Area-minimizing cones in the Heisenberg group H 1
aa r X i v : . [ m a t h . M G ] A ug AREA-MINIMIZING CONES IN THE HEISENBERG GROUP H SEBASTIANO NICOLUSSI GOLO AND MANUEL RITOR´E
Abstract.
We present a characterization of minimal cones of class C and C in the first Heisenberg group H , with an additional set of examples of minimalcones that are not of class C . Contents
1. Introduction 1Plan of the paper 32. Preliminaries 32.1. The Heisenberg group 32.2. Sub-Riemannian perimeter 42.3. Regularity of C area-minimizing surfaces in H
53. Construction of minimal cones 64. Classification results 104.1. Characterization of C minimal cones 10References 111. Introduction
The interest towards Geometric Measure Theory in the Heisenberg groupgrew drastically in the last decades, see for instance [9, 6, 2, 12] and the referencestherein. Despite many deep results, fundamental questions still remain open, themain difficulty being that sets of finite perimeter may not be rectifiable sets in theRiemannian sense.In the effort to understand minimal surfaces in the first Heisenberg group, we arepresenting a characterization of minimal cones of class C and C . Furthermore,we also provide a set of examples of minimal cones that are not of class C .Complete minimal surfaces of class C have been classified in [11]. We provide aself-contained classification of minimal cones of class C , as it is a simple exercisein our case. Minimal surfaces of class C have been studied in [8, 7]. Tentatives tostudy minimal surfaces with regularity lower than C can be found in in [10, 14]. Date : August 11, 2020.2000
Mathematics Subject Classification.
NICOLUSSI GOLO AND MANUEL RITOR´E
The construction of minimal cones is the following, see Section 3 for details.Given proper disjoint open subarcs
I, J of the unit circle S ⊂ R , let L be thebisectrix of I . Then consider the family of planar curves made of (see Figure 1):1. rays emanating from 0 and intersecting J ;2. the line L together with half-lines starting from L parallel to the two bound-ary lines of 0 I . I L J
Figure 1.
The configuration of lines in R for given arcs I, J .All these curves in R lift uniquely to horizontal curves in H , whose union forma surface C ( I, J ) ⊂ H with non-empty boundary in general. The lifted curves arethe characteristic curves of C ( I, J ).Similarly, we can construct a surface C ( I ) from a (possibly infinite) family I ofdisjoint arcs of S , see Figure 3. These are minimal cones with different degrees ofregularity. Theorem A.
Let I be a family of disjoint arcs of S .1. The surface C ( I ) is a minimal cone.2. The surface C ( I ) is of class C if and only if I is finite and the closure of S I is S . Theorem A is proven in Propositions 3.3 and 3.4. With these examples at hand,we provide a classification of minimal cones of class C . The classification is basedon the study of the singular set of minimal surfaces, that is, the set of points wherethe tangent plane is horizontal, see [2]. See Section 4.1 for the proof. Theorem B. If S ⊂ H is a minimal cone of class C , then one of the followingpossibilities holds:1. S is a vertical plane, or2. S is the horizontal plane { t = 0 } , or3. S = C ( I , . . . , I k ) for some disjoint open arcs I , . . . , I k in S with S = S kj =1 ¯ I j .These cases can be distinguished by their singular set: empty in the first case, asingle point in the second case, and a finite family of horizontal half-lines startingfrom the vertex in the third case. REA-MINIMIZING CONES IN THE HEISENBERG GROUP H Not all C minimal cones are of class C . In the third class, the only minimalcones of class C are those with k = 2. Theorem C. If S ⊂ H is a minimal cone of class C , then S is a vertical plane,or the horizontal plane { t = 0 } , or rotations about the t -axis of the graph of thefunction t = − xy . Theorem C follows from Theorem 5.1 of [16], where it is proven that the uniqueentire C area-stationary graphs over the plane H in H are Euclidean planes andvertical rotations of graphs of the form t = xy + ( ay + b ), where a and b are realconstants. In case the surface is a cone then a = b = 0. Plan of the paper.
The preliminary Section 2 introduces the main definitionsand properties of the Heisenberg group that we need. The construction of minimalcones that we sketched above is presented in detail in Section 3. Finally, we proveour main results in Section 4. 2.
Preliminaries
The Heisenberg group.
We identify the first
Heisenberg group H with R with coordinates ( x, y, t ) where we set the group operation( x, y, t ) ∗ ( x ′ , y ′ , t ′ ) = ( x + x ′ , y + y ′ , t + t ′ + ( x ′ y − xy ′ )) . The neutral element is (0 , ,
0) and the inverse of ( x, y, t ) is ( − x, − y, − t ). Wechoose the frame of left-invariant vector fields generated by ∂ x , ∂ y and ∂ t at 0 X = ∂ x + y ∂ t , Y = ∂ y − x ∂ t , T = ∂ t . Notice that [
X, Y ] = − T . These vector fields form a basis for the Lie algebra h of H , which is stratified with first layer H = span { X, Y } , the horizontal plane , andsecond layer [ H , H ] = span { T } .With an abuse of language, we denote by C k (Ω; H ) (and C kc (Ω; H )) the space ofsections of class C k (with compact support in Ω) of the left-invariant vector bundlegenerated by H . These sections are vector fields on R .One can easily see that, if V = v X + v Y with v and v smooth functions,then the standard divergence in R applied to V isdiv( V ) = Xv + Y v . If we consider the left-invariant Riemannian metric g on H making X, Y, T andorthonormal basis, div( V ) is also the divergence with respect to the Riemannianmetric g .The left-invariant vector bundle generated by H is the kernel of the contact form ω = d t − y d x + x d y. Lipschitz curves in R can be lifted to H in the following way. Lemma 2.1.
Let γ : [0 , → R , γ ( s ) = ( x ( s ) , y ( s )) , be a Lipschitz curve with γ (0) = 0 . Define t : [0 , → R by t ( s ) = Z s ( y d x − x d y )[ γ ′ ( u )] d u = Z s ( y ( u ) x ′ ( u ) − x ( u ) y ′ ( u )) d u. Then, the curve s ( x ( s ) , y ( s ) , t ( s )) is the only horizontal Lipschitz curve in H starting from (0 , , and projecting to γ . NICOLUSSI GOLO AND MANUEL RITOR´E
Moreover, if A ( γ, s ) = { vγ ( u ) : u ∈ [0 , s ] , v ∈ [0 , } (with the orientation givenby γ ), then t ( s ) = − Z A ( γ,s ) d x ∧ d y, which is called the balayage area spanned by γ .Proof. Notice that a Lipschitz curve η : s ( x ( s ) , y ( s ) , t ( s )) is horizontal if andonly if ω | η ( s ) [ η ′ ( s )] = 0 for almost all s , that is, t ′ = yx ′ − xy ′ . Integrating, we getthe statement. (cid:3) Sub-Riemannian perimeter.
Given a measurable set E ⊂ H and an openset Ω ⊂ H , the perimeter of E in Ω is defined as P ( E ; Ω) := sup (cid:26)Z E div( V ) d L : V ∈ C c (Ω; H ) , | V | ≤ (cid:27) , where L is the Lebesgue measure in R that is, in our chosen coordinate system,a Haar measure of H .A measurable set E ⊂ H has locally finite perimeter if for every bounded openset Ω ⊂ H we have P ( E ; Ω) < ∞ . It turns out (see [6]) that for a locally finiteperimeter, the distributional gradient of the characteristic function E is a vectorvalued Radon measure, that is, there is a positive Radon measure | ∂E | and a unithorizontal vector field ν E : H → H such that ∇ E = ν E | ∂E | . The measure | ∂E | ,and thus ∇ E , is supported on the so-called reduced boundary ∂ ∗ E ⊂ ∂E . Proposition 2.2 ([6]) . Let E ⊂ H be a set with locally finite perimeter and V ∈ C ∞ ( H ; H ) a smooth horizontal vector field, then (2.1) Z E div( V ) d L = − Z ∂ ∗ E h V, ν E i d | ∂E | . As a corollary, we can easily prove the following formula.
Corollary 2.3.
Let V ∈ C (Ω; H ) , φ ∈ C ( H ) and E ⊂ H a set with locally finiteperimeter. Then (2.2) Z E h∇ φ, V i d L = − Z ∂ ∗ E φ h V, ν E i d | ∂E | − Z E φ div( V ) d L . Proof.
First, by group convolution, the relation (2.1) remains true for v of class C .Second, notice that div( φV ) = h∇ φ, V i + φ div( V ). Therefore, on the one hand, Z E h∇ φ, V i d L = Z E div( φV ) − Z E φ div( V ) d L , on the other hand, Z E div( φV ) = − Z ∂ ∗ E φ h v, ν E i d | ∂E | . by (2.1). Putting these two identities together, we get (2.2). (cid:3) We are interested in perimeter minimizers. A measurable set E ⊂ H is a perime-ter minimizer in an open set Ω ⊂ H if, for every F ⊂ H of locally finite perimeterwith E △ F ⋐ Ω, we have P ( E ; Ω) ≤ P ( F ; Ω) . A set is local perimeter minimizer if it is perimeter minimizer in every boundedopen set. A surface S in H is an area-minimizing surface , or just a minimal surface , REA-MINIMIZING CONES IN THE HEISENBERG GROUP H if it coincides with the reduced boundary of a perimeter minimizer. The followingproposition yields a method via calibrations to prove that a given set is perimeterminimizer. Proposition 2.4 ([13, Theorem 2.1]) . Let E ⊂ H be a measurable set, Ω ⊂ H anopen set and v : Ω → H a Borel map. Assume that(i) E has locally finite perimeter in Ω ;(ii) v = ν E | ∂E | -almost everywhere in Ω ;(iii) there exists an open set ˜Ω ⊂ Ω such that | ∂E | (Ω \ ˜Ω) = 0 and v is continuouson ˜Ω ;(iv) div( v ) = 0 in distributional sense in Ω .Then E is a perimeter minimizer in Ω . The vector field v above is called a calibration for ∂ ∗ E . In applications of Propo-sition 2.4, we will give the calibration v by putting together smooth vector fields indifferent domains. The following proposition gives a way to check that the resultingvector field has zero distributional divergence. Notice that condition (2.3) below isautomatically satisfied if v is continuous. Proposition 2.5.
Let { Ω j } j be a family of open disjoint sets with locally finiteperimeter in H such that { ¯Ω j } j is a locally finite cover of H with L ( H \ S Ω j ) =0 . For each j , let V j ∈ C ( ¯Ω j ; H ) be a horizontal vector field of class C on ¯Ω j ( extensible to a C horizontal vector field on a neighborhood of ¯Ω j ) .The distributional divergence of V := P j V j Ω j is zero if and only if div( V j | Ω j ) =0 for every j and (2.3) X j h V j ( p ) , ν Ω j ( p ) i = 0 for X j | ∂ Ω j | -a.e. p ∈ H ,where we set ν Ω j ( p ) = 0 if p / ∈ ∂ ∗ Ω j .Proof. Let φ ∈ C ∞ c ( H ). Using (2.2), we have Z H h∇ φ, V i d L = X j Z Ω j h∇ φ, V j i d L = − X j Z ∂ ∗ Ω j φ h V j , ν Ω j i d | ∂ Ω j | + Z Ω j φ div( V j ) d L ! . The latter expression is zero for every φ ∈ C ∞ c ( H ) if and only if div( V j | Ω j ) = 0 forevery j and (2.3) holds. (cid:3) Finally, the following stability of perimeter minimizers is well known.
Proposition 2.6.
Let { E k } k ∈ N be a sequence of locally perimeter minimizers and E a set of locally finite perimeter such that E k converge locally in L to E . Then E is also locally perimeter minimizer. Regularity of C area-minimizing surfaces in H . Given a C surface S ,the set S ⊂ S is composed of the points p where T p S is horizontal. It is referredto as the singular set of S . Points in S \ S are called regular points . A horizontalline segment is the image in H of an interval in R through a curve of the form s p exp( sv ), for some p ∈ H and v ∈ H . NICOLUSSI GOLO AND MANUEL RITOR´E
Proposition 2.7 ([2, 7, 8]) . If S is a minimal C surface, then S \ S is ruled byhorizontal line segments whose endpoints lie in S . If S is a t -graph then at mostone endpoint lies in S . Given a function u : A → R defined on a domain A ⊂ R , its t -graph is thesurface { ( x, y, u ( x, y )) : ( x, y ) ∈ A } . We always consider a t -graph as boundaryof the subgraph E := { ( x, y, t ) : t ≤ u ( x, y ) , ( x, y ) ∈ A } . The following lemmacharacterize minimal t -graphs of continuous functions. Lemma 2.8.
The t -graph S of a continuous function u : R → R is a minimalsurface if and only if the unit normal of S , extended to H as a Borel vector fieldindependent of t , has zero distributional divergence. Construction of minimal cones
Consider a finite family I , . . . , I k of disjoint open arcs in the unit circle S , andlet J , . . . , J r the open connected arcs in S \ S ki =1 I i . This set could be eventuallyempty if S = S ki =1 I i . Let 2 α i be the length (opening angle) of I i and L i be thebisectrix of the arc I i . I α L I α L I α L J J Figure 2.
An initial configuration with three open arcs I , I , I ¡¡ The conical sector 0 J i in R (the cone of vertex 0 over J i ) is filled with half-lines leaving the origin. The conical sectors 0 I i are filled with pairs of half-linesmaking angle α i with the half-line L i . This way, every point of R can be joinedto some L i or 0 by a unique shortest path that follows these lines. We lift thesepaths to H as in Lemma 2.1. So, we first lift as horizontal curves the half-lines L , . . . , L k and those in the sectors 0 J i , i = 1 , . . . , r , which remain in the plane { t = 0 } . Then, for i = 1 , . . . , k , we lift the half-lines making angle α i with L i tohorizontal half-lines starting from the corresponding lifted line L i .We obtain a surface, which we call C ( I , . . . , I k ), that is the t -graph of a function u : R → R . The following lemma gives an explicit formula in a specific case. Noticethat, up to a rotation of R , the restriction of u to 0 I i is equal to the function u α i on 0 I described below. REA-MINIMIZING CONES IN THE HEISENBERG GROUP H Lemma 3.1 ([15]) . Let α ∈ (0 , π ) and define the open arc I = { (cos( θ ) , sin( θ )) : | θ | < α } ⊂ S . Then C ( I ) is the t -graph of the function (3.1) u α ( x, y ) = ( y ( | y | cot α − x ) if ( x, y ) ∈ I, otherwise . The function u α is continuous, but not C , and has derivatives ∂ x u α ( x, y ) = ( − y if ( x, y ) ∈ I, if ( x, y ) ∈ R \ I ; ∂ y u α ( x, y ) = ( | y | cot α − x if ( x, y ) ∈ I, if ( x, y ) ∈ R \ I. Proof.
The value of the function u α : R → R at a point ( x, y ) is the balayage areaof the curve from (0 ,
0) to ( x, y ) that follows the half-lines singled out in the aboveconstruction. So, if ( x, y ) / ∈ I , then u α ( x, y ) = 0. If ( x, y ) ∈ I and y ≥ x ≥ s ≥ ( x = x + s cos( α ) y = s sin( α ) that is ( x = x − y cot( α ) s = y sin( α ) . So, define u α ( x, y ) as the Balayage area of the curve from (0 ,
0) to ( x, y ) that followsthe x -axis until ( x ,
0) and then follows the line parallel to (cos α, sin α ), that is, u α ( x, y ) = − x s sin( α )2 = y ( y cot α − x ) . Similarly, if ( x, y ) ∈ I and y ≤
0, one finds that u α ( x, y ) = y ( − y cot α − x ) andso (3.1) is proven. (cid:3) Proposition 3.2.
Let I , . . . , I k be a finite set of disjoint open arcs in S and C ( I , . . . , I k ) the associated surface. Then1. C ( I , . . . , I k ) is a conical continuous t -graph with vertex at ;2. C ( I , . . . , I k ) is a C , surface outside the lines ∂J i , with singular set S ki =1 L i . It is not C at points of the singular set unless α = π/ .3. C ( I , . . . , I k ) is area-minimizing.4. The horizontal unit normal of C ( I , . . . , C k ) is continuous.Proof. The surface C ( I , . . . , I k ) is the graph of version of the function (3.1) in eachsector 0 J k , up to a pre-composition with a rotation of the plane. Therefore, thefirst two statements are clear.We prove that C ( I , . . . , I k ) is area-minimizing by presenting a calibration andthus applying Proposition 2.4. Figure 3 helps the understanding. Let v be thehorizontal vector field that is invariant along t and that is equal to the upward unitnormal to C ( I , . . . , I k ) outside the half-lines S ki =1 L i and S ki =1 ∂I j . We claimthat the distributional divergence of v is zero.In fact, the unit normal of C ( I , . . . , I k ) is the upward unit horizontal vectorthat is orthogonal to the horizontal characteristic lines we lifted. Above the sectors0 J j is simply yX − xY √ x + y , which is actually the calibration of the plane { t = 0 } ; inparticular, it is smooth and with zero divergence. Above the other sectors, v has NICOLUSSI GOLO AND MANUEL RITOR´E constant coefficients in the basis (
X, Y ) above the regions between the half-lines L j and the boundaries ∂I j , where it has thus zero divergence.Finally, one easily sees that v satisfies (2.3) above the half-lines ∂I j and thelines L j .We conclude that div( v ) = 0 by Proposition 2.5. (cid:3) Proposition 3.3.
Given a finite set of disjoint open arcs I , . . . , I k in S , theassociated surface C ( I , . . . , I k ) is of class C if and only if S = S kj =1 ¯ I j .Proof. Let u : R → R be the function whose t -graph is C ( I , . . . , I k ). In eachsector 0 I j , the function u is a version of u α j as in (3.1), up to a rotation of theplane.Since ∂ x u α is not continuous along the half-lines 0 ∂I , then we conclude that,if C ( I , . . . , I k ) is of class C , then S = S kj =1 ¯ I j .Next, notice that the derivative of u α along the vector (cos α, sin α ) (or the vector(cos( α ) , − sin( α ))) is continuous in the half-plane { y > } (in the half-plane { y < } ,respectively), and zero along the half-line 0 α, sin α ) (or 0 α, − sin α ),respectively).So, if 0 I and 0 I share a half-line 0 v , where | ˆ v | = 1, then the derivative of u along ˆ v is continuous across 0 v .What remains to be checked is the continuity across 0 v of the derivative of u along the orthogonal direction to ˆ v . Going back to u α , a computation shows that( − sin( α ) ∂ x u α + cos( α ) ∂ y u α ) | ( s cos α,s sin α ) = s and (sin( α ) ∂ x u α + cos( α ) ∂ y u α ) | ( s cos α, − s sin α ) = s, where the derivatives are the continuous limit from inside 0 I . Therefore, thederivative of u along the orthogonal direction to ˆ v are continuous across 0 v . (cid:3) J L I J L I L I Figure 3.
The complete configuration
REA-MINIMIZING CONES IN THE HEISENBERG GROUP H Figure 4.
The calibration of C ( I , . . . , I k )In the special case of two disjoint open intervals I , I such that S = I ∪ I ,the singular line is a horizontal straight line L which complement is foliated by twofamilies of parallel lines making a constant angle with L . This is merely C , exceptin the case α = π/ t ≤ − xy with C ∞ boundary.Via approximation, we can consider also the above cones constructed using in-finitely many arcs. More precisely, let I be a family of disjoint open arcs of S ,possibly countable. For each I ∈ I , let u I be the function whose t -graph is C ( I ).Define(3.2) u I = X I ∈I u I , where the sum is well defined, because for every v ∈ R there is at most one I ∈ I with u I ( v ) = 0. Proposition 3.4.
Given a family I of disjoint open arcs of S , the function u I iscontinuous and its t -graph C ( I ) is a minimal cone. Moreover, if I is infinite, then C ( I ) is not a C surface.Proof. From (3.1), one easily sees that | u α ( v ) | ≤ | v | tan( α ). We deduce that thesum in (3.2) converges uniformly on compact sets. So, u I is continuous and its t -graph is a cone. By Proposition 2.6, C ( I ) is a minimal surface.Finally, if I is infinite, then there are ˆ v ∈ S and a sequence { I k } k ⊂ I so that dist (ˆ v, I k ) → I k also goes to zero. Now, if we consider the function u α in (3.1), we see that its y -derivative is ∂ y u α ( x, y ) = ( | y | cot α − x if ( x, y ) ∈ I, x, y ) ∈ R \ I. In particular, if ( x, y ) ∈ I is close enough to (1 , tan( α )), then ∂ y u α ( x, y ) isarbitrary close to 1, while ∂ y u α (1 ,
0) = −
1. We conclude that for every k there arepoints in 0 I k where some derivative of u I oscillates between 1 and −
1, so ∇ u I isnot continuous at ˆ v . Since ∇ u I remains bounded, C ( I ) is not a C surface. (cid:3) Classification results
Characterization of C minimal cones. This section is devoted to theproof of our main classification result in the C case, Theorem B. Lemma 4.1.
A conical C surface S ⊂ H without singular points is a verticalplane.Proof. For any p = ( p , p , p ) in S out of the vertical axis V we consider thecurve γ ( s ) = ( sp , sp , s p ), whose tangent vector at s = 0 is the horizontal vector γ ′ (0) = p X + p Y = 0. Since 0 is not a singular point, S \ V must be containedin the vertical plane p x − p y = 0 and so is a vertical plane. (cid:3) Lemma 4.2.
Let S ⊂ H be a conical C surface, and let p ∈ S \ { } . Then and p belong to a horizontal half-line contained in S .Proof. We let p = ( p , p , p ) and consider the curve γ ( s ) := ( sp , sp , s p ), whoseimage is contained in S . We trivially have γ ′ ( s ) = p X γ ( s ) + p Y γ ( s ) + 2 tp T γ ( s ) .Since γ (1) = p and p is a singular point, the vector γ ′ (1) is horizontal and so p = 0.This implies that γ ( s ) is a parameterization of a horizontal half-line starting from0. Since dilations preserve the horizontal distribution, γ ( s ) ∈ S for all s ≥ (cid:3) In the following we denote by H the plane t = 0. Lemma 4.3.
Let S ⊂ H be a conical C minimal t -graph. If p ∈ S \ H , then p isa regular point and there are a singular point q ∈ S ∩ H and a horizontal half-line L starting from q and containing p .Proof. Let p = ( x, y, t ) ∈ S with t = 0. We know that the point p is regular byLemma 4.2. By Proposition 2.7, there are ˆ v ∈ H , with | ˆ v | = 1, and s < s = −∞ ) such that γ (( s , + ∞ )) ⊂ S , where γ ( s ) = p exp( s ˆ v ), and s is minimalwith this property. We have two cases.First, if s = −∞ , then there is s ∈ R such that γ ( s ) ∈ H ∩ S . Indeed, if thiswere not the case, the horizontal line γ ( R ) would meet the t -axis in a non-zeropoint contradicting the hypotheses that S is a t -graph and 0 ∈ S . Now, noticethat γ ′ ( s ) = ˆ v is not parallel to dd λ | λ =1 δ λ γ ( s ), but these two vectors are bothhorizontal and tangent to S . Therefore γ ( s ) is a singular point of S and thus,the lemma is proven if we take L = γ ([ s , + ∞ )) if s > s or L = γ (( −∞ , s ]) if s < s .Second, if s > −∞ , then γ ( s ) is a singular point and thus it belongs to H . Thelemma is proven if we take L = γ ([ s , + ∞ )). (cid:3) Lemma 4.4.
Let S ⊂ H be a C minimal surface invariant by dilations centeredat . If S = { } , then S is the horizontal plane { t = 0 } . REA-MINIMIZING CONES IN THE HEISENBERG GROUP H Proof.
Let p = ( x , y , t ) ∈ S and suppose that t = 0. Since p is a regular pointbut no horizontal line passing through p contains 0, then, by Proposition 2.7, thereˆ v ∈ H such that the entire line s p exp( s ˆ v ) is contained in S . Since S is a cone,for every s ∈ R and λ >
0, we have δ λ ( p exp( s ˆ v/λ )) = ( λx , λy , λ t ) exp( s ˆ v ) ∈ S. However, direct computations show that in such a surface 0 is not the only singularpoint, in contradiction with the assumption S = { } . Therefore, t = 0 and so S ⊂ H . Since S is a cone, S = H . (cid:3) Proof of Theorem B.
In case S has no singular points, Lemma 4.1 implies that S is a vertical plane. If S has only 0 as a singular point, then S = H by Lemma 4.4.Finally let us assume that S contains at least two points and that S is invariantby dilations centered at 0. Then 0 is a singular point, and Lemma 4.2 implies that S is a union of horizontal half-lines leaving the origin.Since 0 is a singular point, S can be represented near 0 as the t -graph of a C function and thus, since S is a cone, the whole S is the t -graph of a function u : R → R .If L is one of the singular half-lines leaving the origin and p ∈ L \ { } , then thereis a neighborhood U of p such that U ∩ S = U ∩ L , because T p S = pH = H and S ⊂ H . Therefore, these singular half-lines cannot accumulate and so we have afinite number of them L , . . . , L k , with k ≥ p ∈ S \ H , then p is a regular point and, by Lemma 4.3, there is a half-line L ⊂ S starting from a singular point q ∈ S ∩ H , say q ∈ L j . Then S λ> δ λ L describes S on one side of L j . In other words, there is an arc I j so that L j is onthe boundary of 0 I j , so that u is a version of the function u α | { y ≥ } or u α | { y ≤ } in (3.1) on 0 I j .We conclude that for every j ∈ { , . . . , k } there are two arcs I j and I j , possiblyempty, such that S \ H is the graph of u above the sectors 0 I , j .Notice that u does not have other singular points on 0 I j other than L j . There-fore, if i = j , then L j ∩ I , i = ∅ . It follows that, in fact, I , j are never empty.Indeed, on each side of every L j there are regular points, as we noticed above.However, they cannot belong to a sector of a singular half-line other than L j .Finally, the unit normal to S is constant over each 0 I , j . By Lemma 2.8, itmust have zero distributional divergence, while, by Proposition 2.4, this happensexactly when it reflects across L j , that is, the characteristic lines in I j and in I j meet L j with the same angle, exactly as it happens with the function u α in (3.1).Set I j = I j ∪ I j . It is clear that L j is the bisectrix of I j and that I j ∩ I i = ∅ if i = j . (cid:3) References
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