METRIC NORMAL AND DISTANCE FUNCTION IN THE HEISENBERG GROUP
NICOLA ARCOZZI, FAUSTO FERRARI
Abstract. We study the smoothness of the distance function from a set in the Heisenberg group, with some applications to the mean curvature of a surface.
Contents 1. Introduction 1 2. Notation and preliminaries 3 3. The metric normal and the distance function for a plane 5 4. The metric normal and the oriented metric normal for a smooth surface 7 5. Regularity of the distance function near a smooth surface 11 6. The Hessian of the distance function for a smooth surface 17 7. Mean Curvature 22 8. Examples 25 8.1. Nonvertical planes 25 8.2. The Carnot-Charath eodory sphere and the Hessian of the distance function 27 References 29
1. Introduction In this article we investigate some properties of the function “distance from a surface” in the Heisenberg group. Let H = H1 be R3 with the Heisenberg group structure and let d be the associated Carnot-Charath eodory distance (see Section 2 for the de nitions). If S H is closed, the distance from a point P to S is
dE(P ) = inf d(P, Q) Q E ∈ Here, we consider the case where S = ∂ is the boundary of a C2, open subset of H, in the Euclidean sense. The choice of considering C2 surfaces in the Euclidean sense is motivated by the fact that such surfaces satisfy a internal/external Heisenberg sphere condition at any non- characteristic point, as we shall see below. Most results in this paper can be extended to open C2 surfaces in H. It is convenient to introduce the signed distance from S,
dS(P ) if P (1) S(P ) = ∈ d (P ) if P / ( S ∈ Observe that, given S, the signed distance S is de ned modulo a sign, which corresponds to the choice of one of the two open sets having boundary S, or, equivalently, to the choice of an orientation for S.
Date: 14/10/2003. MSC 49Q15, 53C17, 53C22 1
2 NICOLA ARCOZZI, FAUSTO FERRARI
Our motivation for studying distance functions is twofold. In a recent article [12] and [13] it was shown, in particular, that the function dE is a viscosity solution of the eikonal equation in H, d = 1 |∇H E| (here, is the horizontal gradient) hence of the in nite Laplacian equation, see [5] and [10] ∇H 2 , u = XiuXju(XiXju). ∞ H i,jX=1 It is interesting to have more infromation on particular solutions of these equations. Another reason for investigating dS is that a useful tool for the study of smooth surfaces S in Rn is the fact that the Euclidean distance from S has maximum growth, at least near S, along the segment normal to S. We introduce a de nition of metric normal to a smooth surface which plays precisely this role in the Heisenberg group. The metric normal to S at P , P a point of S, is the set P S containing all points Q such that dS(Q) = d(P, Q). It will be shown that is an arc of aNgeodesic and an explicit formula for S in terms of the parametrization of SNwill NP be given. The available de nition of horizontal vector normal to a surface in H is related to the metric normal, but it does not exhaust the concept. In fact, there exist di erent surfaces passing through a point P , having the same horizontal normal vector, but di erent metric normals, their common feature being that their velocity at P is a multiple of the the horizontal normal vector. Roughly speaking, the possible horizontal normal vectors are in a 1-to-1 correspondence with the group of rotations in H, and for each choice of the horizontal normal vector, the possible metric normals are in a 1-to-1 correspondence with the family of dilations in H. The knowledge of the metric normal allows one, as in Rn, to prove the smoothness of the distance function.
Theorem 1.1. Let S = ∂ be boundary of a Ck open subset of H. Then, for any non- k 1 characteristic point P S there is a neighborhood of P in H such that S is C in . ∈ U U Moreover, if the surface S is the zero set of a smooth function g : H R, then, on the surface S, → g (2) = ∇H . ∇H S g |∇H | A point C of S is characteristic for S if the Euclidean tangent space to S at C coincides with the horizontal ber at C. See Section 2 for precise de nitions. The choice of + or depends on the choice of between the two open sets in H having S as boundary. Here, Ck is in the Euclidean sense. In Section 6 we compute an explicit expression for the horizontal Hessian (see Section 6 for
the de nition) HessH dS of dS in terms of g. As a corollary, we obtain a new characterization of the mean curvature of a surface in S. Recall that, if S is a smooth surface in H de ned by g(x, y, t) = 0 and P is a non-characteristic point of S, then the mean curvature hS(P ) of S {at P [9] is de ned} as in Euclidean space [8], Xg(P ) Y g(P ) (3) h (P ) = X + Y . S g(P ) g(P ) |∇H | |∇H | Below, we assume that g(x, y, t) = t f(x, y), where f is a C2 function. Theorem 1.2. Let S = ∂ be boundary of a C2 open subset of H. Let Q = XXf (Y f)2 (XY f + Y Xf) Xf Y f + Y Y f (Xf)2.
METRIC NORMAL AND DISTANCE FUNCTION IN THE HEISENBERG GROUP1 3
Then, (Y f)2 Xf Y f Xf Y f f 5 Q 4 f 3 Y f f 5 Q 4 f 3 (4) Hess d = |∇H | |∇H | |∇H | |∇H | . H S Y f Xf h(Xf)2 Xf Y f i Xf f 5 Q + 4 f 3 f 5 Q + 4 f 3 |∇H | |∇H | |∇H | |∇H | In particular, for any non-char acteristich P in S, i (5) d (P ) = h (P ). H S S This note is structured as follows. In Section 2 we give notation and some preliminaries. The Section 3 contains some elementary properties of the metric normal and a description of the metric normal to S when S is a Euclidean plane in H. In Section 4 we prove that the metric normal to S at a non-characteristic point is a non-trivial arc of a geodesic, and we deduce in Section 5 that d is smooth in a neighborhood of S characteristic points of S . In Section 6 S { } we compute the horizontal Hessian of dS on S. In Section 7 we show that several de nitions of mean curvature for a surface dS in H, some of them involving dS, are equivalent. Some examples are given in Section 8. We consider the case when = t = 0 is a plane, and we compute { } HessH d in H, not just on . Then, we compute HessH d , where = P : d(P, O) = 1 is the unit sphere at the origin, with respect to the Carnot-Charath eodory distance.{ This amoun} ts, in fact, to compute the horizontal Hessian of dO, the distance function itself. In this paper we considered the Heisenberg group with the lowest dimension. This object is interesting in itself. For instance, it is related with the study of the isoperimetric inequality in the Euclidean plane [11]. More recently, it was realized that sub-Riemannian structures modelled on the lowest dimensional Heisenberg group can be used to model the human visual system, see [3], [4], [17] and the references quoted therein. In particular, [4] contains some material related to curvature in Heisenberg-like structures.
2. Notation and preliminaries In this section, we collect some basic de nitions and known facts about the structure and the geometry of H. There is a vast literature on sub-Riemannian geometry and Carnot groups. Justs to quote a few titles, we refer the reader to [2], [6], [7], [11], [14], [15], [16], [18]. The Heisenberg group H = H1 is the Euclidean space R3 endowed with the noncommutative product (x, y, t) (x0, y0, t0) = (x + x0, y + y0, t + t0 + 2(x0y xy0)). Sometimes it is convenient to think of the elements of H as (z, t) C R. The Carnot- ∈ Charath eodory distance in H is the sub-Riemannian metric that makes pointwise orthonormal the left invariant vector elds X and Y , X = ∂ + 2y∂ , Y = ∂ 2x∂ . x t y t The vector elds X, Y do not commute, [X, Y ] = 4∂ . The distance between two points P t and Q in H is denoted by d(P, Q). The span of the vector elds X and Y is called horizontal distribution, and it is denoted by . The ber of at a point P of H is P = span XP , YP . The inner product in is denotedH by , . H H { } HP h i An important element of H’s structure is the dilation group at the origin : = 0 , { 6 } 2 (z, t) = ( z, t), z = x + iy By left translation, a dilation group is de ned at each point P of H. The Heisenberg group is also endowed with a rotation group, which is useful in simplifying some calculations. For R, let ∈ i R (z, t) = (e , t) be the rotation by around the t-axis. Composing with left translation, one could de ne rotations around any vertical line (x, y) = (a, b). R is an isometry of H and its di erential
4 NICOLA ARCOZZI, FAUSTO FERRARI
acts on the ber as a rotation by . Under the usual identi cation between the Riemannian HO tangent space of H at O, TOH, and the Lie algebra h of H, the di erential of R can be thought of as a rotation on span X, Y , the rst stratum of h. With respect to the basis X, Y , { } { } cos( ) sin( ) dR V = V sin( ) cos( ) whenever V span X, Y . With some abuse of notation, we denote dR by R . ∈ { } The distance between two points P and Q in H is de ned as follows. Consider an absolutely 3 continuous curve in R , joining P and Q, which is horizontal. That is, ˙ (t) = a(t)X (t) + b(t)Y lies in . Its Carnot-Charath eodory length is l ( ), (t) H (t) H 2 2 1/2 lH( ) = (a(t) + b(t) ) dt Z The Carnot-Charath eodory distance between P and Q, d(P, Q), is the in mum of the Carnot- Charath eodory lengths of such curves. The in mum is actually a minimum, the distance between P and Q is realized by the length of a geodesic. By translation invariance, all geodesics are left translations of geodesics passing through the origin. The unit-speed geodesics at the origin [12], [11] are 1 cos( ) sin( ) x( ) = sin( (W )) + cos( (W )) sin( ) 1 cos( ) (6) ( ) = y( ) = sin( (W )) cos( (W )) O, ,W sin( ) t( ) = 2 2 Here, W is a unitary vector in and (W ) [0, 2 ) is unique with the property ˙ (0) = HO ∈ O, ,W W . R, and the geodesic is length minimizing over any interval of length 2 / . In the case = 0,∈the geodesic is a straight line in the plane t = 0 , | | { } x( ) = cos( (W )) , y( ) = sin( (W )) , and we say that the geodesic is straight. If P = (z, t) and z = 0, then there exists a unique length minimizing geodesic connecting P 6 and O. If P = (0, t), t = 0, (i.e., if P belongs to the center of H) then there is a one parameter family of length minimizing6 geodesics joining P and O, obtained by rotation of a single geodesic around the t-axis. We call curvature of the parameter ( ) = . We will see later how ( ) O, ,W O, ,W | | O, ,W is related to the curvature of a surface in H. The notion of curvature extends to all geodesics by left translations, (LP ) = ( ), where LP is left translation by P H in H. Given points ∈ P = (z, t) and P 0 = (z0, t0), they are joined by a unique length minimizing geodesic, unless z = z0. Let P, , = LP O, , The parameter is geometric in the following sense. 2 / is the length of and sgn( ) | | P, ,W is positive if and only if the t-coordinate increases with . Recall that in H the orientation of
the t-axis is an intrinsic notion, unlike the Euclidean space. If P, ,W and P 0, 0,W 0 have an arc in common, then = 0, while no such easy relation exists for the parameter W . To change the orientation of a geo desic, observe that