Research Article Some Surfaces with Zero Curvature in H 2 R

Transcription

1 Applied Mathematics, Article ID 54294, 5 pages Research Article Some Surfaces with Zero Curvature in H 2 R Dae Won Yoon Department of Mathematics Education and RINS, Gyeongsang National University, Jinju , Republic of Korea Correspondence should be addressed to Dae Won Yoon; dwyoon@gnu.ac.kr Received 24 December 203; Accepted 26 February 204; Published 24 March 204 Academic Editor: Chong Lin Copyright 204 Dae Won Yoon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study surfaces defined as graph of the function z=f(x,y)in the product space H 2 R. Inparticular,wecompletelyclassify flat or minimal surfaces given byf(x, y) = u(x) + V(y),where u(x) and V(y) are smooth functions.. Introduction Homogenous geometries have main roles in the modern theory of manifolds. Homogenous spaces are, in a sense, the nicest examples of Riemannian manifolds and have applications in physics []. To underline their importance from the mathematical point of view we roughly cite the famous Thurston conjecture. This conjecture asserts that every compact orientable 3-dimensional manifold has a canonical decomposition into pieces, each of which admits a canonical geometric structure from among the eight maximal simple connected homogenous Riemannian 3-dimensional geometries [2]. The Riemannian product space H 2 R is one of the eight model spaces. Constant mean curvature and constant Gaussian curvature surfaces are one of the main objects which have drawn geometers interest for a very long time. Recently, the study of the geometry of surfaces in H 2 R is growing very rapidly, and the interest is mainly focused on minimal and constant mean curvature surfaces [3 9]. The purpose of this paper is to study surfaces defined as graph of the function z=f(x,y)in the product space H 2 R. InSections4 and 5 we classify minimal and flat surfaces defined as f(x, y) = u(x) + V(y), where u(x) and V(y) are smooth functions. 2. Preliminaries Gaussian curvature,given by g H = (dx2 +dy 2 ) y 2. () The hyperbolic space H 2, with the group structure derived by the composition of proper affine maps, is a Lie group and the metric g H is left invariant. Therefore, the product space H 2 R is a Lie group with the left invariant product metric g= dx2 +dy 2 y 2 +dz 2. (2) On the other hand, an orthonormal basis of left invariant vector fields on H 2 R is E =y x, E 2 =y y, E 3 = (3) z with the only nontrivial commutator relation [E,E 2 ]= E. It follows that the Levi-Civita connection of H 2 R is expressed as E E =E 2, E E 2 = E, E E 3 =0, E2 E =0, E2 E 2 =0, E2 E 3 =0, E3 E =0, E3 E 2 =0, E3 E 3 =0. For any vectors X=x E +y E 2 +z E 3 and Y=x 2 E + y 2 E 2 +z 2 E 3 in H 2 R the cross-product is defined by (4) Let H 2 ={(x,y) R 2 y>0}be the upper half plane model of the hyperbolic plane endowed with the metric, of constant X Y=(y z 2 y 2 z )E +(x 2 z x z 2 )E 2 +(x y 2 x 2 y )E 3. (5)

2 2 Applied Mathematics 3. Graphs in H 2 R Let us consider a surface Σ parametrized by φ (x, y) = (x, y, f (x, y)), (x, y) Ω, (6) where Ω is a domain in H 2 and f:ω R is a smooth function. Then Σ is a surface defined as graph of the function f defined on Ω H 2.Inthiscase,wehave e := φ x = (, 0, f x )= y E +f x E 3, e 2 := φ y = (0,, f y )= y E 2 +f y E 3. It follows that the coefficients of the first fundamental form of Σ are given by E=g(φ x,φ x )=f 2 x + y 2, F=g(φ x,φ y )=f x f y, G=g(φ y,φ y )=f 2 y + y 2. Also, the unit normal vector field U to Σ is given by where (7) (8) U(x,y)= f x ωy E f y ωy E 2 + ωy 2 E 3, (9) ω= y 2 y2 (fx 2 +f2 y )+. (0) By a straightforward calculation, we obtain e e = y 2 E 2 +f xx E 3, e e 2 = y 2 E +f xx E 3, e2 e 2 = y 2 E 2 +f yy E 3, () which imply that the coefficients of the second fundamental form of Σ are L=g( e e,u)= yf xx f y ωy 3, Thus, from (8) and(2) thegaussiancurvaturek and the mean curvature H are, respectively, K= ω 4 y 6 ((yf xx f y )(yf yy +f y ) (yf xy +f x ) 2 ), H= 2ω 3 y 4 (( + y2 f 2 y )f xx y(f 2 x +f2 y )f y 2y 2 f x f y f xy +(+y 2 f 2 x )f yy). (3) Proposition. Let Σ be a surface defined as graph of the function f:ω H 2 R. ThenΣ is a minimal surface if and only if ( + y 2 f 2 y )f xx y(f 2 x +f2 y )f y 2y 2 f x f y f xy +(+y 2 f 2 x )f yy =0. (4) Proposition 2. Let Σ be a surface defined as graph of the function f:ω H 2 R.ThenΣ is flat if and only if (yf xx f y )(yf yy +f y ) (yf xy +f x ) 2 =0. (5) Remark 3. Some examples are satisfying the ODE (4) studied in [7]. Also, examples in Lorentz product space H 2 R can be found in [0]. 4. Minimal Surfaces Defined by f(x,y) = u(x) + V(y) Let Σ be a surface in H 2 R parametrized by φ (x, y) = (x, y, u (x) + V (y)) (6) for all y>0,whereu(x) and V(y) are smooth functions. We suppose that Σ is a minimal surface. Then, from (4)wehave the following minimal surface equation: ( + y 2 (V ) 2 )u y((u ) 2 +(V ) 2 ) V +(+y 2 (u ) 2 ) V =0. (7) In order to solve it, divide first by +y 2 (V ) 2 =0;thenweget y((u ) 2 +(V ) 2 ) u +y 2 (V ) 2 V + +y 2 (u ) 2 +y 2 (V ) 2 V =0, (8) for all x, y Ω. Differentiating with respect to x,weobtain u +2( y2 V yv +y 2 (V ) 2 ) u u =0. (9) M=g( e e 2,U)= yf xy +f x ωy 3, N=g( e2 e 2,U)= yf yy +f y ωy 3. (2) First of all, we suppose that u =0on an open interval; that is, u(x) = ax + b, a, b R.Inthiscase, from(7) weobtain V a 2 y y +a 2 y 2 V +a 2 y 2 (V ) 3 =0. (20)

3 Applied Mathematics 3 We put V (y) = p(y).thenthelastequationcanbewrittenas p Its general solution is given by From this, we thus have y +a 2 y 2 (a2 p+p 3 ) =0. (2) p=± c a +a 2 y 2. (22) c 2 ( + a2 y 2 ) V (y) = ± c a +a 2 y 2 dy, (23) c 2 ( + a2 y 2 ) where c R. Now, we assume that u =0 on an open interval, and divide (9)byu u.itfollowsthat u V yv u u +2y2 =0. +y 2 (V 2 (24) ) Hencewededucetheexistenceofarealnumberk R such that u =2ku u, y 2 V yv = k(+y 2 (V ) 2 ). (25) Let us distinguish the following cases according to k. Case. If k=0,thenu =0and yv V =0.Itfollows that u(x) = a x 2 +b x+c (a =0,b,c R).IfV =0,then V(y) = a 2 (a 2 R). Inthiscase,from(7) weobtaina =0; it is a contradiction. If V =0,thenwegetV(y) = (/2)b 2 y 2 + c 2 (b 2 =0,c 2 R). Insuchcase,(7) ispolynomialequation on x and y. From the coefficients of y 4 and the constant term we have 2a b 2 =0and 2a +b 2 =0, which imply a =0 and b 2 =0. It is a contradiction. Case 2. If k =0, then from the first equation in (25)wehave where d R.Let u =e 2ku+d, (26) u= 2k ( d + ln g) (27) be any solution of (26), where g is a smooth function. Then (26)canberewrittenas We put p=g.then,wehave gg (g ) 2 =2kg 3. (28) dp dg g p=2kg2 p. (29) We again put t=p 2.Inthiscasetheaboveequationbecomes and its general solution is given by Thus, we get dt dg 2 g t=4kg2 (30) t=g 2 (4kg+c ). (3) dg dx =±g 4kg + c. (32) After an integration, we can find g= c 4k tan2 (8k 2 c (±x+c 2 )) c 4k, (33) where c 2 R.Bycombining(27)and(33), we thus have u (x) = 2k [ d + ln ( c 4k tan2 (8k 2 c (±x + c 2 )) c 4k )]. (34) Now,weconsiderthesecondequationin(25). Since y> 0,weyield V + k y 2 y V +k(v ) 2 =0. (35) We put p=v.then,theaboveequationbecomes p + k y 2 y p+kp2 =0. (36) Since k =0, without loss of generality we take k=or k=. Subcase i.letk=.wedothechange p= y + h(y), (37) where h is a nonzero smooth function. Then, (36) canbe rewritten as the form Thus, its general solution is h h=. (38) y h (y) =y(ln y+c ), (39) where c R. So,p = (/y) + (/y(ln y+c )) and from its integration we can obtain where c 2 R. Subcase ii.letk=.weput V (y) = ln (c 2 y ln (y+c )), (40) p= y + h(y), (4)

4 4 Applied Mathematics Figure : A minimal surface defined by (34)and(44). Figure 2: A flat surface defined by (52)and(55). where h is a nonzero smooth function. Then, (36)becomes and its general solution is given by where c R.Thus,wehave h h= (42) y h (y) = y(ln y+c ), (43) V (y) = ln (c 2 y ln (y+c )), (44) where c 2 R.Thesurfacegivenby(34)and(44)isshownin Figure. Consequently, we have the following. Theorem 4. Let Σ be a surface defined as graph of the function f(x, y) = u(x) + V(y). IfΣ is a minimal surface, then Σ is parametrized as where φ (x, y) = (x, y, u (x) + V (y)), (45) () u(x) = ax + b and V(y) = ± (c a +a 2 y 2 / c 2( + a2 y 2 ))dy with a, b, c R,or (2) u(x) = (/2k)[ c 3 + ln((c /4k)tan 2 (8k 2 c (±x + c 2 )) (c /4k))] and V(y) = ± ln(d y ln(y + d 2 )) with k =0,c,c 2,c 3,d,d 2 R. 5. Flat Surfaces Defined by f(x,y) = u(x) + V(y) Let Σ be a surface defined by (6). Assume that Σ is a flat surface. Then, from (5) we have the following flat surface equation: y(yv + V )u (yv + V ) V (u ) 2 =0. (46) In order to solve it, differentiating with respect to x,wehave y(yv + V ) d dx (u ) d dx ((u ) 2 )=0. (47) Thus, there exists a nonzero real number k such that d dx (u ) =k d dx ((u ) 2 ), y(yv + V ) = k. (48) Fromthefirstequationin(48), we get u =k(u ) 2 +c, (49) where c R.Weputp=u,anditfollowsthatweyield dp du = kp2 +c. (50) p From this, the general solution is p=± k e2k(u+c2) c k, (5) where c 2 R. We can assume that c = 0.Fromthelast equation we can easily obtain (see Figure 2) where c 3 R. u (x) =± k (ln ( k(x+c 3 )) + kc 2 ), (52)

5 Applied Mathematics 5 In order to solve the second equation in (48), divide by y 2 and put q=v. Then, we get q + y q= ky 2 (53) and its general solution is given by q= y ( k ln y+d ), (54) [8] B. Nelli and H. Rosenberg, Minimal surfaces in H 2 R, Bulletin of the Brazilian Mathematical Society, vol. 33, no. 2, pp , [9] D. W. Yoon, Minimal translation surfaces in H 2 R, Taiwanese Mathematics,vol.7,no.5,pp ,203. [0] A. L. Albujer, New examples of entire maximal graphs in H 2 R, Differential Geometry and its Applications,vol.26,no.4,pp , where d R. From this, we thus obtain (see Figure 2) V (y) = 2k (ln y)2 +d ln y+d 2, (55) where d 2 R. As a conclusion, we have the following. Theorem 5. Let Σ be a surface defined as graph of the function f(x, y) = u(x) + V(y). IfΣ is a flat surface, then Σ is parametrized as φ (x, y) = (x, y, u (x) + V (y)), (56) where u(x) = ±(/k)(ln( k(x + c )) + kc 2 ) and V(y) = (/2k)(ln y) 2 +d ln y+d with k =0,c,c 2,d,d 2 R. Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper. Acknowledgment This paper was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) fundedbytheministryofeducation,scienceandtechnology (202RAA ). References [] J. Gegenberg, S. Vaidya, and J. F. Vázquez-Poritz, Thurston geometries from eleven dimensions, Classical and Quantum Gravity,vol.9,no.23,pp.L99 L204,2002. [2] W. Thurston, Three-Dimensional Geometry and Topology, vol. 35 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 997. [3] J. M. Espinar, M. M. Rodríguez, and H. Rosenberg, The extrinsic curvature of entire minimal graphs in H 2 R, Indiana University Mathematics Journal,vol.59,no.3,pp ,200. [4] I. Fernández and P. Mira, Harmonic maps and constant mean curvature surfaces in H 2 R, The American Mathematics,vol.29,no.4,pp.45 8,2007. [5] Y. W. Kim, S. E. Koh, H. Shin, and S. D. Yang, Helicoidal minimal surfaces in H 2 R, Bulletin of the Australian Mathematical Society,vol.86,no.,pp.35 49,202. [6] S. Montaldo and I. I. Onnis, Invariant CMC surfaces in H 2 R, Glasgow Mathematical Journal,vol.46,no.2,pp.3 32,2004. [7] S. Montaldo and I. I. Onnis, A note on surfaces in H 2 R, Bollettino della Unione Matematica Italiana B,vol.0,no.3,pp , 2007.

6 Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization