Invariant surfaces in H 2 R with constant (mean or Gauss) curvature

Transcription

1 Proceedings of XIII Fall Workshop on Geometry and Physics Mrcia, September 20 22, 2004 Pbl. de la RSME, Vol. 9 (2005), Inariant srfaces in H 2 R with constant (mean or Gass) cratre Stefano Montaldo 1 and Irene I. Onnis 2 1 montaldo@nica.it 2 onnis@ime.nicamp.br 1 Uniersità degli Stdi di Cagliari, Dipartimento di Matematica Via Ospedale 72, Cagliari, Italy 2 Departamento de Matematica, C.P. 6065, IMECC, UNICAMP , Campinas, SP, Brazil Keywords: Inariant srfaces, mean cratre, Gass cratre, hyperbolic space Mathematics Sbject Classification: 53C42, 53A Introdction Srfaces of reoltion in the Eclidean three dimensional space are the first examples of inariant srfaces. They are inariant nder the action of the one-parameter sbgrop SO(2) of the isometry grop of R 3. Since the beginning of differential geometry of srfaces mch attention has been gien to the srfaces of reoltion with constant Gass cratre or constant mean cratre. The srfaces of reoltion with constant Gass cratre seem to be known to Minding (1839), while those with constant mean cratre hae been classified by Delanay (1841). Aside from rotations, the isometry grop of R 3 incldes the one-parameter sbgrop of translations and that of helicoidal motions. It is rather interesting to note that the classification and explicit parametrisations of helicoidal srfaces in R 3 (inariant nder the action of helicoidal grops) with constant mean cratre hae been achieed only recently, in 1982, by M.P. Do Carmo and M. Dajczer in [3]. In the last decades, de to the deelopment of sitable redction techniqes [1, 4, 5, 7], appeared many works on the stdy of srfaces in a three dimensional manifold which are inariant nder the action of a one-parameter sbgrop of the isometry grop, see, for example, [2, 5, 6, 7, 8, 9, 10, 14]. In this paper we

2 92 Inariant srfaces consider the three dimensional manifold H 2 R, wherebyh 2 we denote, as sal, the half-plane model of the hyperbolic space endowed with the standard metric of constant Gass cratre 1. The space H 2 R is one of the eight Thrston s geometries and its isometry grop is of dimension 4 which is, among the 3-dimensional spaces with non-constant sectional cratre, the greatest possible. This means that the space H 2 R is sfficiently symmetric to motiate the stdy of srfaces which are inariant nder the action of a oneparameter sbgrop of the isometry grop. The paper is diided as follows. In section 2 we smmarize the redction techniqes for inariant srfaces with constant mean cratre (CMC) or constant Gass cratre; in section 3 we describe the isometry grop of H 2 R and the one-parameter grops of isometries; in section 4 we gie the classification of inariant srfaces with constant mean cratre and in section 5 we analyze the inariant srfaces with constant Gass cratre. 2. Redction Techniqes Let (N 3,g) be a three dimensional Riemannian manifold and let X be a Killing ector field on N. Then X generates a one-parameter sbgrop G X of the grop of isometries of (N 3,g). For x N, the isotropy sbgrop G x of G X is compact and the qotient space G X /G x is diffeomorphic to the orbit G(x) = {gx N : g G}. An orbit G(x) is called principal if there exists an open neighborhood U N of x sch that for all orbits G(y), y U, theisotropy sbgrops G y are conjgate. If N/G X is connected, from the Principal Orbit Theorem ([13]), the principal orbits are all diffeomorphic and the reglar set N r, consisting of points belonging to principal orbits, is open and dense in N. Moreoer, the qotient space N r /G X is a connected differentiable manifold and the qotient map π : N r N r /G X is a sbmersion. Definition 1. Let f : M 2 (N 3,g) be an immersion from a srface M 2 into N 3 and assme that f(m) N r. We say that f is a G X -eqiariant immersion, and f(m) ag X -inariant srface of N, if there exists an action of G X on M 2 sch that for any x M 2 and g G X we hae f(gx) =gf(x). A G X -eqiariant immersion f : M 2 (N 3,g) indces on M 2 ariemannian metric, the pll-back metric, denoted by g f and called the G X -inariant indced metric. Letf : M 2 (N 3,g)beaG X -eqiariant immersion from a srface M 2 into a Riemannian manifold (N 3,g) and let endow M 2 with the G X -inariant indced metric g f. Assme that f(m 2 ) N r and that N/G X is connected. Then f indces an immersion f : M/G X N r /G X between the orbit spaces; moreoer, the space N r /G X can be eqipped with a Riemannian

3 Stefano Montaldo and Irene I. Onnis 93 metric, the qotient metric, so that the qotient map π : N r N r /G X is a Riemannian sbmersion. Following [8] we shall describe the qotient metric of the reglar part of the orbit space N/G X. It is well known (see, for example [11]) that N r /G X can be locally parametrized by the inariant fnctions of the Killing ector field X. If {f 1,f 2 } is a complete set of inariant fnctions on a G X -inariant sbset of N r, then the qotient metric is gien by g = 2 i,j=1 hij df i df j where (h ij ) is the inerse of the matrix (h ij )withentries h ij = g( f i, f j ). If we denote by ω(y) = X(y) the olme fnction of the principal orbit G(y) ={gy : g G}, then the mean cratre fnction of f can be expressed in terms of the geodesic cratre of f and of the fnction ω(y) as it is shown in the following Theorem 1 (Redction Theorem [1]). Let H be the mean cratre fnction of f : M 2 N 3 and k g thegeodesiccratreof f : M/G X N r /G X.Then H = k g n (ln ω), where n is the nit normal ector to M/G X in N 3 r /G X Inariant srfaces with constant Gass cratre We first gie a local description of the G X -inariant srfaces of N 3. Let γ :(a, b) R (Nr 3/G X, g) be a cre parametrized by arc length and let γ :(a, b) R Nr 3 be a lift of γ, sch that dπ( γ )=γ. If we denote by φ r,r ( ɛ, ɛ), the local flow of the Killing ector field X, then the map ψ :(a, b) ( ɛ, ɛ) N 3, ψ(t, r) =φ r ( γ(t)), defines a parametrized G X -inariant srface. Conersely, if f : M 2 Nr 3 is a G X -eqiariant immersion, then f defines a cre in (Nr 3/G X, g) thatcan be, locally, parametrized by arc length. The cre γ is generally called the profile cre. The following theorem describe (locally) the inariant srfaces with constant Gass cratre. Theorem 2 ([10]). Let f : M 2 (N 3,g) be a G X -eqiariant immersion, γ :(a, b) R (N 3 r /G X, g) a parametrisation by arc length of f and γ a lift of γ. (i) If the G X -inariant indced metric g f is of constant Gass cratre K, then the fnction ω(t) = X( γ) g satisfies the following differential eqation d 2 ω(t)+kω(t) =0. (1) dt2

4 94 Inariant srfaces (ii) Vice ersa, sppose that Eqation (1) holds with K a real constant. Then, in all points where d(ω 2 )/dt 0,theG X -inariant indced metric g f has constant Gass cratre. By integration of (1) we hae Corollary 3. Let f : M 2 (N 3,g) be a G X -eqiariant immersion which indces a G X -inariant metric g f on M 2 of constant Gass cratre K. Then the norm ω(t) of the Killing ector field X along a lift of the profile cre is: for K =0 gien by ω(t) =c 1 t + c 2 ; for K =1/R 2 > 0 gien by ω(t) =c 1 cos(t/r)+c 2 sin(t/r); for K = 1/R 2 < 0 gien by ω(t) =c 1 cosh(t/r)+c 2 sinh(t/r), with c 1,c 2 R. As we shall show in Section 5 the profile cre of a G X -inariant srface can be parametrized as a fnction of ω. Ths, sing Corollary 3, we can gie the explicit parametrisation of the profile cre. Remark 1. If (N 3,g)=(R 3,can) is the Eclidean three dimensional space, then the Killing ector fields generate either translations or rotations. In the case of translations the qotient space R 3 /G X is R 2 with the flat metric and ω is constant. Ths, from Eqation 1, we see that any cre in the qotient space generates a flat right cylinder. In the case of rotations we can assme, withot loss of generality, that the rotation is abot a coordinate axis, say x 3. Then the Killing ector field is X = x 2 x 1 + x 1 x 2 and the reglar part of the qotient space is R 3 r/g X = {(x 1,x 2,x 3 ) R 3 : x 2 =0, x 1 > 0} with the flat metric. If γ(t) =((t), 0,(t)) R 3 r/g X is a arc length parametrized profile cre of a G X -inariant srface, then the norm of X restricted to the profile cre is ω = (t) and, sing Corollary 3, we find the classical explicit parametrisation of srfaces of reoltion with constant Gass cratre. 3. One-parameter sbgrops of isometries of H 2 R Let H 2 = {(x, y) R 2 : y>0} be the half plane model of the hyperbolic plane endowed with the metric, of constant Gass cratre 1, gien by g H = dx2 + dy 2 y 2. The hyperbolic plane H 2, with the grop strctre deried by the composition of proper affine transformations, is a Lie grop and the metric g H is leftinariant. Then the prodct space H 2 R is a Lie grop with the prodct

5 Stefano Montaldo and Irene I. Onnis 95 strctre L (x,y,z) (x,y,z )=(x, y, z) (x,y,z )=(x y + x, yy,z+ z ) and the left inariant metric gien by the prodct metric g = dx2 + dy 2 y 2 + dz 2. From a direct integration of the Killing eqation L X g =0wehae Proposition 4. The Lie algebra of the infinitesimal isometries of the prodct (H 2 R,g) admits the following bases of Killing ector fields X 1 = (x2 y 2 ) 2 x + xy y ; X 2 = x ; X 3 = x x + y y ; X 4 = z. Let denote by G i the one-parameter sbgrop of isometries generated by X i,by G ij the one-parameter sbgrop of isometries generated by linear combinations of X i and X j and so on. Explicitly we hae that G 1 = {L (t,0,0,0) t R} with ( 2[t(x 2 + y 2 ) 2x] 4y ) L (t,0,0,0) (x, y, z) = (tx 2) 2 + t 2 y 2, (tx 2) 2 + t 2 y 2,z ; G 2 = {L (0,t,0,0) t R} with L (0,t,0,0) (x, y, z) =(x + t, y, z); G 3 = {L (0,0,t,0) t R} with L (0,0,t,0) (x, y, z) =(e t x, e t y,z); G 4 = {L (0,0,0,t) t R} with L (0,0,0,t) (x, y, z) =(x, y, z + t). Remark 2. The integral cres of X 2,X 3 and X 4 are easy to pictre ot. In fact, for t fixed the isometries L (0,t,0,0) L (t,1,0), L (0,0,t,0) L (0,e t,0) and L (0,0,0,t) L (0,1,t) are left translations. The integral cre of X 1, throgh the point p 0 =(x 0,y 0,z 0 ) H 2 R, isl (t,0,0,0) (x 0,y 0,z 0 )=(x(t),y(t),z 0 ), where ( x x(t) 2 + y(t) y0 2 ) y(t) =0. y 0 Therefore, it is a horocycle, in the plane z = z 0, with radis (x y2 0 )/2y 0 and centered at (0, (x y2 0 )/2y 0,z 0 ). In Figre 1 there is a plot of the integral cres of X 1 throgh three different points.

6 96 Inariant srfaces y y x i x Figre 1: Integrale cres of X 1 (left) and of X 12 (right). Two grops G X and G Y, generated by two Killing ector fields X and Y, are conjgate if there exists an isometry ϕ of H 2 R sch that G Y = ϕ 1 G X ϕ. If G X and G Y are conjgate, then the respectiely inariant srfaces are congrent, i.e. isometric with respect to the isometry ϕ of the ambient space. Therefore, we can redce the stdy of the inariant srfaces by analyzing all the conjgate one-parameter grops of isometries. In [12] there is the complete list of the conjgate grops of isometries in H 2 R which gies the following Lemma 5 ([12]). Any srface in H 2 R which is inariant nder the action of a one-parameter sbgrop of isometries G X, generated by a Killing ector field X = i a ix i, is isometric to a srface inariant nder the action of one of the following grops G 24, G 34, G 12, G 124, where G 12 is the one-parameter grop generated by X 12 = X 1 +(X 2 )/2 and G 124 is the one-parameter grop generated by X 12 and X 4. Remark 3. The integral cre of X 12 = X 1 + X 2 /2 throgh the point p 0 = (x 0,y 0,z 0 ) H 2 R is L (t,t/2,0,0) (x 0,y 0,z 0 )=(x(t),y(t),z 0 ), where ( 1+x x(t) 2 + y(t) y0 2 ) y(t)+1=0. y 0 An easy comptation shows that the hyperbolic distance from a point of the integral cre to the point (0, 1,z 0 ) is constant. Therefore, the integral cres of X 12 are geodesic circles centred at (0, 1,z 0)(seeFigre1). 4. Inariant srfaces with constant mean cratre In this section we shall consider only the actions of G 4 and G 124 whichleadto inariant srfaces with a nice geometric description. For a detailed accont of the classifications presented in this section and for the other actions, we refer the reader to [12].

7 Stefano Montaldo and Irene I. Onnis CMC srfaces inariant nder the action of the grop G 4 The grop G 4, generated by the Killing ector field X 4 = z, acts freely on H 2 R, ths the reglar part is the whole space. A complete set of inariant fnctions of X 4 is (x, y, z) =x and (x, y, z) =y. Ths the orbit space is H 2 = {(, ) R 2 >0} and the orbital metric is gien by g H = d2 +d 2. From the Redction Theorem 1 we hae that a 2 cre γ(s) =((s),(s)) in the orbit space H 2, parametrized by arc length, generates a CMC srface if and satisfy the following system { = cos σ, = sin σ, H = σ +cosσ = k g, (2) where σ = σ(s) is the angle between γ and the positie direction, while k g is the geodesic cratre of γ. Now, assming that the mean cratre H is constant and non negatie, we hae that the fnction J(s) = σ/ is constant along any cre γ(s) which is a soltion of system (2). Ths the soltions of (2) are gien by J(s) =k, forsomek R. By a qalitatie analysis of the eqation J(s) = k, we can proe the following Theorem 6. The CMC srfaces in H 2 R, which are inariant nder the action of the sbgrop G 4, are ertical cylinders oer cres of H 2 with constant geodesic cratre. Moreoer: if H =0, they are geodesics of H 2.Inparticlar,if 1. k =0, the cre is an Eclidean ray normal to the line =0; 2. k 0, the cre is an Eclidean semicircle with center on the line =0; if H>0 they are: 1. for H>1 Eclidean circles; 2. for H =1horocycles; 3. for H<1 hypercycles.

8 98 Inariant srfaces Figre 2: Profile cres of G 4 -inariant CMC srfaces: geodesics (left), horocycles (center) and hypercycles (right) Helicoidal CMC srfaces: inariant nder the action of G 124 Introdcing the cylindrical coordinates (r, θ, z) into(h 2 R,g), with r>0 and θ (0,π), the metric g takes the form g = dr2 r 2 sin 2 θ + dθ2 sin 2 θ + dz2, and the Killing ector field X 124 becomes X 124 = X 12 + ax 4 = r Choosing the inariant fnctions (r, θ, z) = r2 +1 r sin θ and cos θ r + r2 1 2r (r, θ, z) =z + a arctan sin θ θ + a z. ( 2r cos θ ) r 2, 1 the orbit space is B = {(, ) R 2 2} and the qotient metric redces to g = d2 ( 2 4) (a 2 1) d2. The system of ODE s that characterizes the profile cre γ(s)ofag 124 -srface is: = 2 4cosσ, = 2 +4(a 2 1) 2 sin σ, 4 (3) σ = H 2 4 sin σ. Remark 4. The Eqation (3) for σ has a singlarity at the bondary of B. This type of singlarity has been dealt extensiely in the literatre (see, for example, [4, 6]). In particlar, soltions that go to the bondary mst enter orthogonally, which means that the generated srface will be reglar at those points.

9 Stefano Montaldo and Irene I. Onnis 99 If H is constant, the fnction J(s) = 2 4sinσ H is constant along any cre γ(s) which is a soltion of System (3). Ths the soltions of this system are gien by 2 4sinσ H = k, k R. As in the preios section a qalitatie analysis of the eqation J(s) = k, gies the following characterization of the profile cres of CMC helicoidal srfaces. Theorem 7. Let Σ H 2 R be a CMC helicoidal srface and let γ be the profile cre in the orbit space. Then we hae the following characterization of γ according to the ale of the mean cratre H and of k. 1. (H > 1) - The profile cre is of Delanay type. Moreoer if k < 2H is of nodary-type; k = 2H is of circle-type; k > 2H is of ndlary-type. 2. (H = 1) - The profile cre is for k< 2 of folim-type; for k = 2 of conic-type; for k> 2 of bell-type. 3. (0 < H < 1) - The profile cre is for k < 2H of bonded folim-type; for k = 2H of helicoidal-type; for k > 2H of bonded bell-type. 4. (H =0)- The profile cre is for k =0a horizontal straight line; for k 0of catenary-type. Remark 5. (i) The plots of the profile cres in Figre 3 and Figre 4 are drown sing the qalitatie analysis of the angle σ with respect to the metric g of the orbit space. (ii) The plots of the profile cres of the helicoidal srfaces with 0 <H<1 are similar to those with H = 1; the only different is that for 0 <H<1the limit of the angle σ, for that goes to infinity, is between 0 and π/2, instead of π/2 as for the helicoidal srfaces with H =1. (iii) We note that some of the inariant srfaces described in Theorem 7 are complete, for example the minimal srface of reoltion (G 12-inariant) generated by a cre of catenary-type. Moreoer, there are interesting examples

10 100 Inariant srfaces Figre 3: Profile cres of helicoidal CMC srfaces with H>1: nodary-type (left), circle-type (center) and ndlary-type (right). Figre 4: Profile cres of helicoidal CMC srfaces with H = 1: folim-type (left), conic-type (center) and bell-type (right). of complete minimal srfaces which are inariant nder the action of the grop G 34. It is proed in [12] that the fnction f : H 2 R, gien by f(x, y) =ln(x 2 + y 2 ), defines a complete minimal graph of H 2 R which is G 34 inariant; ths the Berstein Theorem in H 2 R does not hold. In Figre 5 there is a plot of sch a srface. y x z Figre 5: A complete minimal graph of H 2 R.

11 Stefano Montaldo and Irene I. Onnis Inariant srfaces with constant Gass cratre Let G be a one-parameter grop of isometries among those described in Lemma 5. If we denote, as before, by ω the olme fnction of the principal orbits, we can gie the following local description of the G-inariant srfaces of H 2 R. Theorem 8 ([10]). Let γ =((s),(s)) be a cre in the orbit space (H 2 R/G, g), parametrized by arc length, which is the profile cre of a G-inariant srface in (H 2 R). Then: if G = G 4, the orbit space is H 2 and any cre parametrized by arc length is the profile cre of a flat G 4 -inariant cylinder; if G = G 24, the orbit space is B = {(, ) R 2 cre can be parametrized by (s) = a / ω 2 b 2, (s) = s s 0 a 2 ω 2 ω 2 b 2 [1 a,b R ( ) ωω 2 ω 2 b ]dt; 2 : >0} and the profile if G = G 34, then the orbit space is B = {(, ) R 2 : 0 <<π} and the profile cre can be parametrized by (s) =arcsin ( a / ) ω 2 b 2, a,b R (s) = ] s dt; a 2 ω 2 s 0 ω 2 b 2 [1 (ωω ) 2 (ω 2 b 2 )(ω 2 a 2 b 2 ) if G = G 124, then the orbit space is B = {(, ) R2 : profile cre can be parametrized by (s) =2 ω 2 +1 a 2, (s) = s s 0 ω 2 ω 2 a 2 [1 a R (ωω ) 2 (ω 2 a 2 )(ω 2 +1 a 2 ) 2} and the ] dt. Now if γ is the profile cre of a G-inariant srface in H 2 R with constant Gass cratre, then the explicit parametrisation of γ can be obtained by replacing in Theorem 8 the corresponding expression of the fnction ω, according to the ale of the Gass cratre K, as we hae described in Corollary 3. For example, in the case of the G 12 -inariant srfaces of H2 R, for some ales of ω we hae:

12 102 Inariant srfaces 1) if K = 0, choosing ω(s) = s, we hae the following parametrisation for the profile cre γ(s) =(2 s 2 +1, s 2 +1); 2) if K>0, choosing ω(s) = cos s, the corresponding profile cre is γ(s) =(2 2cos cos 2 s +1, 2 s ( sin s ) arctan ); cos s cos 2 s +1 3) if K<0, taking the fnction ω(s) = sinhs, we obtain γ(s) =(2coshs, 0). Acknowledgments The first athor wishes to thank the organizers of the XIII Fall Workshop on Geometry and Physics, Mrcia, 2004 for their exqisite hospitality and the opportnity of presenting this lectre. The athors also wish to thank the referee for a carefl reading of the paper and sggestions. References [1] A. Back, M.P. do Carmo and W.Y. Hsiang. On the fndamental eqations of eqiariant geometry, npblished manscript. [2] R. Caddeo, P. Pi and A. Ratto. Rotational srfaces in H 3 with constant Gass cratre, Boll. Un. Mat. Ital. B 10 (1996), [3] M.P. do Carmo and M. Dajczer. Helicoidal srfaces with constant mean cratre, Tohok Math. J. 34 (1983), [4] J. Eells and A. Ratto. Harmonic maps and minimal immersions with symmetries, Annals of Mathematics Stdies, 130. Princeton Uniersity Press, Princeton, NJ, [5] W.T. Hsiang and W.Y. Hsiang. On the niqeness of isoperimetric soltions and embedded soap bbbles in non-compact symmetric spaces, Inent. Math. 89 (1989), [6] W.T. Hsiang and W.Y. Hsiang. On the existence of codimension one minimal spheres in compact symmetric spaces of rank 2, J. Diff. Geom. 17 (1982),

13 Stefano Montaldo and Irene I. Onnis 103 [7] W.Y. Hsiang and H.B. Lawson. Minimal sbmanifold of low cohomogeneity, J. Diff. Geom. 5 (1971), [8] C.B.Figeroa,F.MercriandR.H.L.Pedrosa. Inariant srfaces of the Heisenberg grops, Ann. Mat. Pra Appl. 177 (1999), [9] S. Montaldo and I.I. Onnis. Inariant CMC srfaces in H 2 R, Glasg. Math. J. 46 (2004), [10] S. Montaldo and I.I. Onnis. Inariant srfaces in a three-manifold with constant Gassian cratre, J. Geom. Phys. 55 (2005), [11] P.J. Oler. Application of Lie Grops to Differential Eqations, GTM 107, Springer-Verlag, New York, [12] I.I. Onnis. Sperficies em certos espacos homogeneos tridimensionais, Ph.D. Thesis, Uniersidade Estadal de Campinas (Brazil), [13] R.S. Palais. On the existence of slices for actions of non-compact Lie grops, Ann. of Math. 73 (1961), [14] P. Tompter. Constant mean cratre srfaces in the Heisenberg grop, Proc.ofSymp.PreMath.54 (1993),