A characterization of p-minimal surfaces in the Heisenberg group H_1
AA CHARACTERIZATION OF p -MINIMAL SURFACES IN THEHEISENBERG GROUP H HUNG-LIN CHIU AND HSIAO-FAN LIU
Abstract.
In Euclidean 3-space, it is well known that the Sine-Gordon equation was con-sidered in the nineteenth century in the course of investigation of surfaces of constant Gauss-ian curvature K = −
1. Such a surface can be constructed from a solution to the Sine-Gordonequation, and vice versa. With this as motivation, by means of the fundamental theoremof surfaces in the Heisenberg group H , we show in this paper that the existence of a con-stant p -mean curvature surface (without singular points) is equivalent to the existence ofa solution to a nonlinear second order ODE (1.2), which is a kind of Li´enard equations .Therefore, we turn to investigate this equation. It is a surprise that we give a complete setof solutions to (1.2) (or (1.5)) in the p -minimal case, and hence use the types of the solutionto divide p -minimal surfaces into several classes. As a result, we obtain a representationof p -minimal surfaces and classify further all p -minimal surfaces. In Section 9, we providean approach to construct p -minimal surfaces. It turns out that, in some sense, generic p -minimal surfaces can be constructed via this approach. Finally, as a derivation, we recoverthe Bernstein-type theorem which was first shown in [3] (or see [5, 6]). Contents
1. Introduction and main results 12. The Fundamental Theorem for surfaces in H
93. Constant p -mean curvature surfaces 134. Solutions to the Li´enard equation p -minimal surfaces 176. A representaion of p -minimal surfaces 207. Examples of p -minimal surfaces 238. Structures of singular sets of p -minimal surfaces 309. An approach to construct p -minimal surfaces 34References 381. Introduction and main results
In literature, the Heisenberg group and its sub-Laplacian are active in many fields ofanalysis and sub-Riemannian geometry, control theory, semiclassical analysis of quantum
Mathematics Subject Classification.
Key words and phrases.
Key Words: Heisenberg group, Pansu sphere, p-Minimal surface, Li´enard equa-tion, Bernstein theorem. a r X i v : . [ m a t h . DG ] J a n HUNG-LIN CHIU AND HSIAO-FAN LIU mechanics and etc. (cf. [16, 17, 18, 19, 20]). It also has applications on signal analysis andgeometric optics [29, 30, 31]. Research on the sub-Riemannian geometry and its analyticconsequences, in particular geodesics, has been studied widely and extensively in the pastten years (cf. [17, 21, 22, 23, 24, 25, 27, 26, 28]). In this paper, the Heisenberg group isstudied as a pseudo-hermitian manifold. As Euclidean geometry, it is a branch of Kleingeometries, and the corresponding Cartan geometry is pseudo-hermitian geometry.Recall that the Heisenberg group H is the space R associated with the group multipli-cation ( x , y , z ) ◦ ( x , y , z ) = ( x + x , y + y , z + z + y x − x y ) , which is a 3-dimensional Lie group. The space of all left invariant vector fields is spannedby the following three vector fields:˚ e = ∂∂x + y ∂∂z , ˚ e = ∂∂y − x ∂∂z and T = ∂∂z . The standard contact bundle on H is the subbundle ξ of the tangent bundle T H which isspanned by ˚ e and ˚ e . It is also defined to be the kernel of the contact formΘ = dz + xdy − ydx. The CR structure on H is the endomorphism J : ξ → ξ defined by J (˚ e ) = ˚ e and J (˚ e ) = − ˚ e . One can view H as a pseudohermitian manifold with ( J, Θ) as the standard pseudohermitianstructure. There is a natural associated connection ∇ if we regard all these left invariantvector fields ˚ e , ˚ e and T as parallel vector fields. A natural associated metric on H is theadapted metric g Θ , which is defined by g Θ = d Θ( · , J · ) + Θ . It is equivalent to define themetric by regarding ˚ e , ˚ e and T as an orthonormal frame field. We sometimes use < · , · > todenote the adapted metric. In this paper, we use the adapted metric to measure the lengthsand angles of vectors, and so on.A pseudohermitian transformation (or a Heisenberg rigid motion) in H is a diffeomor-phism in H which preserves the standard pseudohermitian structure ( J, Θ). We let
P SH (1)be the group of Heisenberg rigid motions, that is, the group of all pseudohermitian transfor-mations in H . For details of this group, we refer readers to [4], which is the first publishedpaper where the fundamental theorem in the Heisenberg groups has been studied. We saythat two surfaces are congruent if they differ by an action of a Heisenberg rigid motion.The concept of minimal surfaces or constant mean curvature surfaces plays an importantrole in differential geometry to study basic properties of manifolds. There is an analogousconcept in pseudo-hermitian manifolds, which is called p -minimal surfaces. In this paper,we focus on studying such kind of surfaces in the Heisenberg group H . Suppose Σ is a C surface in the Heisenberg group H . There is a one-form I on Σ which is induced fromthe adapted metric g Θ . This induced metric is defined on the whole surface Σ and is calledthe first fundamental form of Σ. The intersection T Σ ∩ ξ is integrated to be a singularfoliation on Σ, called the characteristic foliation. Each leaf is called a characteristic curve.A point p ∈ Σ is called a singular point if at which the tangent plane T p Σ coincides withthe contact plane ξ p ; otherwise, p is called a regular (or non-singular) point. Generically,a point p ∈ Σ is a regular point and the set of all regular points is called the regular part
AMPLE 3 of Σ. On the regular part, we are able to choose a unit vector field e such that e definesthe characteristic foliation. The vector e is determined up to a sign. Let e = J e . Then { e , e } forms an orthonormal frame field of the contact bundle ξ . We usually call the vectorfield e a horizontal vector field. Then the p -mean curvature H of the surface Σ is definedby(1.1) ∇ e e = − He . The p -mean curvature H is only defined on the regular part of Σ. There are two moreequivalent ways to define the p -mean curvature from the point of view of variation and alevel surface (see [3, 32]). We remark that this notion of mean curvature was proposed byJ.-H. Cheng, J.-F. Hwang, A. Malchiodi and P. Yang from the geometric point of view togeneralize the one introduced first by S. Pauls in H for graphs over the xy -plane [33]. Also,in [34], M. Ritor´e and C. Rosales exposed another method to compute the mean curvature ofa hypersurface. If H = 0 on the whole regular part, we call the surface a p -minimal surface.The p -mean curvature is actually the line curvature of a characteristic curve, and hence thecharacteristic curves are straight lines (for the detail, see [3]). There also exists a function α defined on the regular part such that αe + T is tangent to the surface Σ. We call thisfunction the α -function of Σ. It is uniquely determined up to a sign, which depends on thechoice of the characteristic direction e . Define ˆ e = e and ˆ e = αe + T √ α , then { ˆ e , ˆ e } formsan orthonormal frame field of the tangent bundle T Σ. Notice that ˆ e is uniquely determinedand independent of the choice of the characteristic direction e . In [4], it was shown thatthese four invariants, I, e , α, H, form a complete set of invariants for C surfaces in H . We remark that all the resultsprovided in [4] still hold in the C -category. For each regular point p , we can choose asuitable coordinates around p to study the local geometry of such surfaces. Actually, therealways exists a coordinate system ( x, y ) of p such that e = ∂∂x . We call such coordinates ( x, y ) a compatible coordinate system (see Figure 1.1). It isdetermined up to a transformation in (2.19). Notice that the compatible coordinate systemsare dependent on the characteristic direction e .Let Σ ⊂ H be a constant p -mean curvature surface with H = c . Then in terms of acompatible coordinate system ( U ; x, y ), the α -function satisfies the following equation(1.2) α xx + 6 αα x + 4 α + c α = 0 , which is a nonlinear ordinary differential equation and actually is a kind of the so-called Li´enard equations , named after the French physicist Alfred-Marie Li´enard. The Li´enardequations were intensely studied as they can be used to model oscillating circuits. Conversely,in this paper, we show that if there exists a C solution α ( x, y ) to the Li´enard equation (1.2), we are able to construct a constant p -mean curvature surface with H = c and this givensolution α as its α -function. This result is summarized as Theorem 1.1. One motivation of HUNG-LIN CHIU AND HSIAO-FAN LIU
Figure 1.1.
A compatible coordinates with ∂∂x = e this theorem comes from the famous Sine-Gordon Equation (SGE), u tt − u xx = sin( u ) cos( u ) , which is considerably older than the Korteweg de Vries Equation (KdV). It was discoveredin the late eighteenth century to study pseudospherical surfaces, that is, surfaces of Gaussiancurvature K = − R , and it was intensively studied for this reason. It arisesfrom the Gauss-Codazzi equations for pseudospherical surfaces in R and is known as anintegrable equation [11]. In addition, it can also be viewed as a continuum limit [12]. Itssolutions and solitons have been widely discussed by the Inverse Scattering Transform andother approaches.There is a bijective relation between solutions u of the SGE with (cid:61) ( u ) ⊂ (0 , π ) and theclasses of pseudospherical surfaces in R up to rigid motion. If u : R → R is a solutionsuch that sin u cos u is zero at a point u , then the immersed pseudoshperical surface hascusp singularities. For example, the pseudospherical surfaces corresponding to the 1-solitonsolutions of SGE are the so-called Dini surfaces and have a helix of singularities.The study of line congruences give rise to the concept of B¨acklund transformations. Aline congruence L : M → M ∗ is called a B¨acklund transformation with the constant angle θ between the normal to M at p and the normal to M ∗ at p ∗ = L ( p ) and the distance between p and p ∗ is sin θ for all p ∈ M . The classical B¨acklund transformation for the sine-Gordonequation was constructed in the nineteenth century by Swedish differential geometer AlbertB¨acklund by means of a geometric construction [13, 14, 15]. We then are motivated to haveanalogue theorems for Heisenberg groups. Theorem 1.1.
The existence of a constant p -mean curvature surface ( without singularpoints ) in H is equivalent to the existence of a solution to the equation (1.2) . In this article, we sometimes call the
Li´enard equation (1.2) as the
Codazzi-like equa-tion from the geometrical point of view [4]. We would also like to specify that for a graph( x, y, u ( x, y )) to be p -minimal if it satisfies the p -minimal equation (see [3])(1.3) ( u y + x ) u xx − u y + x )( u x − y ) u xy + ( u x − y ) u yy = 0 . This is a degenerate hyperbolic and elliptic partial differential equation.
AMPLE 5
Theorem 1.1 follows from Theorem 1.2, which is another version of the fundamental the-orem for surfaces in H acquired after we make a detailed investigation of the origin versionof the integrability conditions (2.1), and hence is more useful than the previous one in somesense (for the origin version, we refer readers to [4] or Theorem 2.1 of this paper). Theorem 1.2.
Let α ( x, y ) and H ( x, y ) be two arbitrary C functions on a coordinate neigh-borhood ( U ; x, y ) ⊂ R . If they satisfy the following integrability condition (1.4) (cid:18) h ( y ) − (cid:90) Hα (cid:16) e (cid:82) αdx (cid:17) dx (cid:19) H x + e k ( y ) H y = (cid:16) e (cid:82) αdx (cid:17) ( α xx + 6 αα x + 4 α + αH ) , for some functions k ( y ) and h ( y ) in the variable y , then there exists an C -embedding X : U → H ( provided that U is small enough ) such that the surface Σ = X ( U ) has H and α as its p -mean curvature and α -function, respectively, and ˆ e = ∂∂x , ˆ e = a ( x, y ) ∂∂x + b ( x, y ) ∂∂y with a and b defined as (2.15) and (2.18) . In addition, such embeddings are unique, up to aHeisenberg rigid motion. In particular, if H = c , for some constant c , then (1.4) reads (1.2), for each pair offunctions k ( y ) and h ( y ). We hence have the following fundamental theorem for constant p -mean curvature surfaces in H , which implies Theorem 1.1. Theorem 1.3.
Let α ( x, y ) be an arbitrary C -function on a coordinate neighborhood ( U, x, y ) ⊂ R . If α ( x, y ) satisfies the Codazzi-like equation (1.2) , then there exists an C -embedding X : U → H ( provided that U is small enough ) such that the surface Σ = X ( U ) is a con-stant p -mean curvature surface with H = c and the given function α ( x, y ) as its α -function,and ˆ e = ∂∂x . In addition, such a surface depends upon two functions k ( y ) and h ( y ) of y ,which, together with c, α , describe the induced metric with ˆ e = a ( x, y ) ∂∂x + b ( x, y ) ∂∂y . Here a and b are specified as (2.15) and (2.18) with H = c . Theorem 1.2 follows from our detailed study on the integrability condition (see (2.1)) ofthe fundamental theorem (Theorem 2.1) for surfaces in H . Actually, if we let ˆ ω be theLevi-Civita connection associated to the induced metric with respect to the orthonormalframe field { ˆ e , ˆ e } , as specified in Theorem 1.2, then (1.4) means that ˆ ω , α , and H satisfythe integrability condition (2.1). This is equivalent to saying that a, b, α and H satisfy theintegrability condition (2.13) (see Subsection 2.2), which is another version of (2.1). Wetherefore have Theorem 1.2.Given a function α ( x, y ) in a coordinate neighborhood ( U ; x, y ) ⊂ R which satisfies the Codazzi-like equation (1.2), we are able to construct a family of constant p -mean curvaturesurfaces. Therefore, it suggests a good strategy to investigate constant p -mean curvaturesurfaces by means of the Codazzi-like equation (1.2); in particular, p -minimal surfaces. Inthis paper, we will focus on the theory of p -minimal surfaces. Strategically, we first studythe equation (1.2) with c = 0, that is,(1.5) α xx + 6 αα x + 4 α = 0 . For nonlinear ordinary differential equations, it is known that it is rarely possible to find ex-plicit solutions in close form, even in power series. Fortunately, we indeed obtain a completeset of solutions to (1.5) in a simple form (see Section 4). This is the following theorem.
HUNG-LIN CHIU AND HSIAO-FAN LIU
Theorem 1.4.
Besides the following three special solutions to (1.5) , α ( x ) = 0 , x + c , x + c ) , we have the general solution to (1.5) of the form (1.6) α ( x ) = x + c ( x + c ) + c , which depneds on two constants c and c , and c (cid:54) = 0 . In Subsection 5.1, we are able to use the types of the solutions in Theorem 1.4 to divide the p -minimal surfaces into several classes, which are vertical , special type I , special typeII and general type (see Definition 5.1 and 5.2). Each type of these p -minimal surfacesis open and contains no singular points. Generically, each p -minimal surface is an unionof these types of surfaces. And ”type” can be shown to be invariant under an action of aHeisenberg rigid motion. Now for each type, no matter it is special or general, if a function α is given, then Proposition 5.3, 5.4 and 5.5 express the formula for the induced metric a, b (see (5.5), (5.6) and (5.7)), which is a representation of I , on the p -minimal surfaces withthis given α as α -function. From these formulae, we see that such constructed p -minimalsurfaces depends upon two functions k ( y ) and h ( y ) for each given α . Nonetheless, in Section6, we proceed to normalize these invariants to the following normal forms in terms of anorthogonal coordinate system ( x, y ), which is a coordinate system such that a = 0. Suchan coordinate system is determined up to a translation on ( x, y ), thus we call it a normalcoordinate system . Theorem 1.5.
Let Σ ⊂ H be a p -minimal surface. Then, in terms of a normal coordinatesystem ( x, y ) , we can normalize the α -function and the induced metric a, b to be the following normal forms : (1) α = x + ζ ( y ) , and a = 0 , b = α √ α if Σ is of special type I , (2) α = x + ζ ( y ) , and a = 0 , b = | α |√ α if Σ is of special type II , (3) α = x + ζ ( y )( x + ζ ( y )) + ζ ( y ) , and a = 0 , b = | α || x + ζ ( y ) |√ α if Σ is of general type ,for some functions ζ ( y ) and ζ ( y ) , in which ζ ( y ) is unique up to a translation on y , and ζ ( y ) is unique up to a translation on y as well as its image. Therefore, ζ ( y ) constitutes a complete set of invariants for p -minimal surfaces of specialtype I (or of special type II). And both ζ ( y ) and ζ ( y ) constitute a complete set of invari-ants for p -minimal surfaces of general type. We hence give a representaion for p -minimalsurfaces (see Section 6). From Theorem 1.5, together with, Theorem 1.2 and Theorem 1.3, itholds immediately that the following version of fundamental theorem for p -minimal surfacesin H . Theorem 1.6.
Given two arbitrary C functions ζ ( y ) and ζ ( y ) defined on ( c, d ) ⊂ R , and ζ ( y ) (cid:54) = 0 for all y ∈ ( c, d ) (note that ( c, d ) may be the whole line R ), then (1) there exist an open set U ⊂ ( e, f ) × ( c, d ) ⊂ ( R ; x, y ) , for some ( e, f ) ⊂ R , and an C embedding X : U → H such that Σ = X ( U ) is a p -minimal surface of special AMPLE 7 type I ( or of special type II ) with ( x, y ) as a normal coordinate system and ζ ( y ) as its ζ -invariant; (2) there exist an open set U ⊂ ( e, f ) × ( c, d ) ⊂ ( R ; x, y ) , for some ( e, f ) ⊂ R , and an C embedding X : U → H such that Σ = X ( U ) is a p -minimal surface of generaltype with ( x, y ) as a normal coordinate system and ζ ( y ) and ζ ( y ) as its ζ - and ζ -invariants.Moreover, such embeddings in (1) and (2) are unique, up to a Heisenberg rigid motion. Due to Theorem 1.6, for each pair of function ζ ( y ) and ζ ( y ), we define in Subsection 6.3eight maximal p -minimal surfaces in the sense specified in Theorem 1.7. Roughly speaking,it says that any connected p -minimal surface with type is a part of one of these eight classesof p -minimal surfaces. And notice that generically, a p -minimal surface is a union of those p -minimal surfaces with type. Theorem 1.7.
Given two arbitrary functions ζ ( y ) and ζ ( y ) defined on ( c, d ) ⊂ R , and ζ ( y ) (cid:54) = 0 for all y ∈ ( c, d ) ( note that ( c, d ) may be the whole line R ) , then all the eight p -minimal surfaces S − I ( ζ ) , S + I ( ζ ) , S − II ( ζ ) , S + II ( ζ ); and Σ I ( ζ , ζ ) , Σ − II ( ζ , ζ ) , Σ + II ( ζ , ζ ) and Σ III ( ζ , ζ )(1.7) are immersed, in addition, they are maximal in the following sense: • Any connected p -minimal surface of special type I with ζ ( y ) as the ζ -invariant isa part of either S − I ( ζ ) or S + I ( ζ ) . • Any connected p -minimal surface of special type II with ζ ( y ) as the ζ -invariantis a part of either S − II ( ζ ) or S + II ( ζ ) . • Any connected p -minimal surface of type I with ζ ( y ) and ζ ( y ) as the ζ - and ζ -invariants is a part of Σ I ( ζ , ζ ) . • Any connected p -minimal surface of type II with ζ ( y ) and ζ ( y ) as the ζ - and ζ -invariants is a part of either Σ − II ( ζ , ζ ) or Σ + II ( ζ , ζ ) . • Any connected p -minimal surface of type III with ζ ( y ) and ζ ( y ) as the ζ - and ζ -invariants is a part of Σ III ( ζ , ζ ) . As applications of this theory, in Section 8, we give a complete description about thestructures of the singular sets of p -minimal surfaces in the Heisenberg group H . Theorem 1.8.
The singular set of a p -minimal surface is either (1) an isolated point; or (2) a C smooth curve.In addition, an isolated singular point only happens in the surfaces of special type I with ζ = cont., that is, a part of the graph u = 0 contains the origin as the isolated singularpoint. Actually, the result in Theorem 1.8 is just a special one of Theorem 3.3 in [3]. However, wegive a computable proof of this result for p -minimal surfaces. We also have the descriptionabout how a characteristic leaf goes through a singular curve, which is called a ”go through”theorem in [3]. HUNG-LIN CHIU AND HSIAO-FAN LIU
Theorem 1.9.
Let Σ ⊂ H be a C p -minimal surface. Then the characteristic foliation issmooth around the singular curve in the following sense that each leaf can be extended C smoothly to a point on the singular curve. Due to Theorem 1.9, we have the following result.
Theorem 1.10.
Let Σ be a p -minimal surface of type II ( III ) . If it can be C smoothlyextended through the singular curve, then the other side of the singular curve is of type III ( II ) . Theorem 1.10 plays a key point to enable us to recover the Bernstein-type theorem (seeSection 8), which was first shown in the original paper [3] (or see [1, 5, 6]), and says that u ( x, y ) = Ax + By + C, for some constants A, B, C ∈ R , and u ( x, y ) = − ABx + ( A − B ) xy + ABy + g ( − Bx + Ay ) , where A, B are constants such that A + B = 1 and g ∈ C , are the only two classes ofentire smooth solutions to the p -minimal graph equation (1.3). In addition, in Section 7,we present some basic examples which, in particular, help us figure out the Bernstein-typetheorem.Finally, in Section 9, depending on a parametrized curve C ( θ ) = ( x ( θ ) , y ( θ ) , z ( θ )) for θ ∈ R ,we deform the graph u = 0 in some way to construct p -minimal surfaces with parametrization(1.8) Y ( r, θ ) = ( x ( θ ) + r cos θ, y ( θ ) + r sin θ, z ( θ ) + ry ( θ ) cos θ − rx ( θ ) sin θ ) , for r ∈ R . It is easy to see that Y is an immersion if and only if either Θ( C (cid:48) ( θ )) − ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) (cid:54) = 0 or r + ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) (cid:54) = 0 for all θ (see Remark 9.2).In particular, we have Theorem 1.11.
The surface Y defines a p -minimal surface of special type I if the curve C satisfies (1.9) z (cid:48) ( θ ) + x ( θ ) y (cid:48) ( θ ) − y ( θ ) x (cid:48) ( θ ) − ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) = 0 , for all θ , or equivalently (1.10) z ( θ ) = (cid:90) (cid:2) ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) + y ( θ ) x (cid:48) ( θ ) − x ( θ ) y (cid:48) ( θ ) (cid:3) dθ. In addition, the corresponding ζ -invariant reads (1.11) ζ ( θ ) = y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ − (cid:90) (cid:2) x (cid:48) ( θ ) cos θ + y (cid:48) ( θ ) sin θ (cid:3) dθ, where (cid:82) (cid:2) x (cid:48) ( θ ) cos θ + y (cid:48) ( θ ) sin θ (cid:3) dθ is an anti-derivative of the function x (cid:48) ( θ ) cos θ + y (cid:48) ( θ ) sin θ . On the other hand, we also have
Theorem 1.12.
The surface Y defines a p -minimal surface of general type if the curve C satisfies (1.12) z (cid:48) ( θ ) + x ( θ ) y (cid:48) ( θ ) − y ( θ ) x (cid:48) ( θ ) − ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) (cid:54) = 0 , AMPLE 9 for all θ . In addition, the corresponding ζ - and ζ -invariant read (1.13) ζ ( θ ) = y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ − (cid:90) (cid:2) x (cid:48) ( θ ) cos θ + y (cid:48) ( θ ) sin θ (cid:3) dθζ ( θ ) = z (cid:48) ( θ ) + x ( θ ) y (cid:48) ( θ ) − y ( θ ) x (cid:48) ( θ ) − ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) . From this construction, together with Theorem 7.8 which gives parametrizations for p -minimal surfaces of special type II, we conclude that we has generically provided a parametriza-tion for any given p -minimal surface (see the argument in Section 9). Acknowledgments.
The first author’s research was supported in part by NCTS and inpart by MOST 109-2115-M-007-004 -MY3. The second author’s research was supported inpart by MOST 108-2115-M-032-008-MY2.2.
The Fundamental Theorem for surfaces in H In this section, we first review the fundamental theorem for surfaces in the Heisenberggroup H (Theorem 2.1). For the detailed, we refer the reader to [4]. Next we give anotherversion (Theorem 1.2) of this theorem in terms of compatible coordinate systems.2.1. The Fundamental Theorem for surfaces in H . Recall that there are four invari-ants for surfaces induced on a surface Σ from the Heisenberg group H :(1) The first fundamental form (or the induced metric) I , which is the adapted metric g Θ restricted to Σ. This metric is actually defined on the whole surface Σ.(2) The directed characteristic foliation e , which is a unit vector field ∈ T Σ ∩ ξ . Thisvector field is only defined on the regular part of Σ.(3) The α -function α , which is a function defined on the regular part such that αe + T ∈ T Σ, where e = J e .(4) The p -mean curvature H , which is a function on the regular part defined by ∇ e e = − He .These four invariants constitute a complete set of invariants for surfaces in H . That is, if φ : Σ → Σ is a diffeomorphism between these two surfaces which preserves these four in-variants, then φ is the restriction of a Heisenberg rigid motion Φ. We have the integrabilitycondition ˆ ω (ˆ e ) = Hα (1 + α ) / ˆ ω (ˆ e ) = 2 α + α (ˆ e α )1 + α ˆ e H = ˆ e ˆ e α + 6 α (ˆ e α ) + 4 α + αH (1 + α ) / , (2.1)where ˆ e = αe + T √ α , which is nothing to do with the orientation of Σ but a vector field andonly determined by the contact form Θ. Moreover, ˆ e = e , which is the characteristicdirection and is determined up to a sign (If we choose ˆ e such that ˆ e ∧ ˆ e is compatible withthe orientation of Σ then ˆ e is unique). The form ˆ ω is the Levi-Civita connection form with respect to the frame { ˆ e , ˆ e } . We are ready to write down the following fundamentaltheorem (see [4]). Theorem 2.1 ( The Fundamental theorem for surfaces in H ). Let (Σ , g ) be a Rie-mannian -manifold, and let ˆ α, ˆ H be two real-valued functions on Σ . Assume that g , togetherwith ˆ α, ˆ H , satisfies the integrability condition (2.1) , with α, H replaced by ˆ α, ˆ H , respectively.Then for every point p ∈ Σ there exist an open neighborhood U containing p , and an embed-ding X : U → H such that g = X ∗ ( I ) , ˆ α = X ∗ α and ˆ H = X ∗ H . And X ∗ (ˆ e ) defines thefoliation on X ( U ) induced from H . Moreover, X is unique up to a Heisenberg rigid motion. The new version of the integrability condition.
The goal of this subsection is toexpress the integrability condition (2.1) in terms of a compatible coordinate system ( x, y ).We write(2.2) ˆ e = a ( x, y ) ∂∂x + b ( x, y ) ∂∂y , for some functions a and b (cid:54) = 0. We can assume, without loss of generality, that b >
0, thatis, both ∂∂x ∧ ∂∂y and ˆ e ∧ ˆ e define the same orientation on Σ. The two functions a and b area representation of the first fundamental form I . The dual co-frame { ˆ ω , ˆ ω } of { ˆ e , ˆ e } isˆ ω = dx − ab dy, ˆ ω = 1 b dy. (2.3)Then the Levi-Civita connection forms are uniquely determined by the following Riemannianstructure equations d ˆ ω = ˆ ω ∧ ˆ ω ,d ˆ ω = ˆ ω ∧ ˆ ω , (2.4)with the normalized condition(2.5) ˆ ω + ˆ ω = 0 . A computation shows that d ˆ ω = − d (cid:16) ab (cid:17) ∧ dy = − (cid:18) bda − adbb (cid:19) ∧ dy = dyb ∧ bda − adbb = ˆ ω ∧ bda − adbb . (2.6)By comparing with the first equation of the Riemannian structure equations (2.4), we have(2.7) ˆ ω = bda − adbb + a ˆ ω AMPLE 11 for some function a . On one hand, the second equation of (2.4) and the normalized condi-tion (2.5) imply d ˆ ω = ˆ ω ∧ ˆ ω = − ( dx − ab dy ) ∧ (cid:18) bda − adbb + a ˆ ω (cid:19) = (cid:18) − a y + ab b y − a b − ab a x + a b b x (cid:19) dx ∧ dy. (2.8)On the other hand,(2.9) d ˆ ω = (cid:18) d b (cid:19) ∧ dy = − b x b dx ∧ dy. Equations (2.8) and (2.9) yield(2.10) a = b x b − ba y + ab y − aa x + a b b x . Therefore, ˆ ω = bda − adbb + a ˆ ω = ( ba x − ab x ) dxb + ( ba y − ab y ) dyb + (cid:18) b x b − ba y + ab y − aa x + a b b x (cid:19) dyb = (cid:18) ba x − ab x b (cid:19) dx + (cid:18) b x b − aa x b + a b x b (cid:19) dy. (2.11)From the connection form formula (2.11), we haveˆ ω (ˆ e ) = ab x − ba x b = − a x + a b x b , ˆ ω (ˆ e ) = a ( ab x − ba x ) b + b (cid:18) aa x b − b x b − a b x b (cid:19) = − b x b . (2.12)Therefore, in terms of a compatible coordinate system ( U ; x, y ), the integrability condition(2.1) is equivalent to − a x + a b x b = Hα (1 + α ) / , − b x b = 2 α + αα x α ,aH x + bH y = α xx + 6 αα x + 4 α + αH (1 + α ) / . (2.13) The computation of the first fundamental form I . We would like to solve thefirst two equations of (2.13), which are part of the integrability condition. From the secondequation of (2.13), it is easy to see that(2.14) ln | b | = (cid:90) − (cid:18) α + αα x α (cid:19) dx + k ( y )for some function k ( y ) in the variable y , that is | b | = e k ( y ) e (cid:82) − (cid:16) α + ααx α (cid:17) dx , = e k ( y ) e − (cid:82) αdx (1 + α ) , (2.15)where (cid:82) αdx is an anti-derivative of 2 α with respect to x . Throughout this paper, we alwaysassume, without loss of generality, that b >
0. For a , we substitute the second equation of(2.13) into the first one to obtain the first order linear ODE(2.16) a x + a (cid:18) α + αα x α (cid:19) + Hα (1 + α ) / = 0 . To solve a , we choose the integrating factor u = e (cid:82) (cid:16) α + ααx α (cid:17) dx such that the one-form(2.17) u (cid:18)(cid:20) a (cid:18) α + αα x α (cid:19) + Hα (1 + α ) / (cid:21) dx + da (cid:19) is an exact form. Therefore, using the standard method of ODE, one sees that a = e (cid:82) − (cid:16) α + ααx α (cid:17) dx (cid:18) h ( y ) − (cid:90) Hα (1 + α ) / e (cid:82) (cid:16) α + ααx α (cid:17) dx dx (cid:19) , = e − (cid:82) αdx (1 + α ) (cid:18) h ( y ) − (cid:90) Hα (cid:16) e (cid:82) αdx (cid:17) dx (cid:19) , (2.18)for some function h ( y ) in y and (cid:82) Hα (cid:0) e (cid:82) αdx (cid:1) dx is an anti-derivative of Hα (cid:0) e (cid:82) αdx (cid:1) withrespect to x . From (2.18) and (2.15), we conclude that the first fundamental form I (or a and b ) is determined by α and H , up to two functions k ( y ) and h ( y ). We are thus ready toprove a more useful version of fundamental theorem for surfaces (see Theorem 1.2).2.4. The proof of Theorem 1.2.
We define a Riemannian metric on U by regarding { ∂∂x , a ( x, y ) ∂∂x + b ( x, y ) ∂∂y } as an orthonormal frame field, where a and b are specified as(2.15) and (2.18) with h ( y ) and k ( y ) given in (1.4). Then it is easy to see that α and H ,together with a and b , satisfy the integrability condition (2.13), and hence, by the funda-mental theorem for surfaces (see Theorem 2.1), U can be embedded uniquely as a surfacewith H and α as its p -mean curvature and α -function, respectively. In addition, the char-acteristic direction ˆ e = ∂∂x and ˆ e = a ( x, y ) ∂∂x + b ( x, y ) ∂∂y define the induced metric on theembedded surface. We thus complete the proof of Theorem 1.2, which is another version ofthe fundamental theorem for surfaces (see Theorem 2.1 for the original version). AMPLE 13
The Transformation law of invariants.
First of all, we compute the transformationlaw of compatible coordinate systems. Let ( x, y ) and (˜ x, ˜ y ) be two compatible coordinatesystems and φ a coordinate transformation, i.e., (˜ x, ˜ y ) = φ ( x, y ). Then we have φ ∗ ∂∂x = ∂∂ ˜ x ,which means that [ φ ∗ ] (cid:18) (cid:19) = (cid:18) ˜ x x ˜ x y ˜ y x ˜ y y (cid:19) (cid:18) (cid:19) = (cid:18) (cid:19) , where [ φ ∗ ] is the matrix representation of φ ∗ with respect to these two coordinate systems.We then have the coordinates transformation(2.19) ˜ x = x + A ( y ) , ˜ y = B ( y ) , for some functions A ( y ) and B ( y ). Since det [ φ ∗ ] = B (cid:48) ( y ), we immediately have that B (cid:48) ( y ) (cid:54) =0 for all y . Next, we compute the transformation law of representations of the inducedmetrics. Suppose that the representations of the induced metric are, respectively, givenby a, b and ˜ a, ˜ b , that is, ˆ e = a ∂∂x + b ∂∂y = ˜ a ∂∂ ˜ x + ˜ b ∂∂ ˜ y . By (2.19), we have the followingtransformation law of the induced metric as(2.20) ˜ a = a + bA (cid:48) ( y ) , ˜ b = bB (cid:48) ( y ) . Notice that we have omitted the sign of pull-back φ ∗ on ˜ a and ˜ b . Since the p -mean curvatureand α -function are function-type invariants, they transform by pull-back.3. Constant p -mean curvature surfaces In this section, we aim to prove Theorem 1.3 and Theorem 1.1, and then provide a newtool to study the constant p -mean curvature surfaces. More precisely, we will indicate thatone is able to convert the investigation of the constant p -mean curvature surfaces into thestudy of the so-called Codazzi-like equation.3.1.
The proof of Theorem 1.1 and Theorem 1.3.
Let Σ ⊂ H be a constant p -meancurvature surface with H = c . Then, in terms of a compatible coordinate system ( U ; x, y ),the integrability condition (2.13) is reduced to − a x + a b x b = cα (1 + α ) / , − b x b = 2 α + αα x α ,α xx + 6 αα x + 4 α + c α = 0 . (3.1)In other words, there exists the α satisfying the Codazzi-like equation(3.2) α xx + 6 αα x + 4 α + c α = 0 , which is a nonlinear ordinary differential equation. Conversely, given an arbitrary function α ( x, y ) on a coordinate neighborhood ( U, x, y ) ⊂ R which satisfies the Codazzi-like equa-tion (3.2) (or (1.2)). It is easy to see that equation (3.2) is just the equation (1.4) in theconstant p -mean curvature cases. Namely, that α satisfies equation (3.2) means it satisfies(1.4) for arbitrary functions h ( y ) and k ( y ). Therefore, the embedding Σ = X ( U ) in Theorem1.2 is a constant p -mean curvature surface with H = c , the given function α ( x, y ) as its α -function, and ˆ e = ∂∂x . Notice that, in view of (2.15) and (2.18), the constant p -meancurvature surface determined by the given function α usually is not unique. They dependon two functions h ( y ) and k ( y ) in y . We then have Theorem 1.3, and hence Theorem 1.1.This completes the proof of Theorem 1.3 and Theorem 1.1.Throughout the rest of this paper, we are going to apply the tool to the subject of the p -minimal surfaces. 4. Solutions to the
Li´enard equation
Since we bring in a strategy to study p -minimal surfaces by means of the understandingof the Codazzi-like equation (4.1), we focus on studying this equation in this section. Firstof all, we suppose that α is regarded as a function in x and we want to discuss the solutionsto the Codazzi-like equation(4.1) α xx + 6 αα x + 4 α = 0 . This is actually one kind of the so-called
Li´enard equation [2]. Derivation of explicit solutions(see Theorem 1.4) to the equation (4.1) is given below.4.1.
The proof of Theorem 1.4.
Using v = α (cid:48) to see that (4.1) becomes(4.2) v dvdα + 6 αv + 4 α = 0 , which is the second kind of the Abel equation [7, 8]. Apparently, given v (cid:54) = 0, equation (4.2)can be written as(4.3) dvdα = − α (3 v + 2 α ) v . Denote u = v . Then (4.2) or (4.3) becomes(4.4) dudα = 6 αu + 4 α u . We apply Chiellini’s integrability condition (stated in [9, 10]) for the Abel equation to(4.4), which is exactly integrated with k = (cid:54) = 0. It can be checked that ddα (cid:18) α α (cid:19) = ddα (cid:18) α (cid:19) = 43 α = 29 (6 α ) . Let u = α α ω , i.e., ω = α u and ω (cid:54) = 0. The equation (4.4) turns out to be(4.5) dωdα = ωα (2 + 9 ω + 9 ω ) , which is a separable first order ODE, i.e.,(4.6) dωω (2 + 9 ω + 9 ω ) = dαα . Using the method of partial fractions to integrate the left hand side, we have(4.7) (cid:90) (cid:18) ω − ω + 1 + 32(3 ω + 2) (cid:19) dω = (cid:90) dαα , AMPLE 15 which gives(4.8) 12 (cid:18) ln | ω (3 ω + 2) || ω + 1 | (cid:19) = ln | α | + const. The implicit solution to (4.5) is then expressed as(4.9) ω (3 ω + 2)(3 ω + 1) = Cα , provided that ω (cid:54) = 0 , − , − , and C is arbitrary nonzero constant. Hence, with the assump-tion α (cid:54) = 0, we have(4.10) 4 α (3 Cα − u + 4(3 Cα − u + 3 C = 0 , or equivalently,(4.11) 3 C ( α (cid:48) ) + 4(3 Cα − α (cid:48) + 4 α (3 Cα −
1) = 0 . This yields α (cid:48) = 2(1 − Cα ) ± √ − Cα C = (1 − Cα ) ± √ − Cα C , (4.12)for a nonzero constant C . Since we have assumed that v = α (cid:48) is not zero, neither is 1 − Cα .In order to obtain the general solutions, we proceed to solve equation (4.12) by means of thevariable separable method. We rewrite (4.12) as dxC = dα (1 − Cα ) ± √ − Cα = (1 − Cα ) ∓ √ − Cα [(1 − Cα ) ± √ − Cα ][(1 − Cα ) ∓ √ − Cα ] dα = (1 − Cα ) ∓ √ − Cα − Cα (1 − Cα ) dα = (cid:18) − Cα ± Cα √ − Cα (cid:19) dα. (4.13) Case I. If C <
0, we use the trigonometric substitution (cid:112) | C | α = tan θ, − π < θ < π , toget(4.14) (cid:90) dα Cα √ − Cα = − √ − Cα Cα + c , for some c ∈ R . Substituting (4.14) into (4.13), we obtain x + c C = 1 ∓ √ − Cα Cα , that is,(4.15) 2 α ( x + c ) − ∓√ − Cα , for some c ∈ R . Taking the square of both sides and noticing that α (cid:54) = 0, we obtain(4.16) α ( x ) = x + c ( x + c ) + c , for some c , c ∈ R and c <
0. If α satisfies 2 α ( x + c ) − √ − Cα in (4.15), thenwe have α ( x + c ) >
0, and hence (4.16) implies x + c < − (cid:112) | c | or (cid:112) | c | < x + c . On theother hand, if α satisfies 2 α ( x + c ) − −√ − Cα , then we have α ( x + c ) <
0, and wethen obtain that − (cid:112) | c | < x + c < (cid:112) | c | by (4.16). Case II. If C >
0, we use the trigonometric substitution √ Cα = sin θ, − π < θ < π , toget(4.17) (cid:90) dα Cα √ − Cα = − √ − Cα Cα + c , for some c ∈ R . Substituting (4.17) into (4.13), we obtain x + c C = 1 ∓ √ − Cα Cα , that is,(4.18) 2 α ( x + c ) − ∓√ − Cα , for some c ∈ R . Taking the square of both sides and noticing that α (cid:54) = 0, we obtain(4.19) α ( x ) = x + c ( x + c ) + c , for some c , c ∈ R and c > α (cid:48) (cid:54) =0 , α (cid:54) = 0 , ω (cid:54) = 0 , ω (cid:54) = − and ω (cid:54) = − . Now suppose α (cid:48) = 0 on an open interval, then (4.1)immediately implies α = 0 on that interval. And it is easy to see that ω = 0 is equivalent to α = 0. Finally, since ω = α u , we see that ω = − ⇔ α u = − ⇔ α (cid:48) = − α ⇔ α ( x ) = 12( x + c ) . Similarly, we have ω = − ⇔ α ( x ) = 1( x + c ) , for some c ∈ R . We hence complete the proof of Theorem 1.4.4.2. The phase plane.
We remark that when c = 0 (the p -minimal surfaces case), (3.2) isin fact one of the so-called Li´enard equation [2](4.20) α xx = f ( α, α x ) = − (6 αα x + 4 α ) . If we imagine a simple dynamical system consisting of a particle of unit mass moving on the α -axis, and if f ( α, α x ) is the force acting on it, then (4.20) is the equation of motion. Thevalues of α (position) and α x (velocity), which at each instant characterize the state of the AMPLE 17 system, are called its phases, and the plane of the variables α and α x is called the phaseplane . Using v = α x to see that (4.20) can be replaced by the equivalent system(4.21) dαdx = v,dvdx = − (6 αv + 4 α ) . In general a solution of (4.21) is a pair of functions α ( x ) and v ( x ) defining a curve on thephase plane. It follows from the standard theory of ODE that if x is any number and ( α , v )is any point in the phase plane, then there exists a unique solution ( α ( x ) , v ( x )) of (4.21)such that α ( x ) = α and v ( x ) = v . If this solution ( α ( x ) , v ( x )) is not a constant, then itdefines a curve on the phase plane called a path of the system; otherwise, it defines a criticalpoint. Actually, all paths together with critical points form a directed singular foliation onthe phase plane with critical points as singular points of the foliation, and each path lies ina leaf of the foliation. It is easy to see that this foliation is really defined by the vector field V = ( v, − (6 αv + 4 α )) and (0 ,
0) is the only critical point. We express the direction field (orthe directed singular foliation) as Figure 4.1: ( I )( II )( III ) - - α - - v = α ′ Direction Field of a first order system: α ' = v,v' =- α v - α Figure 4.1. direction field V The classification of p -minimal surfaces The classification of p -minimal surfaces. Theorem 1.4 suggests us to divide (lo-cally) the p -minimal surfaces into several classes. In terms of compatible coordinates ( x, y ), the function α ( x, y ) is a solution to the Codazzi-like equation (4.1) for any given y . ByTheorem 1.4, the function α ( x, y ) hence has one of the following forms of special types0 , x + c ( y ) , x + c ( y ) , and general types x + c ( y )( x + c ( y )) + c ( y ) , where, instead of constants, both c ( y ) and c ( y ) are now functions of y . Notice that c ( y ) (cid:54) = 0for all y . We now use the types of the function α ( x, y ) to define the types of p -minimal surfaceas follows. Definition . Locally, we say that a p -minimal surface is(1) vertical if α vanishes (i.e., α ( x, y ) = 0 for all x, y ).(2) of special type I if α = x + c ( y ) .(3) of special type II if α = x + c ( y ) .(4) of general type if α = x + c ( y )( x + c ( y )) + c ( y ) with c ( y ) (cid:54) = 0 for all y .We further divide p -minimal surfaces of general type into three classes as follows: Definition . We say that a p -minimal surface of general type is(1) of type I if c ( y ) > y .(2) of type II if c ( y ) < y , and either x < − c ( y ) − (cid:112) − c ( y ) or x > − c ( y ) + (cid:112) − c ( y ).(3) of type III if c ( y ) < y , and − c ( y ) − (cid:112) − c ( y ) < x < − c ( y ) + (cid:112) − c ( y ).We notice that the type is invariant under an action of a Heisenberg rigid motion andthe regular part of a p -minimal surface Σ ⊂ H is a union of these types of surfaces. Thecorresponding paths of each type of α is marked on the phase plane (Figure 4.1). We expresssome basic facts about p -minimal surfaces with type as follows. • If α vanishes, then it is part of a vertical plane. • The two concave downward parabolas represent α = x + c , x + c respectively. Theone for α = x + c is on top of the one for α = x + c . For surfaces of special type I ,we have that α → (cid:26) ∞ , if x → − c from the right , −∞ , if x → − c from the left;(5.1) and, for surfaces of special type II , we have that α → (cid:26) ∞ , if x → − c from the right , −∞ , if x → − c from the left;(5.2) • The closed curves taken away the origin correspond to the family of solutions α ( x ) = x + c ( x + c ) + c , AMPLE 19 where c , c are constants and c >
0, which are of type I . There exists a zero for α -function at x = − c . For surfaces of type I , we have | α | ≤ √ c , so α is a boundedfunction for each fixed y , that is, along each path on the phase plane, α is bounded.Therefore, there are no singular points for surfaces of type I . • The curves in between the two concave downward parabolas are of type II . The α -function of type II does not have any zeros. For surfaces of type II , it can bechecked that α → (cid:26) ∞ , if x → − c + √− c from the right , −∞ , if x → − c − √− c from the left . (5.3) • The curves beneath the lower concave downward parabolas are of type III . Thereexists a zero for α -function at x = − c . For surfaces of type III , we have α → (cid:26) −∞ , if x → − c + √− c from the left , ∞ , if x → − c − √− c from the right . (5.4) Proposition . Suppose α ( x, y ) = x + c ( y )( x + c ( y )) + c ( y ) , which is of general type . Then theexplicit formula for the induced metric on a p -minimal surface with this α as its α -functionis given by (5.5) a = | α | h ( y ) | x + c ( y ) |√ α , b = | α | e k ( y ) | x + c ( y ) |√ α , for some functions h ( y ) and k ( y ) .Proof. If α = x + c ( y )( x + c ( y )) + c ( y ) , we choose ln | ( x + c ( y )) + c ( y ) | as an anti-derivative of 2 α with respect to x . Simple computations imply e − (cid:82) αdx (1 + α ) = 1 (cid:112) ( x + c ( y )) + (( x + c ( y )) + c ( y )) = | α || x + c ( y ) |√ α . Substituting the above formula into (2.15) and (2.18), equation (5.5) follows. (cid:3)
Similarly, we have
Proposition . Suppose α ( x, y ) = x + c ( y ) , which is of special type I . Then the explicitformula for the induced metric on a p -minimal surface with this α as its α -function is givenby (5.6) a = α h ( y ) √ α , b = α e k ( y ) √ α . Proof.
In order to obtain (5.6), we choose 2 ln | x + c ( y ) | as an anti-derivative of 2 α withrespect to x . (cid:3) Proposition . Suppose α ( x, y ) = x + c ( y ) , which is of special type II . Then the explicitformula for the induced metric on a p -minimal surface with this α as its α -function is givenby (5.7) a = | α | h ( y ) √ α , b = | α | e k ( y ) √ α . Proof.
To have (5.7), we choose ln | x + c ( y ) | as an anti-derivative of 2 α with respect to x . (cid:3) A representaion of p -minimal surfaces Let Σ ⊂ H be a p -minimal surface. We define an orthogonal coordinate system ( x, y )to be a compatible coordinate system such that a = 0, that is, ˆ e = b ∂∂y . Proposition . There always exists an orthogonal coordinate system around any regularpoint of a p -minimal surface Σ .Proof. Suppose that p ∈ Σ is a regular point and ( x, y ) is an arbitrary compatible coordinatesystem around p . Since H = 0, equations (2.15) and (2.18) imply that the ratio − ab = − h ( y ) e g ( y ) is just a function of y . Now we define another compatible coordinates (˜ x, ˜ y ) by(˜ x, ˜ y ) = ( x + A ( y ) , B ( y )) , for some functions A ( y ) and B ( y ) such that A (cid:48) ( y ) = − ab . By the transformation law (2.20)of the representation of the induced metric, we have ˜ a = 0. This means that (˜ x, ˜ y ) areorthogonal coordinates around p . (cid:3) The proof of Theorem 1.5.
It will be suitable to choose an orthogonal coordinatesystem ( U ; x, y ) to study a p -minimal surface. And it is easy to see from (2.19) and (2.20)that any two orthogonal coordinate systems ( x, y ) and (˜ x, ˜ y ) are transformed by(6.1) ˜ x = x + C, ˜ y = B ( y ) , for a constant C and a function B ( y ). That is, the orthogonal coordinate systems aredetermined, up to a constant C on the coordinate x and a scaling on the coordinate y . Thetransformation law of the representation of the induced metric hence reduces to(6.2) ˜ a = a = 0 , ˜ b = bB (cid:48) ( y ) . In terms of orthogonal coordinate systems, the integrability condition hence reads − b x b = 2 α + αα x α ,α xx + 6 αα x + 4 α = 0 . (6.3)Then the α -function determines the metric representation b , and hence a p -minimal surface,up to a positive function e k ( y ) as (2.15) specified (or see (5.5),(5.6) and (5.7) ). Therefore,from the transformation law of the induced metric (6.2), we are able to choose anotherorthogonal coordinate system (˜ x, ˜ y ) = φ ( x, y ) with B satifying e k ( y ) B (cid:48) = 1. That is, we canfurther normalize b such that k (˜ y ) = 0 for each type, no matter it is special or general. Here k (˜ y ) is the function k in the numerator of ˜ b (see (5.5),(5.6) and (5.7)) with α, x, y replaced AMPLE 21 by ˜ α, ˜ x, ˜ y . In fact, for a general type (the special types are similar), it is possible to chooseanother orthogonal coordinate system (˜ x, ˜ y ) = φ ( x, y ) such that φ ∗ ˜ b = | α || x + c ( y ) |√ α . And it is easy to see that such orthogonal coordinate systems are unique up to a translationon the two variables x and y . In other words, there are constants C and C such that(6.4) (˜ x, ˜ y ) = φ ( x, y ) = ( x + C , y + C ) . We call such an orthogonal coordinate system a normal coordinate system . Indeed,since φ ∗ ˜ α = α , Definition 5.1 indicates the following transformation law for c ( y ) and c ( y )functions ˜ c (˜ y ) = c (˜ y − C ) − C , for special type I , ˜ c (˜ y ) = c (˜ y − C ) − C , for special type II, and˜ c (˜ y ) = c (˜ y − C ) − C , ˜ c (˜ y ) = c (˜ y − C ) , for general type , (6.5)where ˜ c (˜ y ) and ˜ c (˜ y ) are with respect to ˜ α . Namely, c ( y ) is unique up to a translationon y , and c ( y ) is unique up to a translation on y and its image as well. We denote thesetwo unique functions c ( y ) and c ( y ) by ζ ( y ) and ζ ( y ), respectively. We then complete theproof of Theorem 1.5.Both two functions ζ ( y ) and ζ ( y ) are invariants under a Heisenberg rigid motion. There-fore we call them ζ - and ζ -invariants, respectively. In terms of ζ and ζ , we thus have theversion of fundamental theorem for p -minimal surfaces in H (Theorem 1.6).6.2. The proof of Theorem 1.6.
Given ζ ( y ), for (1) in Theorem 1.6, we define α, a, b on U by α = 1 x + ζ ( y ) , a = 0 and b = α √ α . Notice that ( e, f ) needs be chosen so that ( e, f ) × ( c, d ) does not contain the zero set of x + ζ ( y ). Then they satisfy the integrability condition (3.1) with c = 0, and hence U togetherwith α, a, b can be embedded into H to be a p -minimal surface with α as its α -function,and the induced metric a, b . Moreover the characteristic direction e = ∂∂x . From the typeof α , this minimal p -surface is of special type I. In view of a = 0 and b = α √ α , we see thatthe coordinates ( x, y ) are a normal coordinate system. Therefore ζ ( y ) is the corresponding ζ -invariant. The uniqueness follows from the fundamental theorem for surfaces in H ortheorem 1.5. This completes the proof of (1) for the special type I. Both proofs of (1) forthe special type II and of (3) are similar with ( e, f ) chosen according to their types. For thespecial type II, note that ( e, f ) needs be chosen so that ( e, f ) × ( c, d ) does not containthe zero set of 2 x + ζ ( y ), and we define α, a, b on U by α = 12 x + ζ ( y ) , a = 0 and b = | α |√ α . For (3), we see that ( e, f ) needs be chosen so that ( e, f ) × ( c, d ) does not contain the zeroset of ( x + ζ ( y )) + ζ ( y ), and we define α, a, b on U by α = x + ζ ( y )( x + ζ ( y )) + ζ ( y ) , a = 0 and b = | α || x + ζ ( y ) |√ α . Thus we complete the proof of Theorem 1.6.We remark that, in terms of normal coordinates ( x, y ), the co-frame formula (2.3) readsˆ ω = dx − ab dy = dx, ˆ ω = 1 b dy, (6.6)and hence the induced metric I (the first fundamental form) reads I = ˆ ω ⊗ ˆ ω + ˆ ω ⊗ ˆ ω = dx ⊗ dx + 1 b dy ⊗ dy, = dx ⊗ dx + (cid:2) ( x + ζ ( y )) + ( x + ζ ( y )) (cid:3) dy ⊗ dy, for special type I ,dx ⊗ dx + (cid:2) x + ζ ( y )) (cid:3) dy ⊗ dy, for special type II ,dx ⊗ dx + (cid:2) ( x + ζ ( y )) + [( x + ζ ( y )) + ζ ( y )] (cid:3) dy ⊗ dy, for general type . (6.7)From (6.7), we see immediately that the induced metric I degenerates on the singular set { ( x, y ) | x + ζ ( y ) = 0 } for surfaces of special type I. Therefore, it won’t be able to extendsmoothly through the singular set. On the other hand, this phenomenon does not happenfor both special type II and general type.6.3. The maximal p -minimal surfaces and the proof of Theorem 1.7. From the proofof Theorem 1.6, it is clear to see that • for the special type I, the open rectangle U in (1) of Theorem 1.6 can be extendedto be either U − I = { ( x, y ) ∈ R | y ∈ ( c, d ) , x + ζ ( y ) < } , or U + I = { ( x, y ) ∈ R | y ∈ ( c, d ) , x + ζ ( y > } , (6.8) which depends on that U is originally contained in U − I or U + I . Notice that the em-bedding X might be just extended to be an immersion. Since both U − I and U + I are connected and simply connected, the immersion X is unique, up to a Heisen-berg rigid motion. We denote these two p -minimal surfaces of special type I by S − I ( ζ ) = X ( U − I ) and S + I ( ζ ) = X ( U + I ). From (6.7), we see that the induced metric I degenerates on the singurlar set { ( x, y ) ∈ R | y ∈ ( c, d ) , x + ζ ( y ) = 0 } . AMPLE 23 • for the special type II, the open rectangle U in (2) of Theorem 1.6 can be extendedto be either U − II = { ( x, y ) ∈ R | y ∈ ( c, d ) , x + ζ ( y ) < } , or U + II = { ( x, y ) ∈ R | y ∈ ( c, d ) , x + ζ ( y > } , (6.9) which depends on that U is originally contained in U − II or U + II . And the embedding X might be extended to be an immersion. Since both U − II and U + II are connected andsimply connected, the immersion X is unique, up to a Heisenberg rigid motion. Wedenote these two p -minimal surfaces of special type II by S − II ( ζ ) = X ( U − II ) and S + II ( ζ ) = X ( U + II ). • when ζ ( y ) > y ∈ ( c, d ), since there exist no zeros of ( x + ζ ( y )) + ζ ( y ) = 0,the open rectangle U in (3) can be extended to be the product space V I = R × ( c, d ) . Since the extended immersion X is unique, up to a Heisenberg rigid motion, wedenote the p -minimal surface of type I by Σ I ( ζ , ζ ) = X ( V I ). • when ζ ( y ) < y ∈ ( c, d ), since the zero set ( x + ζ ( y )) + ζ ( y ) = 0 consists oftwo separated curves defined by x + ζ ( y )+ (cid:112) − ζ ( y ) = 0 and x + ζ ( y ) − (cid:112) − ζ ( y ) = 0,respectively, the open rectangle U in (3) can be extended to be one of the followingthree domains: V − II = { ( x, y ) ∈ R | y ∈ ( c, d ) , x < − ζ ( y ) − (cid:112) − ζ ( y ) } ,V + II = { ( x, y ) ∈ R | y ∈ ( c, d ) , x > − ζ ( y ) + (cid:112) − ζ ( y ) } , and V III = { ( x, y ) ∈ R | y ∈ ( c, d ) , − ζ ( y ) − (cid:112) − ζ ( y ) < x < − ζ ( y ) + (cid:112) − ζ ( y ) } , (6.10) Since the extended immersion X is unique, up to a Heisenberg rigid motion, we denotethese two p -minimal surfaces of type II by Σ − II ( ζ , ζ ) = X ( V − II ) and Σ + II ( ζ , ζ ) = X ( V + II ), and the p -minimal surface of type III by Σ III ( ζ , ζ ) = X ( V III ).We see that ζ -invariant is the only one invariant for p -minimal surfaces of special type,and ζ - and ζ -invariants are the only two invariants for general type. From the above,Theorem 1.7 immediately follows.6.4. Symmetric p -minimal surfaces. A p -minimal surface is called symmetric if ζ -invariantis a constant for the special types; and both ζ - and ζ -invariants are constants for the generaltypes. Since ζ , up to a translation on its image, is an invariant, we presently have Theorem 6.2.
All symmetric p -minimal surfaces of the same special type are locally con-gruent to one another, whereas for the general type, locally there are a family of symmetric p -minimal surfaces, depending on a parameter on R . Examples of p -minimal surfaces Examples of special type I.
The following is a family of p -minimal surfaces. Theyare defined by the graphs of(7.1) u = Ax + By + C, for some real constants A, B and C . It is easy to see that ( − B, A, C ) or ( x, y ) = ( − B, A ) isthe only singular point of the graph of u = Ax + By + C . Lemma . The graph defined by (7.1) is congruent to the graph of u = 0 .Proof. After the action of the left translation by ( B, − A, − C ), we have( B, − A, − C )( x, y, u ) = ( x + B, y − A, u − C − Ax − By )= ( x + B, y − A, . This completes the proof. (cid:3)
Example . The p-minimal surface defined by the graph of u = 0 corresponds to α = r ,where r = (cid:112) x + y . Indeed, consider a surface defined by X : ( x, y ) → ( x, y, . We compute the horizontal normal(7.2) e = ( u x − y ) r ◦ e + ( u y + x ) r ◦ e = − y (cid:112) x + y ◦ e + x (cid:112) x + y ◦ e . Thus(7.3) e = x (cid:112) x + y ◦ e + y (cid:112) x + y ◦ e = x (cid:112) x + y ∂∂x + y (cid:112) x + y ∂∂y . For the α -function, we compute αe + T = α ( − y (cid:112) x + y , x (cid:112) x + y , − ( x + y ) (cid:112) x + y ) + (0 , , . On the other hand, αe + T = EX x + F X y = ( E, F, E and F . Comparing withthe above formula, we have α = 1 (cid:112) x + y , and hence (0 ,
0) is the only singular point. Notice that ( x, y ) is not a compatible coordinatesystem. In terms of the polar coordinates ( r, θ ) with the coordinates transformation x = r cos θ and y = r sin θ , that is, we consider the re-parametrization X : ( r, θ ) → ( r cos θ, r sin θ, . It represents(7.4) X r = (cos θ, sin θ, , X θ = ( − r sin θ, r cos θ, . From (7.3), it is easy to see that e = X r = ∂∂r , and thus ( r, θ ) is a compatible coordinatesystem. For the α -function and the induced metric a and b , we solve the equation αe + T √ α = aX r + bX θ to get e = ( − sin θ, cos θ, − r ) from (7.2), and to obtain α = 1 r , a = 0 , b = α √ α , AMPLE 25 which, from the formula of b , implies that the polar coordinates ( r, θ ) is a normal coor-dinate system . Since α = r , the surface X is of special type I with the ζ ( θ ) = 0 as ζ -invariant, and hence X is a symmetric p -minimal surface. Figure 7.1.
The charac-teristic direction field of X Figure 7.2. e = ∂∂x for x > − g (cid:48) ( y )In view of Theorem 6.2, we immediately have the following theorem. Theorem 7.3. A symmetric p -minimal surface of special type I is locally congruent tothe graph of u = 0 .Proof. This is because that ζ is constant for a symmetric p -minimal surface of special typeI . And the function ζ , up to a constant, is a complete invariant. (cid:3) Theorem 7.4.
In terms of normal coordinates ( x, y ) , the induced metric (the first fun-damental form) on a p -minimal surface of special type I degenerates on the singular set { ( x, y ) | x = − ζ ( y ) } . Therefore if it is not symmetric, then it will never smoothly extendthrough the singular set.Proof. In terms of normal coordinates, a p -minimal surface of special type I is given by afunction ζ ( y ), in which we have α = 1 x + ζ ( y ) , a = 0 , b = α √ α . Therefore, ˆ e = a ∂∂x + b ∂∂y = α √ α ∂∂y . This is equivalent to that the induced metric I is degenerate on the singular set on which α blows up. Actually, we have I = ˆ ω ⊗ ˆ ω + ˆ ω ⊗ ˆ ω = dx ⊗ dx + (cid:18) α α (cid:19) dy ⊗ dy, where (ˆ ω , ˆ ω ) is the dual co-frame of (ˆ e , ˆ e ). If it is not symmetric, and suppose that it canbe smoothly extended beyond the singular set, then the singular set is actually a singularcurve and the induced metric must be non-degenerate. This completes the proof. (cid:3) Examples of special type II.
The other family of p -minimal surfaces are defined bythe graph of(7.5) u = − ABx + ( A − B ) xy + ABy + g ( − Bx + Ay ) , for some real constants A and B such that A + B = 1 and g ∈ C ( R ). Lemma . The graph defined by (7.5) is congruent to the graph of u = xy + g ( y ) .Proof. Since A + B = 1, the matrix (cid:18) A B − B A (cid:19) defines a rotation on R . Let (cid:18) XY (cid:19) = (cid:18) A B − B A (cid:19) (cid:18) xy (cid:19) , we have XY = ( Ax + By )( − Bx + Ay )= − ABx + ( A − B ) xy + ABy , which implies u = XY + g ( Y ) . This completes the proof. (cid:3)
Example . We now study the example of the graph of u = xy + g ( y ). We consider aparametrization of the graph defined by X : ( x, y ) → ( x, y, xy + g ( y )) , then we have(7.6) X x = (1 , , y ) = ˚ e , X y = (0 , , x + g (cid:48) ( y )) . The horizontal normal e is taken to be e = ( u x − y ) D ˚ e + ( u y + x ) D ˚ e − ( u x − y ) D ˚ e − ( u y + x ) D ˚ e = x + g (cid:48) ( y ) D ˚ e = ˚ e , if 2 x + g (cid:48) ( y ) > − x + g (cid:48) ( y ) D ˚ e = ˚ e , if 2 x + g (cid:48) ( y ) < , where D = | x + g (cid:48) ( y ) | . Combining with (7.6), we then have(7.7) e = − J e = ˚ e = X x = ∂∂x . AMPLE 27
We proceed to compute the α -function, a and b in terms of ( x, y ), which is a compatiblecoordinate system. First, that αe + T √ α = aX x + bX y gives1 √ α [ α (0 , , − x ) + (0 , , a, b, ay + b ( x + g (cid:48) ( y ))) . It immediately shows(7.8) a = 0 , b = α √ α , and α = 12 x + g (cid:48) ( y ) . (i) Thus, from the formula of b , the coordinates ( x, y ) is a normal coordinate system on the part where 2 x + g (cid:48) ( y ) >
0, and we have ζ ( y ) = g (cid:48) ( y ). It is easy to see that the graphof u = xy + g ( y ), for g ∈ C ( R ), is just the maximal surface S + II ( g (cid:48) ( y )) when we restrict tothe domain { ( x, y ) | y ∈ R , x + g (cid:48) ( y ) > } .(ii) For the other part with 2 x + g (cid:48) ( y ) <
0, we have b <
0. Therefore, instead of ( x, y ),the new coordinate system (˜ x, ˜ y ) = ( x, − y ) is a normal coordinate system (notice that thecompatible coordinates are chosen so that b > α, a and b read(7.9) a = 0 , b = − α √ α > , and α = 12 x + g (cid:48) ( − ˜ y ) < , and hence ζ (˜ y ) = g (cid:48) ( − ˜ y ). Here (cid:48) means the derivative with respect to y .(iii) For the other part with 2 x + g (cid:48) ( y ) <
0, we can say something more. If, instead of ∂∂x , we choose − ∂∂x as the characteristic direction, that is, e = − ∂∂x , then the coordinates(˜ x, ˜ y ) = ( − x, − y ) lead to the normal coordinate system for the part with 2 x + g (cid:48) ( y ) <
0. Aaa result, we consider the re-parametrization of the surface X : (˜ x, ˜ y ) → ( − ˜ x, − ˜ y, ˜ x ˜ y + g ( − ˜ y ))such that e = ∂∂ ˜ x = X ˜ x . Similarly, from αe + T √ α = aX ˜ x + bX ˜ y , we have(7.10) a = 0 , b = α √ α > , and α = 12˜ x + ∂ ˜ g∂ ˜ y (˜ y ) > , where ˜ g (˜ y ) is defined by ˜ g (˜ y ) = g ( − ˜ y ). Thus (˜ x, ˜ y ) is the normal coordinate system andwe have ζ (˜ y ) = ∂ ˜ g∂ ˜ y (˜ y ).Let R ( g ( y )) and L ( g ( y )) be the part of the surface with 2 x + g (cid:48) ( y ) > x + g (cid:48) ( y ) < R ( g ( y )) = S + II ( g (cid:48) ( y )) and L ( g ( y )) = S − II ( g (cid:48) ( y )). Then, compairing with (7.8) and (7.10), weimmediately have the following proposition, due to Theorem 1.6. Proposition . The surface L ( g ( − y )) (cid:0) or S − II ( − g (cid:48) ( y )) (cid:1) and R ( g ( y )) (cid:0) or S + II ( g (cid:48) ( y )) (cid:1) arecongruent to each other. They in fact differ by an action of the Heisenberg rigid motion ( x, y, t ) → ( − x, − y, t ) . Theorem 7.8.
Any p -minimal surface of special type II is locally a part of the surfacedefined by u = xy + g ( y ) for some g ∈ C ( R ) , up to a Heisenberg rigid motion. In addition,it is symmetric if and only if g ( y ) is linear in the variable y . Therefore, any symmetric p -minimal surface of special type II is locally a part of the surface defined by the graph of u = xy , up to a Heisenberg rigid motion. Proof.
Any p -minimal surface of special type II locally has the following normal representa-tion(7.11) a = 0 , b = | α |√ α , α = 12 x + ζ ( y ) , in terms of normal coordinates ( x, y ). Therefore, comparing with (7.8), the proof is finished ifwe choose g such that g (cid:48) ( y ) = ζ ( y ). Moreover, it is symmetric if and only if ζ ( y ) = g (cid:48) ( y ) =constant, i.e., g is linear in y . (cid:3) Examples of type I, II and III.
We consider the surface Σ ∈ H defined on R by(7.12) X : ( s, t ) → ( x, y, z ) = ( s cos θ ( t ) , s sin θ ( t ) , t ) . Then it can be calculated that X s = (cos θ ( t ) , sin θ ( t ) , , X t = ( − sθ (cid:48) sin θ ( t ) , sθ (cid:48) cos θ ( t ) , . Notice that ˚ e | (0 , ,z ) = ∂∂x , ˚ e | (0 , ,z ) = ∂∂y and θ ( t ) = θ ( z ), i.e., θ is independent of x and y .We rewrite X s as X s = cos θ ( t ) ∂∂x + sin θ ( t ) ∂∂y (7.13) = cos θ ( t )(˚ e ( X ) − y ∂∂z ) + sin θ ( y )(˚ e ( X ) + x ∂∂z )(7.14) = cos θ ( t )(˚ e ( X ) − x sin θ ∂∂z ) + sin θ ( t )(˚ e ( X ) + x cos θ ∂∂z )(7.15) = cos θ ( t )˚ e ( X ) + sin θ ( t )˚ e ( X ) ∈ ξ, (7.16)which is a vector tangent to the contact plane. Then we choose e = X s , and hence e = J e = − sin θ ( t )˚ e ( X ) + cos θ ( t )˚ e ( X ). We compute (cid:53) e e = − ( e θ ( t ))(cos θ ( t )˚ e ( X ) − sin θ ( t )˚ e ( X ))= 0 , (cid:18) ∵ e θ ( t ) = dθ ( t ) ds = 0 (cid:19) . (7.17)This implies that such surface defined by (7.12) has p -mean curvature H = 0. We proceed towork out the α -function α , and a and b . By definition, it is defined by a function satisfyingthat αe + T ∈ T Σ, that is,(7.18) α ( − sin θ ˚ e + cos θ ˚ e ) + T = EX s + F X t , for some functions E, F . Similar to that we rewrite X s as a linear combination of ˚ e , ˚ e and ∂∂z , we can express X t as(7.19) X t = ( − sθ (cid:48) sin θ )˚ e + ( sθ (cid:48) cos θ )˚ e + ( s θ (cid:48) + 1) ∂∂z . Combining (7.18) and (7.19) and notice that X s = e , we obtain that E = 0 , F = s θ (cid:48) ( t )+1 and hence α = sθ (cid:48) ( t ) s θ (cid:48) ( t ) + 1 , a = 0 , b = F √ α . AMPLE 29 If θ (cid:48) ( t ) = 0, then we have α = 0. However, if θ (cid:48) ( t ) (cid:54) = 0, then α , a and b read(7.20) α = ss + θ (cid:48) ( t ) , a = 0 , b = αs √ α θ (cid:48) ( t ) , which means that ( s, t ) is an orthogonal coordinate system, but not normal. In particular,if θ (cid:48) ( t ) > t , then one sees that the p -minimal surface has no singularities. From(7.20), we conclude that this surface is of type I if θ (cid:48) ( t ) > t . On the other hand, if θ (cid:48) ( t ) < t , then it is either of type II on which s > (cid:113) − θ (cid:48) ( t ) ( t ) or s < − (cid:113) − θ (cid:48) ( t ) ( t );or type III on which − (cid:113) − θ (cid:48) ( t ) ( t ) < s < (cid:113) − θ (cid:48) ( t ) ( t ).Finally, we can further take the coordinates (˜ s, ˜ t ) = ( s − C, θ ( t )) to normalize b such that itonly depends on s and α , then we have(7.21) α = ˜ s + C (˜ s + C ) + θ (cid:48) ( θ − (˜ t )) , a = 0 , b = α (˜ s + C ) √ α , for some constant C , and hence(7.22) ζ (˜ t ) = C, ζ (˜ t ) = 1 θ (cid:48) ( θ − (˜ t )) . Examples of type I.
There is another surface of type I . Consider a surface Σ ∈ H defined on R by(7.23) X : ( s, t ) → ( x, y, z ) = (cos s + (sin s ) t, sin s − (cos s ) t, t ) , accordingly we obtain X s = ( − sin s + (cos s ) t, cos s + (sin s ) t,
0) and X t = (sin s, − cos s, . Suppose that ηX s + ζX t ∈ ξ . Then 0 = θ ( ηX s + ζX t ) = ηθ ( X s ) + ζθ ( X t ). It is easy to get θ ( X s ) = 1 + t and θ ( X t ) = 0, and hence we have η = 0. We conclude that X t = (sin s, − cos s,
1) = (sin s )˚ e − (cos s )˚ e ∈ ξ ∩ T Σ , which means that ( t, s ) is a compatible coordinate system. Taking e = X t , we see e = J e = (cos s )˚ e + (sin s )˚ e , and then ∇ e e = 0. Thus the p -mean curvature H = 0. For α, a and b , we solve ˆ e = αe + T √ α = aX t + bX s , to get α = t t , a = b = 1 √ t + 3 t + 1 . Therefore it is of type I . We can further normalize the invariants in terms of the newcoordinates (˜ t, ˜ s ) = ( t − s + C, s ) such that(7.24) ˜ α = ˜ t + ˜ s − C (˜ t + ˜ s − C ) + 1 , ˜ a = 0 , ˜ b = ˜ α (˜ t + ˜ s − C ) √ α > . As a consequence, we have(7.25) ζ (˜ s ) = ˜ s, up to a constant C ; and ζ (˜ s ) = 1 . Figure 7.3.
Helicoid
Figure 7.4.
Conicoid8.
Structures of singular sets of p -minimal surfaces In this section, we assume that Σ ⊂ H is a p -minimal surface. Proposition . Let p be a singular point of a p -minimal surface Σ . Then there must bea characteristic line approaching this point p .Proof. Suppose no characteristic line approaches this point p , we would like to find a con-tradiction. Firstly, by [3], there exists a small neighborhood of p whose intersection with thesingular set is contained in a smooth curve Γ p . And if the neighborhood is small enough,then, on one side of the curve Γ p , we are able to find a compatible coordinate system ( U ; x, y )such that p is contained on the boundary of U . Notice that, by our assumption, p does not lieat the end of any leaf of the foliation defined by e = ∂∂x . Thus the image of the map definedon U by ( x, y ) → ( α, α x ) is bounded on the phase plane (see Figure 4.1). Therefore, wehave that lim ( x,y ) → p α ( x, y ) is finite, which is a contradiction since p is a singular point. (cid:3) The proof of Theorem 1.8.
Due to Proposition 8.1, it suffices to show this theoremfor a p -minimal surface of some type. For general type (notice that there are no singularpoints for type I ), we choose a normal coordinate system ( x, y ) such that α , and a, b read(8.1) α ( x, y ) = x + ζ ( y )( x + ζ ( y )) − c ( y ) , and a = 0 , b = | α || x + ζ ( y ) |√ α , where c ( y ) is a positive function of the variable y such that ζ ( y ) = − c ( y ). Then thesingular set is the graphs of the functions(8.2) x = − ζ ( y ) ± c ( y ) . AMPLE 31
By (6.7), the induced metric I (or the first fundamental form) on the regular part reads(8.3) I = dx ⊗ dx + (cid:2) ( x + ζ ( y )) + [( x + ζ ( y )) + ζ ( y )] (cid:3) dy ⊗ dy. Now we use the metric to compute the length of the singular set { ( − ζ ( y ) ± c ( y ) , y ) } , where y belongs to some open interval. Let γ ± ( y ) = ( − ζ ( y ) ± c ( y ) , y ), which is a parametrizationof the singular set. Then the square of the velocity at y is | γ (cid:48)± ( y ) | = | − ζ (cid:48) ( y ) ± c (cid:48) ( y ) | + (cid:2) ( x + ζ ( y )) + [( x + ζ ( y )) + ζ ( y )] (cid:3) = | − ζ (cid:48) ( y ) ± c (cid:48) ( y ) | + c ( y ) ,> y, (8.4)where we have used the fact that ( x + ζ ( y )) = c ( y ) on the singular set. Formula (8.4)shows that the parametrized curve γ ± ( y ) of the singular set has a positive length.Similarly, for special type II , in terms of a normal coordinate system ( x, y ), we have(8.5) α = 12 x + ζ ( y ) , and a = 0 , b = | α |√ α . In this case, equation (6.7) suggests that the induced metric I on the regular part reads(8.6) dx ⊗ dx + (cid:2) x + ζ ( y )) (cid:3) dy ⊗ dy. Let γ ( y ) = ( − ζ ( y )2 , y ) be a parametrization of the singular set { ( − c ( y )2 , y ) } , where y belongsto some open interval. Then the square of the velocity at y is(8.7) | γ (cid:48) ( y ) | = ( ζ (cid:48) ( y )) (cid:2) x + ζ ( y )) (cid:3) > , where we have used the fact that 2 x = − ζ ( y ) on the singular set. Again, formula (8.10)shows that the parametrized curve γ ( y ) of the singular set has a positive length.Finally, for special type I , in terms of a normal coordinate system ( x, y ), we have(8.8) α = 1 x + ζ ( y ) , and a = 0 , b = α √ α . If γ ( y ) = ( − ζ ( y ) , y ) is a parametrization of the singular set { ( − ζ ( y ) , y ) } for y inside someopen interval, then (6.7) indicates that the induced metric I on the regular part reads(8.9) dx ⊗ dx + (cid:2) ( x + ζ ( y )) + ( x + ζ ( y )) (cid:3) dy ⊗ dy and the square of the velocity at y is(8.10) | γ (cid:48) ( y ) | = ( ζ (cid:48) ( y )) + (cid:2) ( x + ζ ( y )) + ( x + ζ ( y )) (cid:3) = ( ζ (cid:48) ( y )) , where we have used the fact that x = − ζ ( y ) on the singular set. From formula (8.10), wesee that the length of γ ( y ) depends on the value ζ (cid:48) ( y ) is zero or not, which implies that thesingular set is either an isolated point or a smooth curve of positive length. In addition, thesingular set as an isolated point happens if and only if ζ is a constant, that is, the surfaceis part of a plane. We thus completes the proof of this theorem 1.8. The proof of Theorem 1.9.
Around the singular point p , we may assume that thesurface is represented by a graph z = u ( x, y ). Let X be a parametrization of the p -minimalsurface around p defined by X ( x, y ) = ( x, y, u ( x, y )). Then X x = (1 , , u x ) = ∂∂x + u x ∂∂t = ˚ e + ( u x − y ) ∂∂t ; X y = (0 , , u y ) = ∂∂y + u y ∂∂t = ˚ e + ( u y + x ) ∂∂t , (8.11)which yields I ( X x , X x ) = 1 + ( u x − y ) , I ( X y , X y ) = 1 + ( u y + x ) , I ( X x , X y ) = ( u x − y )( u y + x ) , where I is the induced metric (first fundamental form) on the surface. Now we choose ahorizontal normal as follows e = − ( u x − y )˚ e + ( u y + x )˚ e D , where D = (( u x − y ) + ( u y + x ) ) / . Then(8.12) e = ( u y + x )˚ e − ( u x − y )˚ e D is tangent to the characteristic curves.We first claim that either u xx ( p ) (cid:54) = 0 or ( u xy + 1)( p ) (cid:54) = 0. Let f ( x, y ) = u x − y and let( x ( s ) , y ( s )) be a parametrization of the singular curve passing through p . Notice that wemay assume, w.l.o.g., that the x -axis past p is transverse to the singular curve, i.e., y (cid:48) (cid:54) = 0.Since f ( x ( s ) , y ( s )) = 0, taking derivative with respect to s gives u xx x (cid:48) + ( u xy − y (cid:48) = 0.Therefore, ( u xy − p ) = 0 if u xx ( p ) = 0, and hence ( u xy + 1)( p ) = 2.If u xx ( p ) (cid:54) = 0, we turn to compute the angle ζ between e and X x . First, from (8.11), wehave I ( e , X x ) = | e || X x | cos ζ = (1 + ( u x − y ) ) / cos ζ. On the other hand, using (8.12) to get I ( e , X x ) = ( u y + x ) D .
Combining the above two formulae, we obtaincos ζ = u y + xD (cid:112) u x − y ) = u y + xu x − yDu x − y (cid:112) u x − y ) = ± u y + xu x − y (cid:113) u y + xu x − y ) (cid:112) u x − y ) , (8.13)where the sign ± depends on that the sign of u x − y is positive or not. By the mean valuetheorem, it is easy to see (or see [3]) that(8.14) lim q → p + u y + xu x − y = u xy + 1 u xx ( p ) = lim q → p − u y + xu x − y , AMPLE 33 and thus(8.15) lim q → p + cos ζ = u xy +1 u xx ( p ) (cid:114) (cid:16) u xy +1 u xx ( p ) (cid:17) = − lim q → p − cos ζ, where lim q → p + (lim q → p − ) means that q → p from the side in which u x − y is positive (negative).If ( u xy + 1)( p ) (cid:54) = 0, similar computations give the angle η between e and X y by(8.16) cos η = − ( u x − yu y + x ) ± (cid:113) u x − yu y + x ) (cid:112) u y + x ) , thus(8.17) lim q → p + cos η = − u xx u xy +1 ( p ) (cid:114) (cid:16) u xx u xy +1 ( p ) (cid:17) = − lim q → p − cos η, where lim q → p + (lim q → p − ) means that q → p from the side in which u y + x is positive (negative).From (8.14), it is easy to see that both u x − y and u y + x differ by a sign on the differentside of the singular curve which is defined by u x − y = 0 and u y + x = 0. Therefore, from theformula of e (see (8.12)), together with (8.15) and (8.17), we conclude that the characteristicvector field e differs by a sign on the different side of the singular curve when approachingthe singular point p . This completes the proof of theorem 1.9.8.3. The proof of Theorem 1.10.
In terms of normal coordinates ( x, y ), the surface Σ isrepresented by two functions ζ ( y ) and ζ ( y ). Since it is of type II, we have ζ ( y ) < α = x + ζ ( y )( x + ζ ( y )) + ζ ( y ) , a = 0 , b = | α || x + ζ ( y ) |√ α , on which either x + ζ ( y ) > (cid:112) − ζ ( y ) or x + ζ ( y ) < − (cid:112) − ζ ( y ). The induced metric is(8.19) I = dx ⊗ dx + 1 b dy ⊗ dy. We assume that Σ lies on the part x + ζ ( y ) > (cid:112) − ζ ( y ) (the proof for the case that Σlies on the part x + ζ ( y ) < − (cid:112) − ζ ( y ) is similar). Suppose, in addition, that Σ can besmoothly extended beyond the singular curve x + ζ ( y ) − (cid:112) − ζ ( y ) = 0. By theorem 1.9, thecoordinates ( x, y ) can be extended beyond the singular curve to be compatible coordinates.Then the α -function on the other side of the singular curve must be one of the following(1) x + ζ ( y ) − √ − ζ ( y ) , which is of special type I;(2) x +2( ζ ( y ) − √ − ζ ( y )) , which is of special type II;(3) α = x + ζ ( y )( x + ζ ( y )) + ζ ( y ) , which is of general type,for x + ζ ( y ) < (cid:112) − ζ ( y ). The induced metric on this other part is(8.20) I = dx ⊗ dx − ab dx ⊗ dy − ab dy ⊗ dx + (1 + a ) b dy ⊗ dy. Compare (8.19) and (8.20), and notice that I is smooth around the singular curve, we have a = 0 , b = | α || x + ζ ( y ) |√ α , with α = x + ζ ( y )( x + ζ ( y )) + ζ ( y ) . That is, case (1) and (2) for α do not happen. Therefore, theextended coordinates beyond the singular curve are also normal coordinates. And the formulaof α shows that the part on the other side of the singular curve is of type III. This completesthe proof of Theorem 1.10.8.4. The proof of the Bernstein-type theorem.
In this subsection, we are going to showthat (7.1) and (7.5) are the only entire smooth p -minimal graphs. Suppose that Σ ⊂ H isan entire p -minimal graph. First of all, since it is a graph, we notice that there is nowhereat which α is zero. Next, we claim the following lemma. Lemma . The induced singular characteristic foliation of Σ doesn’t contain a leaf alongwhich α is of general type, that is, in terms of normal coordinates around the leaf, α is ageneral solution of the Codazzi-like equation ( see (4.1)) .Proof. Suppose not, we assume that the induced singular characteristic foliation of Σ containssuch a leaf. Then there will be a piece of the surface (a neighborhood) around the leaf suchthat this piece is of general type. Suppose that this piece is of type I or of type III, then theentireness and the phase plane (Figure 4.1) indicate that the α -function must be extendedso that it has a zero at somewhere. This is a contradiction. Therefore this piece (of generaltype) must be of type II. Again, since it is entire, this piece can be smoothly extendedthrough the singular curve. By theorem 1.10, it contains a piece of type III, which lies onthe other side of the singular curve. This is also a contradiction, as we argue above. Wehence complete the proof of Lemma 8.2. (cid:3) From Lemma 8.2, we know that an entire p -minimal graph is either of special type I or ofspecial type II. If it is of special type II, Theorem 7.8 and Lemma 7.5 ensure that Σ is one ofthe graphs in (7.5). If it contains a piece of special type I, then this piece must be symmetricby Theorem 7.4. Therefore, by Theorem 7.3 and Lemma 7.1, the surface Σ must be one ofthe graphs in (7.1). We therefore complete the proof of the Bernstein-type theorem.9. An approach to construct p -minimal surfaces In this section, we provide an approach to construct p -minimal surfaces. It turns out that,in some sense, generic p -minimal surfaces can be constructed by this approach, particularly,other those p -minimal surfaces of special type I. This approach is to perturb the surface u = 0 in some way. Recall we choose the parametrization of u = 0 by X : ( r, θ ) → ( r cos θ, r sin θ, , r > , where each half-ray l θ : r → ( r cos θ, r sin θ,
0) with a fixed angle θ is a Legendrian straightline. Therefore, the image of the action of each Heisenberg rigid motion on l θ is also aLegendrian straight line. Let C be an arbitrary curve C : θ → ( x ( θ ) , y ( θ ) , z ( θ )) , θ ∈ R . Thenfor each fixed θ and r >
0, the curve defined by L C ( θ ) ( l θ ) : r → ( x ( θ ) + r cos θ, y ( θ ) + r sin θ, z ( θ ) + ry ( θ ) cos θ − rx ( θ ) sin θ ) AMPLE 35 is a Legendrian straight line. Here L C ( θ ) is the left translation by C ( θ ). Therefore, the unionof these lines constitutes a p -minimal surface with a parametrization Y given by(9.1) Y ( r, θ ) = ( x ( θ ) + r cos θ, y ( θ ) + r sin θ, z ( θ ) + ry ( θ ) cos θ − rx ( θ ) sin θ ) . This surface depends on the curve C ( θ ) = ( x ( θ ) , y ( θ ) , z ( θ )). We have the following proposi-tion about the surface Y . Proposition . The coordinates ( r, θ ) are compatible coordinates for Y . In terms of thiscoordinate system, the α -invariant and the induced metric read a = − ( x (cid:48) ( θ ) cos θ + y (cid:48) ( θ ) sin θ ) α [ r + ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ )] √ α b = α [ r + ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ )] √ α , (9.2) and (9.3) α = r + ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ )[ r + ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ )] + Θ( C (cid:48) ( θ )) − ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) , where Θ( C (cid:48) ( θ )) = z (cid:48) ( θ ) + x ( θ ) y (cid:48) ( θ ) − y ( θ ) x (cid:48) ( θ ) .Proof. We make a straightforward computation for the invariants α, a and b . First we have Y r = (cos θ, sin θ, y ( θ ) cos θ − x ( θ ) sin θ )= cos θ ˚ e ( Y ( r, θ )) + sin θ ˚ e ( Y ( r, θ )) . (9.4)From the construction of Y , we have e = Y r . Thus(9.5) e = J e = − sin θ ˚ e ( Y ( r, θ )) + cos θ ˚ e ( Y ( r, θ )) , whereas we have Y θ = ( x (cid:48) ( θ ) − r sin θ, y (cid:48) ( θ ) + r cos θ, z (cid:48) ( θ ) + r ( y (cid:48) ( θ ) cos θ − y ( θ ) sin θ − x (cid:48) ( θ ) sin θ − x ( θ ) cos θ )) . If we let(9.6) Y θ = A ˚ e ( Y ( r, θ )) + B ˚ e ( Y ( r, θ )) + C ∂∂z , for some functions
A, B and C . Then straightforward computations show that A = x (cid:48) ( θ ) − r sin θ, B = y (cid:48) ( θ ) + r cos θ,C = z (cid:48) ( θ ) − x (cid:48) ( θ ) y ( θ ) + y (cid:48) ( θ ) x ( θ ) + 2 r ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) + r . (9.7)We recall that the three invariants α, a and b are related by(9.8) αe + T √ α = aY r + bY θ . If we substitute (9.4), (9.5) and (9.6) into (9.8), and compare the corresponding coefficients,we then obtain (9.2) and (9.3). (cid:3)
Remark . Let D = y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ . By (9.4),(9.6) and (9.7), we have0 = Y r ∧ Y θ ⇔ B cos θ − A sin θ = 0 , C cos θ = 0 , C sin θ = 0 ⇔ r + D = 0 , C = 0 ⇔ r + D = 0 , Θ( C (cid:48) ( θ )) + 2 rD + r = 0 , by (9.7) , ⇔ r + D = 0 , r = − D ± (cid:112) D − Θ( C (cid:48) ( θ )) ⇔ r + D = 0 , Θ( C (cid:48) ( θ )) − D = 0 . We conclude that Y is an immersion if and only if either Θ( C (cid:48) ( θ )) − D (cid:54) = 0 or r + D (cid:54) = 0for all θ , where Θ( C (cid:48) ( θ )) = z (cid:48) ( θ ) − x (cid:48) ( θ ) y ( θ ) + y (cid:48) ( θ ) x ( θ ).Formula (9.3) suggests the following. That Y defines a p -minimal surface of special typedepends on that Θ( C (cid:48) ( θ )) − ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) vanishes or not. We immediately havethe Theorem 1.11 and Theorem 1.12.9.1. The proof of Theorem 1.11.
From (9.3), the function α reads(9.9) α = 1 r + ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) . Therefore, the surface Y defines a p -minimal surface of special type I . For the ζ -invariant,we proceed to normalize the three invariants α, a and b by the process presented in Section6. First we choose another compatible coordinates (˜ r = r + A ( θ ) , ˜ θ = B ( θ )), for some A ( θ ) , B ( θ ), such that ˜ a = 0. From the transformation law of the induced metric (2.20), thiscan be chosen so that(9.10) A (cid:48) ( θ ) = − ab = x (cid:48) ( θ ) cos θ + y (cid:48) ( θ ) sin θ, or equivalently,(9.11) A ( θ ) = (cid:90) (cid:2) x (cid:48) ( θ ) cos θ + y (cid:48) ( θ ) sin θ (cid:3) dθ. If we further choose B such that ˜ θ = B ( θ ) = θ then, in terms of the compatible coordinates(˜ r, ˜ θ ), the three invariants read˜ a = 0˜ b = ˜ α √ α ˜ α = 1˜ r + y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ − A ( θ ) . (9.12)From the formula for ˜ b , we see that (˜ r, ˜ θ = θ ) is a normal coordinate system. We thereforeacquire the ζ -invariant of Y as ζ ( θ ) = y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ − A ( θ ) . This completes the proof.
AMPLE 37
The proof of Theorem 1.12.
Applying the same process of normalization we showedin the proof of Theorem 1.11, we normalize the three invariants, in terms of the normalcoordinates (˜ r = r + A ( θ ) , ˜ θ = θ ) with A ( θ ) specified as (9.11), to be˜ a = 0 , ˜ b = ˜ α [ r + ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ )] √ α , ˜ α = ˜ r − A ( θ ) + ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ )[˜ r − A ( θ ) + ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ )] + Θ( C (cid:48) ( θ )) − ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) . (9.13)By the formula for ˜ b , we see that (˜ r, ˜ θ = θ ) are normal coordinates. Therefore, we obtainthe ζ - and ζ -invariants of Y as ζ ( θ ) = − A ( θ ) + ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) ,ζ ( θ ) = z (cid:48) ( θ ) + x ( θ ) y (cid:48) ( θ ) − y ( θ ) x (cid:48) ( θ ) − ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) . This completes the proof.Comparing Theorem 1.11 and Theorem 1.12, it is convenient to regard surfaces of specialtype I as surfaces of general type with ζ -invariant vanishing. Now given two arbitrary func-tions ζ and ζ , we solve equation system (1.13) for a smooth curve C ( θ ) = ( x ( θ ) , y ( θ ) , z ( θ )).Since system (1.13) is equivalent to the following system(9.14) (cid:40) ζ (cid:48) ( θ ) = y (cid:48)(cid:48) ( θ ) cos θ − x (cid:48)(cid:48) ( θ ) sin θ − (cid:0) x (cid:48) ( θ ) cos θ + y (cid:48) ( θ ) sin θ (cid:1) ,ζ ( θ ) = z (cid:48) ( θ ) + x ( θ ) y (cid:48) ( θ ) − y ( θ ) x (cid:48) ( θ ) − ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) , which is underdetermined. Therefore, the solutions always exist. For example, we can solvethe first equation of (9.14) for ( x ( θ ) , y ( θ )) and then solve for z ( θ ) from the second one. Itturns out that we can find a smooth curve C such that the corresponding p -minimal surface Y has the two given functions ζ and ζ as its ζ -and ζ -invariants. If ζ = 0, then Y is of specialtype I. We thus conclude, together with Theorem 7.8 in which states parametrizations for p -minimal surfaces of special type II, that we generically has provided a parametrization forany given p -minimal surface with type. In particular, we give a parametrization presentationfor the eight classes of maximal p -minimal surfaces constructed in Subsection 6.3.Finally, we would like to point out that these p -minimal surfaces constructed in Theorem1.11 and Theorem 1.12 are all immersed surfaces at least (in some cases, they are embedded).This is because that ˜ b (cid:54) = 0 for all points. In particular, formula (1.13) says that if ˜ α → b → | Θ( C (cid:48) ( θ )) − ( y (cid:48) ( θ ) cos θ − x (cid:48) ( θ ) sin θ ) | , which is not zero. Example . If we take C to be the curve C ( θ ) = (0 , , z ( θ )) with z (cid:48) ( θ ) (cid:54) = 0, then Y ( r, θ ) = ( r cos θ, r sin θ, z ( θ )) . Taking the new coordinates ( s, t ) = ( r, z ( θ )), we recover the surface of general type inSubsection 7.3 (see Figure 7.3 for the case z ( θ ) = θ ). Example . If we take C to be the curve C ( θ ) = ( − sin θ, cos θ, θ ), then Y ( r, θ ) = ( − sin θ + r cos θ, cos θ + r sin θ, r ) . The surface of type I in Subsection 7.4 (see Figure 7.4) can be recovered by taking a rotationby π about the z -axis. References [1] V. Barone Adesi, F. Serra Cassano, and D. Vittone,
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Rotationally invariant Hypersurfaces with constant mean curvature in theHeisenberg group H n , The Journal of Geometric Analysis, v.16, n.4 pp. 703-720 (2006). Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300, R.O.C.
Email address : [email protected] Department of Mathematics, TamKang University, New Taipei City 25137, Taiwan, R.O.C.
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