The Gauss Map and Second Fundamental Form of Surfaces in R 3

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1 Geometrae Dedcata 81: 181^192, # 2000 Kluwer Academc Publshers. Prnted n the Netherlands. The Gauss Map and Second Fundamental Form of Surfaces n R 3 J. A. GAè LVE* and A. MARTIè NE* Departamento de Geometr a y Topolog a, Facultad de Cencas, Unversdad de Granada, Granada, Span; e-mal: {jagalve, amartne}@golat.ugr.es (Receved: 9 February 1999; n nal form: 11 August 1999) Communcated by K. Strambach Abstract. Gven a surface S,amapN from S to S 2 and a conformal structure on S, we solve the problem of the exstence and unqueness of an mmerson x: S!R 3 wth a Gauss map N such that the conformal structure on S s the nduced by the second fundamental form. Mathematcs Subject Class catons (2000): 53A05, 53A07, 58E20. Key words: Gauss map, second fundamental form,weerstrass representaton, harmonc maps. 1. Introducton The propertes of the Gauss map on a submanfold n R n and the extent to whch the Gauss map determnes the mmerson of the submanfold have been of great nterest n Dfferental Geometry (see [1, 7^10, 13^16]). The exstence and unqueness of an mmerson from a surface or a hypersurface nto R n wth a gven metrc (or conformal structure) and a gven Gauss map has been studed by several authors. D. A. Hoffman, R. Osserman and K. Kenmotsu ([8^10]) researched the exstence and unqueness problem for surfaces. They proved there exsts a conformal nonmnmal mmerson from a smply-connected surface n R n wth a gven Gauss map f and only f a set of dfferental equatons dependng on the conformal structure and the Gauss map s sats ed. The unqueness problem for hypersurfaces was also studed by K. Abe and J. Erbacher (see [1]). Our object n ths paper s to study the propertes of the Gauss map of a surface mmersed n R 3, partcularly those related to the geometry of the mmerson and the conformal structure determned by ts second fundamental form. The man problem consdered s the exstence and unqueness of an mmerson x : S!R 3 from a surface S wth prescrbed conformal structure that yelds a gven Gauss map and for whch the second fundamental form s a conformal metrc on S. Among the results we obtan are the followng: * Research partally supported by DGICYT Grant No. PB

2 182 J. A. GAè LVE AND A. MARTIè NE (a) For a smply-connected surface wth non-ero Gauss curvature, the exstence of such mmerson s equvalent to that the Gauss map s a local dffeomorphsm and a thrd-order dfferental equaton nvolvng the conformal structure and the Gauss map s sats ed. Moreover, we recover the mmerson by a representaton smlar to the Enneper^Weerstrass formula. (Theorems 3 and 5). (b) If the set of ponts where the Gauss curvature vanshes has an empty nteror then x s unquely determned, up to smlartes, by the Gauss map and the conformal structure gven by the second fundamental form. Otherwse, f the hypothess about the Gauss curvature s not sats ed, the mmerson s, n general, non-unque. (Corollary 4 and Remark 5). (c) A hypersurface n R n wth non-degenerate second fundamental form has constant Gauss^Kronecker curvature f and only f ts Gauss map s harmonc from the hypersurface wth the metrc gven by ts second fundamental form. Thus, surfaces wth non-ero constant Gauss curvature can be recovered from each harmonc local dffeomorphsm nto the unt sphere usng the above mentoned formula, (Corollares 2, 3 and Theorem 6). E. A. Ruh and J. Vlms proved n [15] smlar results nvolvng constant mean curvature and the conformal structure nduced by the rst fundamental form. Some of these results wll be used n [5] n order to estmate the heght, area, curvature and enclosed volume of a surface wth postve constant Gauss curvature n R 3 boundng a planar curve. 2. Surfaces n R SURFACES WITH POSITIVE GAUSS CURVATURE Let S be a smooth surface and x : S!R 3 an mmerson wth postve Gauss curvature. Then some deleted neghborhood n p of any pont p on S les to one sde of the tangent plane T p S to S at p. A smooth unt normal vector eld N : S!S 2 R 3 s obtaned by assgnng at each pont p of S the unt normal vector tothesamesdeoft p S as n p. Ths orents S and makes the quadratc form s assocated wth the second fundamental form de ned by s p v; w ˆh dn p v ; w; p 2 S; v; w 2 T p S; nto a postve de nte metrc. Here <; > s the usual nner product n R 3. Throughout Secton 2:1, S wll be consdered as a Remann surface wth the conformal structure nduced by s. Let ˆ u v be a conformal parameter, E ˆhx u ; x u ; F ˆhx u ; x v ; G ˆhx v ; x v ; e ˆ s x u ; x u ; 0 ˆ s x u ; x v ; e ˆ s x v ; x v ; 1 where e > 0 and, for nstance, x u Then the Wengarten equatons (see

3 THE GAUSS MAP AND SECOND FUNDAMENTAL FORM OF SURFACES IN R pp. 154^155 n [2] or p. 143 n [17]) state that N u ˆ e EG F 2 Gx u Fx v ; N v ˆ e EG F 2 Fx u Ex v : 2 Let us denote by g : S!C [ f1g the composton of the usual stereographc projecton wth N, thats, g ˆ N 1 N 2 = 1 N 3 ; 3 where N ˆ N 1 ; N 2 ; N 3.Wewllalsocallg the Gauss map of the mmerson. Thus we have: THEOREM 1. Let x : S!R 3 be an mmerson wth postve Gauss curvature, K, g : S!C [ f1g ts Gauss map and ˆ u v a conformal parameter. ˆ 1 g 2 g 1 g 2 g ; ˆ 1 g 2 g 1 g 2 g ; ˆ 2 gg gg 2 4 where x ˆ x 1 ; x 2 ; x ˆ and by bar we wll denote the complex conjugaton. Proof. Use (3) @ ˆ N 1 N 2 1 N 3 N 1 N 2 1 N N 1 N 2 N 1 N 2 N 3 N 1 N 2 N 1 N 2 N 5 where ˆ ˆ Now use (1), (2), N ˆ xu ^ x v jx u ^ x v j ; 6 and K ˆ e 2 = EG F 2 to express the second and thrd terms wthn brackets n (5) as products of rst partals wth respect to u or v of the components of x. After sm-

4 184 J. A. GAè LVE AND A. MARTIè NE pl caton, ˆ 1 N N 1 N 2 1 N 3 N 1 N 2 x 1 x x 1 x 2 : 7 Snce hn; N ˆ1, (3) gves N 1 ˆ g g ; N 2 ˆ g g ; N 3 ˆ : 8 Then (7) can be wrtten 4g 2 ˆ 2g p K x1 x 2 ; 4g 2 ˆ 2g p K x1 x 2 ; or equvalently, 2 g g 2 g 2 ˆ K x1 x 2 ; 9 2 g g 2 g ˆ K x1 x 2 2 : 10 The two rst equatons of (4) are obtaned from (9) and the conjugated equaton of (10). Now, snce (6) gves hx u ; N ˆhx v ; N ˆ0, one has x 3u ˆ N 1x 1u N 2 x 2u N 3 ; x 3v ˆ N 1x 1v N 2 x 2v N 3 ; and usng (8) x 3 ˆ g g 1 gg x 1 g g 1 gg x 2: 11 The thrd equaton of (4) follows from (9), (10) and (11). & A straght computaton gves us

5 THE GAUSS MAP AND SECOND FUNDAMENTAL FORM OF SURFACES IN R COROLLARY 1. Wth the above notaton, the rst and second fundamental forms of the mmerson are gven, respectvely, by ds 2 4 ˆ K 2 g g d 2 g g g g jdj 2 g g d 2 ; s ˆ 4 jg j 2 jg j 2 2 jdj2 : Now, we study the structure equatons of the mmerson. THEOREM 2. If x : S!R 3 s an mmerson wth postve Gauss curvature K, then the Gauss map g sats es: g 4K g 2g g ˆ K g K g : E Moreover, the Gauss curvature s determned, up to multplcaton by postve constants, by the Gauss map. Proof. From Theorem 1, x ˆ x ˆ 1; 2; 3fandonlyf 8K g g3 g g g g 3 g g K 1 g 2 g 1 g 2 g 12 K 1 g 2 g 1 g 2 g 4K 1 g 2 g 1 g 2 g ˆ 0; 8K g g3 g g g g 3 g g K 1 g 2 g 1 g 2 g 13 K 1 g 2 g 1 g 2 g 4K 1 g 2 g 1 g 2 g ˆ 0; 8K g 2 g g g 2 g g K gg gg K gg gg 14 K gg gg ˆ 0: If we take (12) mnus (13) plus 2g tmes (14), then we obtan (E). Moreover, (E) and ts conjugated equaton yeld g log K g log K g ˆ 4 g 2g g ; g log K g log K g ˆ 4 g 2g g : Now, jg j 2 jg j 2 > 0gves log K ˆ 4 gg jg j 2 jg j 2 g g g g 2g g gg : L

6 186 J. A. GAè LVE AND A. MARTIè NE From (L) t s clear that K s determned, up to multplcaton by postve constants, by g. & Remark 1. By Lemma 5 n [12], K > 0 s constant on S f and only f < x ; x > s holomorphc. (Use the fact that ds 2 and s satsfy the Coda^Manard equatons on p. 235 of [2] or p. 144 of [17].) Snce Theorem 3 n [12] shows that < x ; x > s holomorphc on S f and only f N : S!S 2 s harmonc (as de ned n [3], [4] or [12]), t follows from Theorem 2 that N : S!S 2 s harmonc f and only f g g 2g g ˆ 0; holds wherever g 6ˆ 1 on S. Remark 2. If x : S!R 3 s an mmerson wth constant K > 0, then computaton based on Theorems 1 and 2, or formula (9) from [11] gves D s x ˆ 2N; where D s ˆ 1=e D s the Laplacan for s and D s the usual Laplacan n the u,v-plane. Thus, n a smlar way to constant mean curvature (see, for nstance, [6]), the exstence of a smply-connected surface wth constant Gauss curvature K > 0 and boundary a Jordan curve G R 3 s equvalent to solve the followng Plateau problem: x : O!R 3 such that p (a) Dx ˆ 2 K xu ^ x v, (b) det x uu x vv ; x u ; x v ˆ0ˆ det x uv ; x u ; x v (conformalty), (c) 3 s an admssble representaton of the Jordan curve G. where O s the unt dsk and det the usual determnant. Remark 3. The equaton (L) s equvalent to (E). And (E) s sats ed f and only f (12), (13) and (14) are sats ed. THEOREM 3. Let S be a smply connected Remann surface and N : S!S 2 R 3 a dfferentable map. Then, there exsts an mmerson x : S!R 3 wth Gauss map N and such that the conformal structure on S s the nduced one by the second fundamental form f and only f jg j 2 jg j 2 > 0; 4 jg j 2 jg j 2 g g g g 2g g gg ˆ 0; where g s as n (3). Moreover, the mmerson s unque, up to a smlarty

7 THE GAUSS MAP AND SECOND FUNDAMENTAL FORM OF SURFACES IN R transformaton of R 3 and t can be recovered usng the equatons x 1 ˆ x 2 ˆ x 3 ˆ Re 2 1 g 2 g 1 g 2 g d c 2 1 ; Re 2 1 g 2 g 1 g 2 g d c 2 2 ; gg gg Re 4 2 d c 3 ; 17 where log K ˆ Re 8 jg j 2 jg j 2 gg g g g g 2g g gg d l; c 1 ; c 2 ; c 3 ; l are real constants and the ntegrals are taken along a path from a xed pont to a varable pont. Proof. If S s a Remann surface wth conformal structure gven by the second fundamental form of an mmerson x : S!R 3,thenK > 0, so that log K 2 log K =@@ must both be real. The result follows from Theorems 1, 2 and Corollary 1. Conversely, snce S s smply connected, there exsts j : S!R, such that K ˆ e j, satsfyng (L) f and only f (27) s sats ed. Now, from Remark 3, t s easy to check that (L) (or equvalently, (E)) s the complete ntegrablty condton for (5). Moreover, f x 1 ; x 2 : S!R 3 aretwommersonsasabovewthgausscurvature K 1 ; K 2, respectvely, then log K 1 ˆ log K 2 and K 1 ˆ rk 2 for some postve constant r. Thus x 2 ˆ p p r x1 and x 2 ˆ r x1 c, c 2 R 3. & COROLLARY 2. Let S be a smply connected Remann surface. Then S can be mmersed n R 3 wth constant Gauss curvature and the conformal structure on S s gven by ts second fundamental form f and only f there exsts a harmonc local dffeomorphsm from S to S 2. Proof. If x : S!R 3 s an mmerson wth constant Gauss curvature K,snceSsa Remann surface wth conformal structure gven by the second fundamental form, then K must be postve. Consequently, N s a local dffeomorphsm and, from Remark 1, N : S!S 2 must be harmonc. Conversely, f N : S!S 2 s a harmonc local dffeomorphsm, then (usng N for N, f necessary, to make jg j > jg j for the g obtaned from (3)), both (26) and (27) are sats ed, so the mmerson can be calculated usng (17). Fnally, from Theorem 3 log K ˆ 0, so that K must be a postve constant. &

8 188 J. A. GAè LVE AND A. MARTIè NE 2.2. SURFACES WITH NEGATIVE GAUSS CURVATURE Let S be an orentable smooth surface and x : S!R 3 an mmerson wth negatve Gauss curvature. Then s s a Lorent metrc and S can be consdered as a Lorent surface (see [17], p. 13). Now, we choose a unt normal vector eld N on S compatble wth proper s-null coordnates u; v, sothat, s x u ; x u ˆ0; f ˆ s x u ; x v > 0; s x v ; x v ˆ0: Any other proper s-null coordnates ^u, ^v are related to u; v by ^u ˆ ^u u, ^v ˆ ^v v wth ^u 0 u ^v 0 v > 0. The Wengarten equatons now take the form f N u ˆ EG F 2 Fx f u Ex v ; N v ˆ EG F 2 Gx u Fx v : 18 Proceedng as n Secton 2:1, use of (1), (3), (6), (18) and K ˆ f 2 = EG F 2 gves so that x 1 u ˆ 2 Im 1 g2 g u 2 x 1 v ˆ 2 Im 1 g2 g v 2 x 2 u ˆ 2 Re 1 g2 g u 2 x 2 v ˆ 2 Re 1 g2 g v 2 Im gg x 3 u ˆ 4 u Im gg 2 x 3 v ˆ 4 v 2 E ˆ 4g ug u 2 G ˆ 4g vg v 2 F ˆ 2 g ug v g v g u 2 g u g f ˆ 2 v g v g u 2 : Settng x 1 uv ˆ x 1 vu and x 2 uv ˆ x 2 vu, one obtans the followng analog of Theorem 2. THEOREM 4. Let S be an orented surface, x : S!R 3 an mmerson wth negatve Gauss curvature K, then the Gauss map must satsfy: g 4K g uv 2g u g v ˆ K u g v K v g u : Moreover, the Gauss curvature s determned, up to postve constants, by the Gauss map. Remark 4. Fact 3 and Lemma 8 from [12] show that K < 0sconstantonS f and only f < x u x u > and < x v ; x v > depend only on u and v respectvely. (Here agan,

9 THE GAUSS MAP AND SECOND FUNDAMENTAL FORM OF SURFACES IN R use the fact that ds 2 and s satsfy the Coda^Manard equatons.) Lemma 6 and Theorem 4 from [12] then show that K < 0 s constant on S f and only f N : S!S 2 s harmonc. Thus N : S!S 2 s harmonc on S f and only f g g uv 2g u g v ˆ 0 holds wherever g 6ˆ 1. THEOREM 5. Let S be a smply connected Lorent surface and N : S!S 2 R 3 a dfferentable map. Then, there exsts an mmerson x : S!R 3 such that the structure gven by the second fundamental form s the one gven on S and N ts Gauss map f and only f g u g v g u g v > 0; 1 Im g v g u g v g u g uv 2jg v j 2 Im gg u v 1 Im g u g v g u g u g uv 2jg u j 2 Im gg v ; v where g s as n (3). Moreover, the mmerson s unque, up to a smlarty transformaton of R 3. And t can be calculated as follows: x 1 ˆ 2 Im 1 g2 g u 2 du 2 Im 1 g2 g v 2 dv c 1 ; x 2 ˆ 2 Re 1 g2 g u 2 du 2 Re 1 g2 g v 2 dv c 2 ; Im gg x 3 ˆ 4 u 2 du 4 Im gg v 2 dv c 3 ; where 8 log K ˆ Im g g u g v g u g u g uv 2jg u j 2 Im gg v du v 8 Im g v g g u g v g u g uv 2jg v j 2 Im gg u dv l; v c 1 ; c 2 ; c 3 ; l are real constants and the ntegrals are taken along a path from a xed pont to a varable pont. COROLLARY 3. Let S be a smply connected Lorent surface. Then S can be mmersed n R 3 wth constant Gauss curvature and the conformal structure on S s gven by the second fundamental form f and only f there exsts a harmonc local dffeomorphsm from S to S 2. Moreover, the mmerson can be calculated as n the above theorem.

10 190 J. A. GAè LVE AND A. MARTIè NE 2.3. UNIQUENESS OF THE IMMERSION Now we wll prove the followng unqueness result. COROLLARY 4. Let S be a connected, orented surface, w : S!R 3,=1,2,two mmersons wth the same Gauss map and conformal structure of the second fundamental form. If the set S 0 ˆfp2S= dn p s not njectveg has an empty nteror then the mmersons agree, up to a smlarty of R 3. Proof. Let <; > be the nduced metrc on S by the mmerson w, that s, <; > ˆ w <; >, ˆ 1; 2. So, snce S 0 s a closed set, f q 62 S 0 then from theorems 3 and 5 there exsts a smply connected, open neghbourhood of q on whch w 1 ˆ m q w 2 b q, where m q 6ˆ0andb q 2R 3 are constants. Snce the nteror of S 0 s empty and <; > 1 q ˆm q 2 <; > 2 q for all q 2 S S 0, the above equalty s true everywhere. Moreover, m 2 s a dfferentable functon such that dm 2 ˆ 0onS S 0. Therefore, m 2 s constant and m s constant on each connected component of S S 0 and equal to r or r, wthr 6ˆ 0. If p 2 S 0 then there s a neghbourhood U of p such that m s constant on U \ S S 0. Otherwse, there would exst two sequences of ponts p m, p n whch tend to p and dw 1 pm ˆ r dw 2 pm, dw 1 pn ˆ r dw 2 pn. So, we obtan, r dw 2 p ˆ dw 1 p ˆ r dw 2 p. Thus we can assume m ˆ r on S and snce d w 1 rw 2 ˆ0the proof s completed. & Remark 5. If the set of ponts, where dn p s not njectve, has no empty nteror then the corollary does not reman true. For nstance, we can consder the mmersons w 1 u; v ˆ 2cos u; 1 sn u; v ; 2! cos u 2 sn u w 2 u; v ˆ ; ; v ; 2 14 cos 2 u 4 sn 2 u 14 cos 2 u 4 sn 2 u q wth the same Gauss map and s 1 ˆ cos 2u 3 s 2. But, from the expresson of w 1 and snce w 2 u; v 2S 1 R, t s clear that there does not exst a smlarty j such that w 1 ˆ j w Hypersurfaces of Constant Gauss ^Kronecker Curvature It s known that the Gauss^Kronecker curvature of a hypersurface s ero f and only f the second fundamental form s degenerate everywhere. Now, we prove that f the second fundamental form s non-degenerate, then the Gauss^Kronecker curvature s constant f and only f the Gauss map s harmonc for the metrc gven by the second fundamental form.

11 THE GAUSS MAP AND SECOND FUNDAMENTAL FORM OF SURFACES IN R THEOREM 6. Let M n be an orentable n-manfold and x : M n!r n 1 an mmerson wth non-degenerate second fundamental form. We consder M n wth the metrc s, then the Gauss-Kronecker curvature of the mmerson s constant f and only f thegaussmapn: M n!s n s harmonc. Moreover, n that case the Laplacan of N for s s gven by D s N ˆ nhn, where H s the mean curvature of the mmerson. Proof. Let E 1 ;...; E n be an orthonormal movng frame for s n a neghbourhood of a pont p 2 M n,thats, s E ; E j ˆE d j,wthe ˆ1andd j the Kronecker delta, such that re s E j p ˆ0. Let us calculate hd s N; E j at p: hd s N; E j ˆX E he E N ; E j ˆX E hr E r E N; E j ˆ X E E hr E N; E j hr E N; r E E j 19 ˆ X E s E ; r E E j ; where r s the Lev-Cvta connecton of R n 1. If we denote by g kl ˆhE k ; E l and g lk the nverse matrx of G ˆ g kl,thenwe have and hr E N; E l ˆ s E ; E l ˆ E d l ; r E N ˆ X E g l E l : l Snce the Le bracket E ; E j Š p ˆ0, usng Kosul formula, we obtan from (19) and (20) hd s N; E j ˆX ;l g l hr E E j ; E l ˆ1 2 E l he ; E j ˆ 1 2 X ;l X ;l g l E he j ; E l E j he ; E l g l E j g l ˆ1 2 trace G 1 E j G : The Gauss^Kronecker curvature K sats es jkj ˆ1=det G and E j log det G ˆ Ej det G det G ˆ trace G 1 E j G ; where det denotes the usual determnant n R n. Thus hd s N; E j ˆ 1 2 E j log jkj : Therefore, N s harmonc f and only f K s constant.

12 192 J. A. GAè LVE AND A. MARTIè NE Moreover, from (21) hd s N; N ˆX E he E N ; N ˆ X E hr E r E N; N ˆ X E E hr E N; N hr E N; r E N ˆ X g l he l ; r E N ˆ X g l E d l ;l ;l ˆ X g E ˆ nh: & References 1. Abe, K. and Erbacher, J.: Isometrc mmersons wth the same Gauss map, Math. Ann. 215 (1975), 197^ do Carmo, M. P.: Dfferental Geometry of Curves and Surfaces, Prentce-Hall, Eells, J. and Lemare, L.: A report on harmonc maps, Bull. London Math. Soc. 10 (1978), 1^ Eells,J.andLemare,L.:Anotherreportonharmoncmaps,Bull. London Math. Soc. 20 (1988), 385^ Ga lve, J. A. and Mart ne, A.: Estmates n surfaces wth postve constant Gauss curvature, to appear n Proc. Amer. Math. Soc. 6. Hldebrandt, S.: On the Plateau problem for surfaces of constant mean curvature, Comm. Pure Appl. Math. 23 (1970), 97^ Hoffman, D. A. and Osserman R.: The geometry of the generaled Gauss map, Mem. Amer. Math. Soc Hoffman, D. A. and Osserman, R.: The Gauss map of surfaces n R n, J. Dfferental Geom. 18 (1983), 733^ Hoffman, D. A. and Osserman, R.: The Gauss map of surfaces n R 3 and R 4, Proc. London Math. Soc. (3) 50 (1985), 27^ Kenmotsu, K.: Weerstrass formula for surfaces of prescrbed mean curvature, Math. Ann. 245 (1979) 89^ Klot, T.: Some uses of the second conformal structure on strctly convex surfaces, Proc. Amer. Math. Soc. 14 (1963), 793^ Mlnor, T. K.: Harmonc maps and classcal surface theory n Mnkowsk 3-space, Trans. Amer. Math. Soc. 280 (1983), 161^ Osserman, R.:ASurvey of Mnmal Surfaces, Van Nostrand-Renhold, New York, Ruh, E. A.: Asymptotc behavor of non-parametrc mnmal hypersurfaces, J. Dfferental Geom. 4 (1970), 509^ Ruh,E.A.andVlmsJ.:Thetenson eldofthegaussmap,trans. Amer. Math. Soc. 149 (1970), 569^ Wener, J. L.: The Gauss map for surfaces: Part 2. The Eucldean case, Trans. Amer. Math. Soc. 293 (1986), 447^ Wensten, T.: An Introducton to Lorent Surfaces, de Gruyter, Berln, 1996.