All solutions of the CMC-equation in H n R invariant by parabolic screw motion

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1 All solutios of the CMC-equatio i H R ivariat by parabolic screw motio Maria Ferada Elbert ad Ricardo Sa Earp March 21, 2012 Abstract I this paper, we give all solutios of the costat mea curvature equatio i H R that are ivariat by parabolic screw motio ad we give the full descriptio of their geometric behaviors. Some of these solutios give examples of o-trivial etire stable horizotal graphs that are ot vertical graphs. Itroductio I the last years, the aciet theory of miimal ad costat mea curvature surfaces has bee revisited for ew ambiet spaces, oe of them beig H 2 R, where H 2 is the hyperbolic space (see for istace [D1], [M-O], [N-R], [O], [R] ad the refereces therei). More geerally, dealig with greater dimesios, the study of hypersurfaces i H R has also bee pursued (see for istace [B-SE], [D2],[B-C-C]). The costructio of examples is a importat tool that cotributes to the uderstadig of what happes i this ew ambiet spaces. I this paper, we search for costat mea curvature (CMC) hypersurfaces i H R that are ivariat by parabolic screw motios, i.e., ivariat by a (-1)-parameter group of isometries such that each elemet is give by a compositio of a parabolic traslatio i H followed by a vertical traslatio. I fact, we give all solutios of the costat mea curvature equatio i H R that are ivariat by parabolic screw motio ad we give the full descriptio of their geometric behaviors. Keywords ad seteces: costat mea curvature, parabolic screw motios, complete graph Mathematical Subject Classificatio: 53C42, 53A10, 35J15. Both authors are partially supported by Faperj ad CNPq. 1

2 We use the half-space model of the hyperbolic space H = {(x 1,..., x 1, x = y) R y > 0}. We begi with a curve t = λ(y) that after the actio of such a group gives rise to a vertical graph. After imposig the mea curvature equatio, we are able to caracterize the geeratig curves λ(y) depedig o the value of H. From the results about λ(y) we costruct may families of iterestig examples of complete CMC hypersurfaces i H R, icludig embedded ad stable oes. Besides givig explicit examples we prove the existece of families of hypersurfaces. These existece results are briefly stated below. Theorem A: There exists a family of o-etire horizotal miimal complete graphs i H R ivariat by parabolic screw motios. The asymptotic boudary of each graph of this family is formed by two parallel (-1)-plaes. Theorem B: i) For each H, 0 < H < ( 1), there exists a family of complete horizotal H-CMC graphs i H R ivariat by parabolic screw motios. ii) For each H, 0 < H ( 1), there exists a family of complete o-embedded H-CMC hypersurfaces i H R ivariat by parabolic screw motios. Theorem C: For each H, H > 1, there exists a family of complete o-embedded ad t-periodic H-CMC hypersurfaces i H R ivariat by parabolic screw motios. Theorems A,B ad C are restated with more details i sectios 3 ad 4, where we give good descriptios of the behavior of the hypersurfaces. We poit out that the traslatios hypersurfaces give i Theorems A ad Theorem B i) are stable (see Propositio 2.2). Some of them are examples of complete stable CMC-hypersurfaces that are either etire vertical graphs or the trivial examples, amely, cyliders over totally umbilic (-1)- submaifolds of H {0}. I [SE], the secod author studies the case = 2. I the followig papers, ad i the refereces therei, the iterested reader will fid related topics [SE-T1],[SE-T2]. 1 Prelimiaries We use the half-space model of the hyperbolic space H ( 2), i.e., we cosider H = {(x 1,..., x 1, x = y) R y > 0} 2

3 edowed with the metric g H := dx dx dy 2 y 2. O H R, with coordiates (x 1,..., x 1, y, t), we cosider the product metric g = g H + dt 2. A vertical graph of a real fuctio u defied over Ω H is the set G = {(x, u(x)) H R x Ω}. A computatio shows that whe u is C 2 ad we choose the orietatio give by the upper uit ormal, the mea curvature H of the graph G is give by div ( ) u (2 )u y y2 u 2 y 1 + y 2 u = H 2 y. (1.1) 2 Here, div,, ad. deote quatities i the Euclidea metric of R. For a proof see, for istace, [SE-T1], Propositio (3.1). We otice that i H R we ca also defie the horizotal graph, y = g(x 1,..., x 1, t), of a real ad positive fuctio g (see [SE]). We search for costat mea curvature (CMC) hypersurfaces that are ivariat by parabolic screw motios, that is, a parabolic traslatio i H followed by a vertical traslatio. We recall that a parabolic traslatio ca be idetified with a horizotal Euclidea traslatio i this model for H. We begi with a geeratig curve t = λ(y) i the yt-vertical 2-plae of H R. The iduced metric i this vertical 2-plae is Euclidea. We fix (l 1,..., l 1 ) R 1. After a parabolic traslatio composed with a vertical traslatio gives rise to a vertical graph t = u(x 1,..., x 1, y) = λ(y) + l 1 x l 1 x 1. (1.2) Imposig the mea curvature equatio (1.1) to this graph we obtai that λ ad (l 1,..., l 1 ) satisfy ( λ (y) (2 )λ y2 l 2 + y 2 (λ ) (y)) (y) 2 y 1 + y 2 l 2 + y 2 (λ ) 2 (y) = H (1.3) y 2 where l 2 = l l. 2 Equivaletly we may write ( y 2 λ (y) (2 )y y2 l 2 + y 2 (λ ) (y)) 1 λ (y) y2 l 2 + y 2 (λ ) 2 (y) = Hy. 3

4 If we suppose that H is costat, itegrate ad coveietly rearrage the terms we obtai λ (y) 1 + y2 l 2 + y 2 (λ ) 2 (y) = d( 1)y 1 H, (1.4) y( 1) where d is the costat that comes from itegratig. After aother rearragemet ad itegratio we get y 1 + ξ2 l λ(y) = 2 (d( 1)ξ 1 H) ( ) dξ. (1.5) 2 ξ( 1) 1 d( 1)ξ 1 H Sice λ depeds also o d ad l we write λ(y, d, l) istead of λ(y) wheever coveiet. 1 2 Some geeral facts The fuctio λ(y, d, l) is give by (1.5) ad we ca suppose that d 0. The case d < 0 is obtaied from ( this by vertical ) traslatios or symmetries. I order to simplifly otatio, we set g = d( 1)y 1 H ad write The 1 λ(y) = y λ (y) = 1 + ξ2 l 2 g ξ dξ. (2.1) 1 g2 1 + y2 l 2 g y 1 g 2, (2.2) where we are assumig that 1 g 2 > 0. We otice that the sig of λ (y) is that of g ad that, whe d 0, both vaish at y = τ 1/ 1 0, where τ 0 = A computatio shows that H. d( 1) λ (y) = (1 + y2 l 2 )g y g(1 g 2 ) (1 + y 2 l 2 ) 1/2 y 2 (1 g 2 ) 3/2 which helps studyig the behavior of the curve t = λ(y, d, l). For otatio purposes, we set η(y) = (1 + y 2 l 2 )g y g(1 g 2 ), i.e., the umerator of λ (y). We poit out that the sig of η(y) is that of λ (y). The followig lemma will be useful. Lemma 2.1. If H 0 the λ > 0. 4

5 Proof: We eed to study the sig of η(y). Sice g = d( 1)y 2 > 0, we easily see that η(y) > 0 whe g 0. Now we see what happes whe g > 0. Sice, i this case the lemma follows. η(y) = (1 + y 2 l 2 )g y g(1 g 2 ) > (1 + y 2 l 2 )g y g = (1 + y 2 l 2 )d( 1)y 1 dy 1 + H 1 ( 2)dy 1 + H 1, We ow aalyse the behavior of λ(y) ear y = 0. If we write λ(y) = y 1 + ξ2 l 2 g y ξ 1 g dξ = 2 f(ξ) ξ dξ, we ca see that f(y) is differetiable i y = 0. By observig the expasio H f(y) = ( 1) 1 ( ) + f (0)y + o(y) H 2 1 ear 0 we ca coclude that λ(y) has a log behavior whe y 0. Now, we study the behavior of λ(y) ear a zero of 1 g 2. We otice that these zeroes oly exist for d > 0. For simplicity purposes, we itroduce the variable v = dξ 1, d > 0. I this ew variable, the itegral give by (1.5) becomes dy l ( 2 v 2/( 1) d) (( 1)v H) Near oe of the zeroes of as ) 2 dv. (2.3) v( 1) 1 2 ( v H 1 1 ( ) 2, v H 1 say v0 = (1+ H ), we ca rewrite the itegral 1 dy 1 h(v) v0 v dv, with a fuctio h that is differetiable at v 0. We ca the see that this itegral, ad a fortiori the itegral i (1.5), coverge to a fiite value ear this zero of 1 g 2. The same happes for the other zero. Hece the graph of λ(y, d, l) is vertical at both zeroes of 1 g 2. A computatio shows that the (Euclidea) curvature of the curve i the yt-vertical 2-plae is fiite at these poits. Propositio 2.2. Ay H-CMC or miimal horizotal graph ivariat by parabolic traslatios (l = 0) is stable. 5

6 Proof: Let us cosider the killig field (for a defiitio, see for istace [J], page 52) geerated by the homotheties i each slice H {t} w.r.t. the origi of the slice (a hyperbolic isometry of the slice). Notice that the geeratig curve is a horizotal smooth graph i the yt-vertical 2-plae (x 1 = x 2 =... = x 1 0). The the trajectories of the killig field cut the graph trasversally at oe poit at most, provig that it is a killig graph ad, a fortiori, provig the propositio, sice killig graphs are stable (see [B-G-S], Theorem 2.7). 3 The miimal examples I this sectio, we cosider the particular case H = 0. We fid some explicit examples give by elemetary formulas or itegral formulas. Preservig the otatio ad the method stated i the prelimiaries, we set our first result. Propositio 3.1. (Miimal vertical graphs) The miimal vertical graphs ivariat by parabolic screw motios geerated by a curve t = λ(y) are, up to traslatios or symmetries, of oe of the followig types: a) t = l 1 x l 1 x 1, y > 0. b) t = λ(y, d, l) + l 1 x l 1 x 1, y (0, (1/d) 1 1 ), where λ is icreasig ad ) 1 1 strictly covex i the iterval ad is vertical at y = ( 1 d particular case l = 0, we obtai the graph give by the elemetary formula t = 1 1 arcsi(dy 1 ) + κ. (see Figure 1). For the Proof: The case H = 0 ad d = 0: If we impose i (1.4) that H ad d vaish, we obtai that λ κ = costat. I this case, the vertical graphs obtaied are t = κ + l 1 x , l 1 x 1, y > 0. The case H = 0 ad d 0: I this case, (1.5) gives, up to vertical traslatios or symmetries, λ(y, d, l) = y 0 dξ ξ 2 l 2 1 (dξ 1 ) 2 dξ, 6

7 with d > 0. Chagig for the variable v = dξ 1 we obtai ) 2 λ(y, d, l) = 1 dy (l( vd ) 1 1 dv. (3.1) v 2 We the have t = λ(y, d, l) + l 1 x l 1 x 1, y (0, (1/d) 1 1 ), with λ give by (3.1). By the results of Sectio 2 applied to this case we coclude that it ( is icreasig ad strictly covex i the iterval (0, (1/d) 1 1 ) ad is vertical at y = 1 ) 1 1. d d=1, = 3, H=0, l=0 Figure 1: d = 1, = 3, H = 0, l = 0. For the special case l = 0, we ca itegrate the above formula ad obtai a explicit family of examples, amely, the graphs of t = 1 1 arcsi(dy 1 ) + κ. Theorem 3.2. The curves of Propositio 3.1 b) geerate a family of o-etire horizotal miimal complete graphs i H R ivariat by parabolic screw motios. The asymptotic boudary of each graph of this family is formed by two parallel (-1)-plaes (see Figure 2). 7

8 Proof: We kow that the fuctio λ(y, d, l) give by b) of Propositio (3.1) is icreasig ( ad covex i the iterval (0, (1/d) 1 1 ) ad is vertical at y = 1 ) 1 1. We also kow that d the Euclidea curvature is fiite at y = (1/d) 1 1. If we set t0 = λ ca the glue together the graph with its Schwarz reflectio give by t = λ(y, d, l) + l 1 x l 1 x 1 2t 0 λ(y, d, l) + l 1 x l 1 x 1, ((1/d) 1 1, d, l ), we i order to obtai a horizotal miimal complete graph ivariat by parabolic screw motios defied over {(x 1,..., x 1, t) R 1 R l 1 x l 1 x 1 t 2t 0 + l 1 x l 1 x 1 }. The asymptotic boudary of this example is formed by two parallel hyperplaes. The stability comes from Propositio 2.2. d=1, = 3, H=0, l=0 Figure 2: d = 1, = 3, H = 0, l = 0. 4 The H-CMC examples I this sectio we treat the case H 0. We recall that we are usig the orietatio give by the upper uit ormal. We first deal with the hypothesis d = 0. The case H 0 ad d = 0: Imposig that d = 0, (1.5) becomes λ(y, l) = H y 1 + ξ2 l 2 ( 1) 1 ( ) dξ (4.1) H 2 ξ 1 8

9 ad aturally imposes the restrictio H < 1. Itegratig (4.1) we obtai H t = ( 1) 1 ( ( ) 1/2 1 + l2 y ) 1 + l 2 y 2 + l κ, l 0, H l2 y which give a explicit family of etire vertical graphs of mea curvature H ( ) H < 1, defied over H R, that are also complete horizotal graphs (see Figure 3). Figure 3: d = 0, = 4, H = 1/2, l = 1. For the case l = 0, we obtai the explicit examples (see Figure 4) t = H ( 1) 1 ( ) l(y) + κ. H 2 1 The case H 0 ad d > 0: We ca distiguish four cases, depedig o the values of H, as oe ca see below. For each case, we come back to Sectio 2 ad try to uderstad the behavior of the curve t = λ(y, d, l). By observig that both y ad 1 g 2 should be positive i (2.1), we see that y, besides beig positive, should be i the iterval (y 0, y 1 ), where y 0 = ( τ 0 1 d) 1/ 1 ad y 1 = ( τ d) 1/ 1. Now we deal with each case separately. H ( 1) : There is o solutio, sice τ d 0. 9

10 Figure 4: d = 0, = 3, H = 1/3, l = 0. ( 1) < H < 0: Sice, for this case, τ 0 1 < 0 we see that λ(y, d, l) is defied o d (0, y 1 ). We also kow that λ is icreasig (see (2.2)), is vertical at y 1 ad has a log behavior ear 0. Now, we should uderstad the sig of λ (y), i.e., the sig of η(y). It is easy to see that η(y) < 0 ear 0 ad η(y) > 0 ear y 1. A computatio shows that η (y) = 2l 2 g y 2 + (1 + y 2 l 2 )g y + 3g 2 g. Sice g > 0, g > 0 ad g 0, we see that η (y) > 0, which implies that η(y) = 0 oly oce i (0, y 1 ). The, λ(y, d, l) is cocave ear 0 ad becomes covex after some poit i (0, y 1 ) (see Figure 5). d=1, = 4, H= 1/4, l=1 Figure 5: d = 1, = 4, H = 1/4, l = 1. 0 < H ( 1) : I this case, λ(y, d, l) is defied o (0, y 1), is decreasig o 10

11 ( 0, τ0 1/ 1 ) ad icreasig o ( τ 0 1/ 1, y 1 ) (see (2.2)). By Lemma 2.1, it is strictly covex ad by the results of Sectio 2, λ is vertical at y 1 ad has a log behavior at y = 0 (see Figure 6). H > ( 1) : I this case, λ(y, d, l) is defied o (y 0, y 1 ), is decreasig o ( y 0, τ 0 1/ 1 ) ad icreasig o ( τ 0 1/ 1, y 1 ) (see (2.2)). By Lemma 2.1, λ is strictly covex i the iterval ad by the results of Sectio 2, is vertical at y 0 ad y 1. We claim that λ(y 0 ) > λ(y 1 ) (see Figure 7). I order to prove the claim, we proceed as follows. Istead of workig with (1.5) we use (2.3), where the itegrad is defied i a iterval (v 0, v 1 ). Let us set σ 0 for the middle poit of this iterval. We first otice the the fuctio 1 + l ( ) 2 v 2/( 1) d v has egative derivative, beig therefore decreasig. Now, we explore the symmetries of the fuctios (( 1)v H) ad 1 ( v 1) H 2 to coclude that the itegrad is egative i (v 0, σ 0 ) ad positive i (σ 0, v 1 ), but with a greater modulus for the egative part. After itegratio, we see that the claim is true. d=1, = 4, H=1/4, l=1 Figure 6: d = 1, = 4, H = 1/4, l = 1. 11

12 We ow collect the results we obtaied for H 0 i the followig propositios ad theorems. Propositio 4.1. (H-CMC vertical graphs, 0 < H 1) The H-CMC vertical graphs ivariat by parabolic screw motios geerated by a curve t = λ(y) are, up to traslatios or symmetries, of oe of the followig types: [ H 1 ( ) ] 1/2 a) t = + l2 y 2 1+l + l 2 y l 1 x l 1 x 1, ( 1) 1 ( 1) H 2 l 0, y > 0, H < 1 (see Figure 3). b) t = H ( 1) 1 ( H 1 1+l 2 y 2 +1 ) 2 l(y), y > 0, H < 1 (see Figure 4). c) t = λ(y, d, l) + l 1 x l 1 x 1, for ( 1) < H < 0. I this case, λ(y, d, l) is defied o (0, y 1 ), is icreasig, is vertical at y 1 ad has a log behavior ear 0. Also, λ(y, d, l) is cocave ear 0 ad becomes covex after some poit i (0, y 1 ) (see Figure 5). d) t = λ(y, d, l) + l 1 x l 1 x 1, for 0 < H ( 1). I this case, λ(y, d, l) is defied o (0, y 1 ) is decreasig o ( 0, τ ) 1/ 1 0 ad icreasig o ( ) τ 1/ 1 0, y 1. It is strictly covex, is vertical at y 1 ad has a log behavior at y = 0 (see Figure 6). Propositio 4.2. (H-CMC vertical graphs, H > 1) The H-CMC vertical graphs ivariat by parabolic screw motios geerated by a curve t = λ(y) are, up to traslatios or symmetries, of the form t = λ(y, d, l) + l 1 x l 1 x 1, where λ(y, d, l) is defied o (y 0, y 1 ), is decreasig o ( y 0, τ ) 1/ 1 0 ad icreasig o ( ) τ 1/ 1 0, y 1. λ is strictly covex i the iterval ad is vertical at y0 ad y 1, with λ(y 0 ) > λ(y 1 )(see Figure 7). Theorem 4.3. (Complete H-CMC hypersurfaces, 0 < H 1 ) a) For each H, 0 < H < 1, the curves of Propositio 4.1 a) geerate a family of complete etire vertical graphs of mea curvature H i H R, that are also complete stable horizotal graphs (see Figure 3). 12

13 d=1, = 3, H=1, l=1 Figure 7: d = 1, = 3, H = 1, l = 1 b) For each H, 0 < H < 1, the curves of Propositio 4.1 b) geerate a family of complete etire vertical graphs of mea curvature H i H R, that are also etire stable horizotal graph (see Figure 4). c) For each H, ( 1) < H < 0, the curves of Propositio 4.1 c) geerate a family of etire horizotal H-CMC graphs i H R ivariat by parabolic screw motios (see Figure 8). d) For each H, 0 < H ( 1), the curves of Propositio 4.1 d) geerate a family of complete o-embedded H-CMC hypersurfaces i H R ivariat by parabolic screw motios (see Figure 9). Proof: The proofs of a) ad b) are trivial. For the other two cases we ca, similar to what we have doe i Theorem 3.2, glue the graphs give i c) ad d) of Propositio 4.1 with their Schwarz reflectio. Stability comes from the well-kow fact that vertical H-CMC graphs are stable. Notice that each parabolic ivariat H-CMC horizotal graph is stable (Propositio 2.2). 13

14 d=1, = 4, H= 1/4, l=1 Figure 8: d = 1, = 4, H = 1/4, l = 1 ad its Schwarz reflectio. d=1, = 4, H=1/4, l=1 Figure 9: d = 1, = 4, H = 1/4, l = 1 ad its Schwarz reflectio. 14

15 Theorem 4.4. (Complete H-CMC hypersurfaces, H > 1 1 ) For each H, H >, the curves of Propositio (4.2) geerate a family of complete o-embedded ad t-periodic H-CMC hypersurfaces i H R ivariat by parabolic screw motios (see Figure 10). Proof: Similar to what we have doe i Theorem 3.2, we ca first take the curve give by Propositio (4.2) ad glue the graph with its Schwarz reflectio. The, we coveietly traslate upwards ad dowwards the obtaied curve successively i order to obtai, after gluig, a complete hypersurface. We recall that the curve obtaied i Propositio (4.2) is vertical at y 0 ad y 1 d=1, = 3, H=1, l=1 Figure 10: d = 1, = 3, H = 1, l = 1, with Schwarz reflectio ad traslatio. Refereces [B-SE] Bérard P., Sa Earp R.; Examples of H-hypersurfaces i H R ad geometric applicatios. Mat. Cotem. 34, (2008). [B-C-C] Bérard, P.; Castillo, P.; Cavalcate, M.; Eigevalue estimates For hypersurfaces i H R ad applicatios, Pacific J. of Math. 253, (2011). [B-G-S] [D1] Barbosa, L.; Gomes, J.; Silveira, A.; Foliatio of 3-dim space forms by surfaces with costat mea curvature, Bol. Soc. Bras. Math. 18 (1987), Daiel, B.; Isometric immersios ito 3-dimesioal homogeeous maifolds, Comm. Math. Helv. 82 (1) (2007),

16 [D2] [J] Daiel, B.; Isometric immersios ito S R ad H R ad applicatios to miimal surfaces, Tras. AMS, 361 (12) (2009), Jost, J.; Riemaia Geometry ad Geometric Aalysis, Uiversitext-Spriger Fourth Editio (2005). [M-O] Motaldo, S.; Ois, I.;Ivariat CMC surfaces i H 2 R, Glasgow Math J. 46 (2004), [N-R] Nelli, B.; Roseberg, H.; Miimal Surfaces i H 2 R Bull. Braz. Math. Soc. 33, (2002) [O] Ois., I.; Ivariat surfaces with costat mea curvature i H 2 R, Aali di Matematica Pura ed Applicata 187 (2008), [R] Roseberg, H.; Miimal Surfaces i M 2 R, Illiois Jour. of Math. 46 (4), (2002), [SE] Sa Earp, R.: Parabolic ad hyperbolic screw motio surfaces i H 2 R, J. Aust. Math. Soc. 85 (2008), [SE-T1] Sa Earp, R.; Toubiaa, E.: Miimal Graphs i H R ad R +1, Aals Ist. Fourrier 60 (7) (2010), [SE-T2] Sa Earp, R.; Toubiaa, E.: Screw motios surfaces i H 2 R ad S 2 R, Illiois Jour. of Math 49 (4) (2005), Maria Ferada Elbert Uiversidade Federal do Rio de Jaeiro - UFRJ ferada@im.ufrj.br Ricardo Sa Earp PUC, Rio de Jaeiro earp@mat.puc-rio.br 16