B(p, r). Clearly Ω*

*r*

*is contained in a fan-shaped region with vertex p and*

*aperture θ*

*r*

*→ 0. Set Γ*

*r*

*= ∂Ω*

*r*

*∩ ∂B(p, r), then*

*|Γ**r**| ≤ rθ**r* (4.50)

*as before. Let n*_{r}*denote the outer normal to ∂Ω*_{r}*and observe that n*_{r}*⊥ ν**H** ^{⊥}* along

*γ*

_{1}

*and γ*

_{2}. Consequently

*g(r) :=*

*∂Ω*_{r}

(*∇*0*U )*^{⊥}*· n**r**ds =*

Γ_{r}

(*∇*0*U )*^{⊥}*· n**r**ds*

=*−*

Γ_{r}

*(q(s)− p + o(r)) ·q(s)− p*

*r* *ds = (−r + o(r))|Γ**r**|.*

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

Ω_{r}

div(*∇*0*U )*^{⊥}*dx*_{1}*dx*_{2}=*−*

*r*
0

*|Γ**s**| 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) = cr*^{2}*+ o(r*^{2}*) for some c > 0, in contradiction with (4.50).*

**Step 4 (ψ**^{−1}*is continuous): We argue by contradiction. Assume there is a sequence*
*(q** _{j}*)

*⊂ ∂B(p, δ) converging to ˜q so that*

*θ*_{j}*:= θ*_{(p,q}_{j}_{)}*→ θ := θ**(p,q)* (4.52)
*with q= ˜q ∈ ∂B(p, δ). Without loss of generality we may assume θ**j* *≥ θ**j+1**≥ θ.*

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