and p2+ q2 = 1. Under these simplifications, terms involving Xi(p) and Xi(q) cancel and one is left with terms involving components of ∇rL:
∇e2e2, νLL= pL pL+ pr l
!
X1(rL) + qL qL+ qr l
!
X2(rL)− ˜X3(rL).
We conclude by rewriting the expression Xi(rL) in terms of Xir
l
.
4.3 Horizontal geometry of hypersurfaces in H
In this section we want to examine the behavior of the second fundamental form (4.4) and curvatures (4.5), (4.6) as L → ∞. As expected, the horizontal compo-nents have well-defined limits which are natural candidates for sub-Riemannian analogs of these classical differential geometric quantities. The vertical components are unbounded, corresponding to the blow-up of the curvature of (H, gL).
Before initiating this analysis we take a moment to introduce a fundamental notion in the study of sub-Riemannian submanifold geometry.
Definition 4.4. Let S ⊂ H be a C1 surface defined as in (4.2). The characteristic set of S is the closed set
Σ(S) ={x ∈ S : ∇0u(x) = 0}. (4.8) In other words, Σ(S) is the set of points where the tangent space is purely hori-zontal.
Note that Σ(S) is nowhere dense in S, as follows from the Frobenius inte-grability theorem. In fact, the surface measure of Σ(S) is equal to zero. For more precise statements on the size of Σ(S), see the notes to this chapter.
Note that r, rLand ˜X3all converge to zero as L→ ∞ to zero at rates on the order of L−1/2. On the other hand, qL → q, pL → p, lL→ l, and e2 → 0. Hence the Riemannian normal νL converges to the so-called horizontal normal
νH =
2 i=1
Xiu
|∇0u|Xi∈ L∞(S\ Σ(S)) (4.9) in the complement of the characteristic set. Note that νH is simply the projection of νL onto the horizontal subbundle. The vectors νH and e1 (see (4.3)) form an orthonormal frame of the horizontal subbundle.
Direct computation shows that the Gauss curvatureKL diverges as L→ ∞ (similarly to the behavior of the sectional, Ricci and scalar curvatures discussed in Section 2.4.2). Indeed
lim
L→∞
KL
L =−1 4.
70 Chapter 4. Horizontal Geometry of Submanifolds
This reflects the fact that νL → p X1+ q X2 as L → ∞, i.e., the tangent plane to S tends, as L→ ∞, towards a vertical plane. As (2.33) and (2.34) show, the curvature of such planes computed with respect to gL equals −L/4.
Surprisingly, while the Gauss curvature does not have a limit as L→ ∞, the mean curvature presents a rather different behavior. The following lemma is an immediate consequence of (4.5).
Lemma 4.5. Let S be a C2 regular surface defined as in (4.2). Then lim
L→∞Trace IIL= X1p + X2q (4.10) at noncharacteristic points.
Definition 4.6. Let S⊂ H be a C2regular surface, given as a level set of a function u. The horizontal mean curvature of S at a noncharacteristic point is1
H0= X1p + X2q, where p = p/l, q = q/l, l =
p2+ q2, and (p, q) = (X1u, X2u).
We can write the horizontal mean curvatureH0 in several ways. First,
H0=
2 i=1
Xi
Xiu
|∇0u|
. (4.11)
A direct computation shows that the horizontal mean curvature can also be ex-pressed via the identity
H0|∇0u| = Lu − L∞u
|∇0u|2, (4.12)
where Lu = X12u + X22u is the Heisenberg Laplacian of u and
L∞u =
2 i,j=1
XiuXjuXiXju
is the Heisenberg infinite Laplacian of u : H → R. Finally, if u(x) = x3− f(|z|) and we let r =|z|, then
H0=−
1
4r2f+(fr)3 ((f)2+14r2)32
. (4.13)
In Section 6.4 we will see a derivation of the horizontal mean curvature as a first variation of the perimeter functional among all horizontal perturbations.
1We note that some authors define the mean curvature as 1
2(X1p + X2q).
4.3. Horizontal geometry of hypersurfaces in H 71
Example 4.7. Both the plane {x3 = 0}, and the saddle surfaces {x3 = ±x12x2} inH have vanishing horizontal mean curvature away from the characteristic locus Σ ={o}.
Definition 4.8. A C2 regular surface S ⊂ H is a horizontal minimal surface if it has vanishing horizontal mean curvature along its noncharacteristic locus.
Lemma 4.5 immediately implies:
Theorem 4.9. Let S ⊂ H be a C2 regular surface and let IIL be its second funda-mental form computed with respect to gL. If
lim
L→∞Trace IIL= 0 then S is a horizontal minimal surface.
Example 4.10. The horizontal mean curvature of the Euclidean sphere {(z, x3) :
|z|2+ x23 = R2} diverges near the characteristic locus {(0, 0, ±R)} at a rate pro-portional to|z|−1. In fact,
H0= 2(4 + R2)
|z|(4 + x23)3/2.
A similar phenomenon holds for the Euclidean paraboloid (and sub-Riemannian cone) Pα={(z, x3) : x3= α|z|2}, whose horizontal mean curvature diverges near the characteristic locus {o} at a rate proportional to |z|−1:
H0=− 4α
√1 + 16α2 · 1
|z|.
Example 4.11. The horizontal mean curvature of the Kor´anyi sphere {(z, x3) :
|z|4+ 16x23= R4}, on the other hand, tends to zero near the characteristic locus {(0, 0, ±R)} at a rate proportional to |z|. In fact,
H0=3|z|
R2.
Example 4.12. The horizontal mean curvature of the CC sphere ∂Bcc(o, R) = {x ∈ H : d(x, o) = R} can be computed via (4.13). From (2.23) one easily de-duces the parametric representation|z| = A(c) := (2/c) sin(cR/2), x3 = B(c) :=
(cR− sin(cR))/(2c2) for ∂Bcc(o, R). With x3 = f (|z|), f = B ◦ A−1, a simple computation gives
H0=1
2 · c/2
sin(cR/2)· sin(cR)− cR cos(cR) sin(cR/2)− (cR/2) cos(cR/2). It can be shown thatH0∼ |z|−1 as x = (z, x3) approaches the x3-axis.
72 Chapter 4. Horizontal Geometry of Submanifolds
The following example is crucial in the study of the isoperimetric profile of H (see Chapter 8).
Example 4.13. Choosing
f (r) = fR(r) =±1 4(r
R2− r2+ R2arccos r/R),
one easily computes that the horizontal mean curvature of the boundaries of the bubble sets B(o, R) defined in Section 2.3 is equal to the constant 2/R.
4.3.1 Horizontal geometry of hypersurfaces in H
nIn this subsection, we repeat the analysis of the preceding sections for hypersur-faces in the higher-dimensional Heisenberg groups. Again, we study the limit as L→ ∞ of the horizontal part of the second fundamental form of a hypersurface S in (Hn, gL). In contrast with the previous section, where we computed very ex-plicitly using a specific frame, we use here only basic properties of the Levi-Civita connection to accomplish our analysis.
Let S ={x ∈ Hn: u(x) = 0} be a C2regular hypersurface. The characteristic set Σ(S) is defined as before: it consists of all points x ∈ S where the horizontal space H(x) and the tangent space TxS agree.
Consider left invariant vector fields ˜X1, . . . , ˜X2n+1 and a Riemannian metric gLinR2n+1 as in Section 2.4.5. Let|∇Lu|2=2n+1
i=1 ( ˜Xiu)2and observe that the vector
νL= 1
|∇Lu|
2n+1
i=1
X˜iu ˜Xi
is the unit normal to S in (Hn, gL).
As before, we restrict attention to noncharacteristic points. Let
νH := lim
L→∞νL= 1
|∇0u|
2n i=1
XiuXi.
Then
νL= αLνH+ βLX˜2n+1, (4.14) where αL=νL, νHL and βL=νL, ˜X2n+1L.
Lemma 4.14. limL→∞αL= 1 and βL = O(1/√
L) on S\ Σ(S).
Proof. αL =νL, νHL = |∇|∇0u|
Lu| and βL = νL, ˜X2n+1L = X˜|∇2n+1u
Lu| = O(1/√ L).
Set
e2n= βLνH− αLX˜2n+1
4.3. Horizontal geometry of hypersurfaces in H 73
and observe that e2n is unit and orthogonal to νL, hence tangent to S. Choose horizontal tangent vector fields e1, . . . , e2n−1 so that {e1, . . . , e2n−1, e2n} is an orthonormal frame of T S. The second fundamental form of S in (Hn, gL) has entries
IIijL =∇eiνL, ejL
for i, j = 1, . . . , 2n. In view of (4.14) we can write
IIijL = αLhLij+ βLvLij (4.15) for i, j = 1, . . . , 2n− 1, where
hLij =∇eiνH, ejL, i, j = 1, . . . , 2n− 1, (4.16) is the so-called horizontal second fundamental form, and
vijL =∇eiX˜2n+1, ejL, i, j = 1, . . . , 2n− 1 is its vertical complement.
Remark 4.15. In view of Proposition 4.1 it is clear that the terms hLij are actually independent of L. We therefore omit the superscript L in what follows. Note also that in general the coefficients vijL do not vanish as L→ ∞.
Proposition 4.16. The matrix (vijL) is anti-symmetric.
The proof is an easy consequence of the following elementary lemma.
Lemma 4.17. %
∇UV, ˜X2n+1
&
L
=−1 2
%
[V, U ], ˜X2n+1
&
L
(4.17) for all orthonormal horizontal vectors U, V .
Proof. (4.17) is a direct consequence of basic properties of the Levi-Civita connec-tion. Here we present a proof using coordinate frames. Write U =2n
i=1aiX˜i and V =2n
i=1biX˜i. Observe that [U, ˜X2n+1] =
2n l=1
al[ ˜Xl, ˜X2n+1]− ( ˜X2n+1al) ˜Xl
=−
2n l=1
( ˜X2n+1al) ˜Xl,
while [V, ˜X2n+1] is given by the same expression with bl replacing al. A direct computation yields
[U, ˜X2n+1], VL=−
2n l=1
bl( ˜X2n+1al) and [V, ˜X2n+1], U = −
2n l=1
al( ˜X2n+1bl).
74 Chapter 4. Horizontal Geometry of Submanifolds
Thus
[U, ˜X2n+1], VL+[V, ˜X2n+1], UL=−
2n l=1
alX˜2n+1bl+ blX˜2n+1al
=− ˜X2n+1U, V L = 0.
The result now follows from the orthogonality of U and V and the Kozul identity.
Theorem 4.18. Let S⊂ Hn be a C2 regular hypersurface. Then
lim
L→∞IIijL= hij+ hji
2 (4.18)
for i, j = 1, . . . , 2n− 1, at noncharacteristic points.
Proof. Since IIL is symmetric, (4.15) and Proposition 4.16 yield IIijL= IIijL+ IIjiL
2 = αLhij+ hji
2 + βLvLij+ vLji
2 = αLhij+ hji
2 . (4.19)
(4.18) now follows from (4.19) and Lemma 4.14.
In other words, the second fundamental form IIL converges as L → ∞ to the symmetrized horizontal second fundamental form
(II0)∗ = (h∗ij),
where h∗ij = 12(hij+ hji). It is now natural to introduce some sub-Riemannian analogs for classical notions of curvature.
Definition 4.19. Let S ⊂ Hn be a C2 regular hypersurface and denote by (II0)∗ its symmetrized horizontal second fundamental form, defined at noncharacteristic points. The horizontal principal curvatures of S are ki= hii, i = 1, . . . , 2n− 1, the horizontal mean curvature of S is
H0= Trace(II0)∗=
2n−1 i=1
ki,
and the horizontal Gauss curvature of S isK0= det(II0)∗.
With this notation in place, we can define the analogue of constant mean curvature (and hence minimal) surfaces:
Definition 4.20. A C2 regular hypersurface S⊂ Hn is called a horizontal constant mean curvature surface (CMC) ifH0is constant along the noncharacteristic locus, and is called a horizontal minimal surface if H0 = 0 along the noncharacteristic locus. We will refer to the class of CMC surfaces with horizontal mean curvature ρ with the notation CM C(ρ).
4.3. Horizontal geometry of hypersurfaces in H 75
Lemma 4.5 extends to Hn as follows:
Corollary 4.21. Let S ⊂ Hn be a C2 regular hypersurface. Then HL → H0 at noncharacteristic points.
Remark 4.22. Observe that
HL = divgL(νL) =
2n+1
i=1
X˜i
X˜iu
|∇Lu|
. (4.20)
A direct computation shows that
HL|∇Lu| =
2n i,j=1
δij−X˜iu ˜Xju
|∇Lu|2
X˜iX˜ju.
It is clear that both sides of this equation converge (at all points, characteristic or not), yielding
lim
L→∞HL|∇Lu| =
H0|∇0u| = Lu −|∇L∞0uu|2 on S\ Σ(S),
Lu on Σ(S), (4.21)
where Lu = 2n
i=1Xi2u and L∞u = 2n
i,j=1XiuXjuXiXju denote the sub-La-placian and infinite sub-Lasub-La-placian inHn, respectively. In view of this consideration we can extend the functionH0|πH(ν1)| from S \ Σ(S) to all of S as a continuous function. Here we denote by πH the orthogonal projection of Lie algebra vectors onto the horizontal bundle.
4.3.2 Horizontal second fundamental form and the Legendrian foliation
In this section we want to give a more extrinsic definition of the horizontal second fundamental form and relate it to Legendrian foliations.
Let S = {x ∈ Hn : u(x) = 0} be a C2 hypersurface with characteristic set Σ(S). For every point x∈ S \ Σ(S) the intersection of the horizontal plane H(x) with the tangent space TxS defines a (2n−1)-dimensional horizontal tangent space HxS; we denote by HS the corresponding horizontal tangent subbundle. We recall that νH denotes the horizontal normal to S, and choose a g1-orthonormal frame {e1, . . . , e2n−1} for HS. The vectors {e1, . . . , e2n−1, νH} form a g1-orthonormal frame for HHn|S. For any x∈ S, let Πi(x) be the 2-plane spanned by the vectors ei(x) and νH(x), and define the curve γi,x = S∩ Πi(x) with γi,x(0) = x and γi,x (0) = ei(x). Note that γi,xis not necessarily horizontal away from zero. Let us parametrize the horizontal component of γi,x by arc length in the g1-metric, i.e., we require thatπγi,x , πγi,x 1= 1.
76 Chapter 4. Horizontal Geometry of Submanifolds
Proposition 4.23. Let S ⊂ Hn be a C2 regular hypersurface and denote by (hij) its horizontal second fundamental form as defined in (4.16). Then
hij(x) = at noncharacteristic points. In particular,
H0 =
This follows immediately from the definition of hij.
InH, HS is a line bundle, and the corresponding flow lines are Legendrian curves γ1,x= γ = (γ1, γ2, γ3). We call this family of curves the Legendrian foliation of S. As above we assume that (γ1, γ2) is parameterized by arc length. Since the metric induced by g1 on HS is the pull-back of the usual Euclidean metric in the plane, it is easy to see that n = πνH is a unit normal for the planar curve (γ1, γ2), and πe1= (γ1, γ2)(0) = i n. In terms of a defining function u for S,
γ= (qX1− pX2)◦ γ = (X2uX1− X1uX2)|∇0u|−1◦ γ. (4.24) The second fundamental form II = (II11) takes a very simple form
II11= where we have denoted by k the Euclidean curvature of the planar curve γ.
Alternatively, we can follow a more explicit approach: The curvature vector
k = k i (γ1, γ2) of (γ1, γ2) is given by
In conclusion, we have proved the following:
Proposition 4.24. Let S be a C1,1 surface inH, and let γ = (γ1, γ2, γ3) be a curve in the Legendrian foliation of S\ Σ(S). Then the curvature k of γ at (γ1(t), γ2(t)) equals the horizontal mean curvatureH0 of S at (γ1(t), γ2(t), γ3(t)).