1 The classification of Monotone sets in the Heisenberg group

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1 1 The classification of Monotone sets in the Heisenberg group after J. Cheeger and B. Kleiner [1] A summary written by Jeehyeon Seo Abstract We show that a monotone set E in the Heisenberg group H is either empty, H or a half space up to a set of measure zero. 1.1 Introduction Theorem 1. The Heisenberg group H (with its usual Carnot group structure) does not admit a bi-lipschitz embedding into L 1. In the proof of the above theorem, we use the structure of monotone sets in H. Definition 2. A measurable subset E R is monotone if it is equivalent to measurable subset F where F and F c are connected. Every monotone set in R may be represented by an equivalent set of the form the empty set, a ray or R. We can see that a measurable set is monotone if and only if the characteristic function of the set is essentially monotone. Definition 3. A subset E R n is precisely monotone if for every line L, E L and E c L are connected. Thus, a precisely monotone set in R n is a convex set with convex complement. We can easily check that a precisely monotone set in R n is either empty set, R n or C E C, where C is an open half-space. We recall the Heisenberg group is R 3 with horizontal distribution =span {X = y, Y= + x }. Every Lie group projects onto its abelianization. x 2 t y 2 t In the case of H, the map π : H H/[H, H] = R 2 corresponds to the map π((a, b, c)) = (a, b). A line in H is a horizontal path (i.e., one tangent to ) that projects to a straight line in R 2. Let L(H) be the collection of all lines in H. (L(H) has a natural smooth structure.) The collection of all pairs of points that can be joined by a line is denoted by hor(h) H H. The collection of all lines passing through a point p H is a horizontal plane (centered at p). A vertical plane is the set π 1 (L), where L is a line in the plane. A half-space is a connected component of H \ P, where P is a plane. 1

2 Definition 4. A subset E H is precisely monotone if for every line L L(H), E L and E C L are connected. Definition 5. A subset E H is monotone if for almost every line L L(H), E L is a monotone subset of L R. That means E L is monotone in R for almost every L L(H). Thus E L and its complement are connected up to a null set within L for almost every L L(H). Theorem 6 (Theorem 4.3 of [1]). If E H is precisely monotone subset, then either E =, E = H, or C E C for some (open) half-space C H. By using the proof of Theorem 6, we will show the main theorem: Theorem 7 (Theorem 5.1 of [1]). If E H is a monotone set, then modulo a null set, either E =, E = H, or E is a half space. 1.2 Outline for the Proof of Theorem 6 We need to understand the (topological) boundary of E to classify precisely monotone sets in the Heisenberg group. So, we have to find the interior of E. The following proposition is the key tool to find interior of E. Proposition 8 (Proposition 4.6 of [1]). Suppose E is precisely monotone in H, L L(H), p L and Σ E is a surface intersecting L transversely at a point q L \ {p}. Then: (1) If p / Int(E c ), then the open segment (p, q) L is contained in Int(E). (2) If p E c, then the connected component of L \ {q} not containing p lie in Int(E). Sketch of the proof: Definition 9 (submersion). Let M and N be smooth manifolds and f : M N be differential map. The map f is submersion at p M if its differential is a surjective linear map. D p f : T p M T f(p) N 2

3 For x H, v 1, v 2, we define a map Γ : H 2 H by sending Γ(x, v 1, v 2 ) to the end points of γ(x, v 1, v 2 ) where γ(x, v 1, v 2 ) is a broken horizontal line with vertices x, xexp(v 1 ) and xexp(v 1 )exp(v 2 ). For x H, we define Γ x : 2 H by Γ x (v 1, v 2 ) = Γ(x, v 1, v 2 ). Then Γ x is submersion near any pair (v 1, v 2 ) 2 with v 1 + v 2 0. We orient the line L in the direction from p to q. Choose a point y L separated from p by q. Then by transversality, any path close to qy intersects Σ. For any point z (p, q), we want to find a neighborhood of z which is contained in E. Choose v such that z = p exp(v)exp(v). Then we claim the following by precisely monotonicity of E and transversality: There exists ɛ > 0 such that if x E, v 1, v 2 satisfy max(d H (x, p), v 1 v, v 2 v ) < ɛ then Γ x (v 1, v 2 ) E. Since Γ x (v 1, v 2 ) is a submersion, the derivative of Γ x is maps the tangent space at (v 1, v 2 ) in 2 onto the tangent space of Γ x (v 1, v 2 ) in H. In particular, Γ x sends an open neighborhood of (v, v) to an open neighborhood of z if we choose x E B(p, ɛ). Therefore, we can complete part of (1) of proposition. The proof of part (2) is similar. 3

4 From the above proposition, we can prove Theorem 6 by proceeding the following steps: 1. Lemma 4.8 of [1]: If L L(H) contains more than one point of E, then L E. 2. Lemma 4.9 of [1]: E is the union of lines. 3. Lemma 4.11 of [1]: Either E plane or E = H. 4. Lemma 4.12 of [1]: The case E = H does not occur. 5. Lemma 4.13 of [1]: If E, then it is a plane and C E C, where C is a connected component of H \ E. 1.3 Outline for the proof of Theorem 7 The proof of Theorem 7 will follow the proof of Theorem 6 closely. The main difference comes from that the condition of monotonicity holds for almost every line L L(H) and up to null set within L. Thus, we need to modify the proof of Theorem 6 by considering positive measure subset of lines in L(H) instead of single line L. Also, the measure theoretic boundary µ E is used instead of the topological boundary E that we used before. In this section, we denote E as a fixed monotone set in H. Definition 10. The support of a measurable set E H, spt(e), is the set of point x H such that µ(b r (x) E) > 0 for all positive r. The measuretheoretic boundary µ E is spt(e) spt(e c ). The measure-theoretic interior of E, Int µ (E), is H\ spt(e c ). A subset of R is monotone if and only if µ E contains at most one point. Definition 11. A line L L(H) is monotone if E L is monotone subset of L R.(Almost every line is monotone if E is monotone). A pointed line (L, p) is monotone if p L(H), the line is monotone and p is either in Int µ (E L) (E L) or Int µ (E c L) (E c L). A direction v \ {0} is monotone if almost every line tangent to v is monotone. Theorem 12 (Fubini). Suppose (X, M, µ) and (Y, N, ν) are σ -finite measure spaces. If f L 1 (X Y ) then f x L 1 (ν) for µ -almost every x and 4

5 f y L 1 (µ) for ν -almost every y. Further the functions x f Y xdν and y f y dν are in L 1 (µ) and L 1 (ν) respectively and X [ ] [ ] f d(µ ν) = f(x, y) dν(y) dµ(x) = f(x, y) dµ(x) dν(y). X Y X Y As an example of a simple consequence of Fubini s theorem, consider the following situation: Let X = [0, 1] [0, 1] and E is a full measure subset of X. Then for almost every x [0, 1], E ({x} [0, 1]) has full measure by Fubini Theorem. Since E is monotone, almost every line L L(H) is monotone. When we consider the smooth fibration L L(H) P ( ) which send a line L tangent to [v] to [v] in the projectivation of the horizontal space, then fibers are lines which are parallel to the given direction. Thus, we can conclude that almost every direction v is monotone. Now we consider the space of pointed lines L(H) whose fiber is R. Since E is monotone, there is a full measure set of monotone lines in L(H).Thus for almost every point p in the fiber over a monotone line implies a monotone pointed line (L, p) by Fubini Theorem. Lemma 13 (Lemma 5.4 of [1]). (1) Almost every direction v is monotone (2) Almost every pointed line is monotone Proposition 14 (Proposition 5.8 of [1]). Suppose L L(H), p L and {Σ α } α A is an admissible family of surfaces. Assume that For all α A, the surface Σ α intersect L transversely in a single point. There is a positive measurable subset A 1 A such that for almost every α A,Σ α L has full measure in Σ α. For some q L \ p, there is α 0 spt(a 1 ) such that Σ α0 L = {q} Then: (1) If p spt(e), then the open segment (p, q) Int µ (E). (2) If p spt(e c ), then the connected component of L\{q} not containing p lies in Int µ (E) Lemma 15 (Lemma 5.9 of [1]). For almost every triple in the set of triples (x, v 1, v 2 ) H such that {max(d H (x, p), v 1 v, v 2 v ) < ɛ} for some positive ɛ, the following statements hold: (1) (L x,v1, x), (L x,v1, x exp v 1 ), (L x exp v1,v 2, x exp v 1 )and(l x exp v1,v 2, x exp v 1 exp v 2 ) are monotone. 5 Y X

6 (2) There is a point w 1 L x,v1 E, close to q, such that (L x,v1, w 1 ) is monotone. (3) There is a point w 2 L x exp v1,v 2 E, close to q, such that (L x exp v1,v 2, w 2 ) is monotone. Here, we denote L x,v the line passing through x tangent to v and q is the element in the above proposition. By this lemma, we have proposition 14 which has a similar role as Proposition 8. Thus, we can prove Theorem 7 by using essentially same outline as the proof of Theorem 6: 1. If L L(H) contains more than one point of µ E, then L µ E. 2. µ E is the union of lines. 3. Either µ E plane or µ E = H. 4. The case µ E = H does not occur. 5. If µ E, then it is a plane and C E C up to sets of measure zero, where C is a connected component of H \ µ E. References [1] J. Cheeger and B. Kleiner, Metric differentiation, Monotonicity and maps to L 1. [2]. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, 91. AMS, Providence, RI, Jeehyeon Seo, University of Illinois at Urbana-Champaign seo6@illinois.edu 6