Linearization and explicit solutions of the minimal surface equation

Publicacions Ma emá iques, Vol 36 (1992), 39-46 . LINEARIZATION AND EXPLICIT SOLUTIONS OF THE MINIMAL SURFACE EQUATION A bs ac ALEXANDER G . REZNIKOV We show ha he appa a us o suppo unc ions, usually used in con ex su aces heo y, leads o he linea equa ion Ah+2h = 0 de- sc ibing locally ge ms o minimal su aces . He e A is he Laplace- Bel ami ope a o on he s anda d wo-dimensional sphe e . I explains he exis en e o he sum ope a ion o minimal su - aces, in oduced ecen ly . In 4-dimensional space he equa ion Oh + 2h = 0 becomes inequali y whe e e he Gauss cu a u e o a minimal hype su ace is nonze o . 0 . In oduc ion Recen ly g ea p og ess was achie ed in he in es iga ion and con- s uc ion o examples o minimal su aces in R3 [1]-[3] . The Gauss map usually plays a signi ican ole and i s singula i ies in a sense con ol opology i he su ace is comple e [4] . I was also no iced [5], [6] ha he e exis s a "sum" ope a ion M l + M2 o wo mini nal su aces Ml, M2 . I may seem o be s ange, o he usual o m o he minimal su ace equa ion is essen ially nonlinea . T ue, gi en a con o mal minimal map R2 D U x> M C R3 we ha e a linea equa ion áx = 0 [7] . Howe e , he condi ion o con o mali y is nonlinea i sel . In his pape , we show ha appa a us o suppo unc ions usually used in con ex su ace heo y leads o he linea and comple ely in e- g adle equa ion o minimal su aces in R3 . We a e able o w i e down an explici o mula desc ibing locally all minimal su aces wi h non anishing cu a u e which is qui e di e en om he Weie s ass desc ip ion . We hope ou me hod will be use ul in global p oblems, oo . I au oma ically implies he exis en e o he sum ope a ion . The main pa o his wo k was done du ing he au ho 's isi o Li huania in 1987 . I wish o hank P o esso F . Weiksa o ui ul dis- cussions . I also wish o hank he e e ee o his e y aluable ema ks, in pa icula , o indica ing o me ha he ela ion o equa ion (6) o minimal su aces was independen ly s a ed in [7] . 40  A .G . REZNIKOV 1 . The main equa ion Le M be a smoo h o ien ed hype su ace in RN and G : M -> SN-1 be i s Gauss map . Then [7] G is a local di eomo phism whe e e he Gauss cu a u e o M is nonze o . F om now en, we assume ha his condi ion holds a e e y poin o M . Then G becomes a co e ing o e i s image G(M) . Le U C G(M) be a simply-connec ed p ope domain, hen G -1 (U) is a disjoin union o open Vi, i E I, and GI S .: Vi - U is a di eomo phism which we call Gi . We supply S N-1 wi h he canonical me ic o cu a u e 1 . De ini ion . By suppo unc ion hi : U- . R we mean ( 1 )  hi(n) = (Gi 1(n),n) . Lemma 1 .  The unc ion hi(n) de e mines G7 1 (n) in he ollowing way : (2)  G~ 1 (n) = h i (n)n + g ad h i (n) , whe e g ad hi(n) is compu ed in e ms o he me ic o SN -1 and looked a as a ec o in RN . (3)  hi (Gi(y)) = (y,Gi(y)) . I X E T~, :SN- 1 hen, di e en ia ing (3) along X we ob ain o P oo .. Le G~ 1 (n) = x . Fo yE Vi we ha e by (1) (g ad h i (G i (x)) , Gi . X) = (X, G i (x)) + (x, Gi . X ) Bu (X, Gi(x)) = 0 by he de ini ion o he Gauss map, so (4)  ((x - g ad hi(Gi(x))) , Gi*X) = 0 . By he nondegene acy condi ion, Gi . maps isomo phically T x M on o TG ( x )S N-1 . The la e space coincides wi h TM as a subspace o HN so, o some i E R, we ha e x - g ad hi (Gi (x)) = pCi (x) , G~ 1 (n) = pn + g ad hi(n) Taking scala p oduc wi h n and accoun ing (1) and (g ad hi(n), n) = 0 we ob ain (2) . MINIMAL SURFACE EQUATION  4 1 Lemma 2 . Le A(x) be he second undamen al ope a o in T,,M . Then (5)  A(x) = (hi(n)E+Hesshi(n)) -1 . He e n = Gj(x), E is he iden i y ope a o in T,,S N-1 =T x M and Hessh i (n) is he Hessian ope a o [8] on he sphe e S N-1 . P ooL Deno e o a momen Fi = Gi 1 on U . By (2) we ha e Fi(n) _ hi(n)n+ g ad hi(n) . Le Z E T n SN -1 . By he de ini ion o he Hessian ope a o , Hess hi(n) (Z) = (Oz g ad hi) (n), whe e 0 is he Le i-Ci i a connec ion on he sphe e . Fo any ec o ield on he sphe e we ha e [8] Vz = z + ( , Z)n, whe e ¿ deno es usual di e en ia ion in RN . So by (2), (Hess hi(n))(Z) =F i , Z - (g ad hi(n), Z)n - hi(n)Z + (g ad hi(n), Z)n = Fi . Z - hi (n) Z , o Fi, Z = h i (n) Z + (Hess h i (n)) (Z) . Le Y = Fi, Z, hen Z= G, Y . By de ini ion o A, A(x)Y =G i ,Y = Z, which p o es (5) . Theo em 1 . Suppose N = 3 and M is minimal . Then o any p ope simply-connec ed domain U C G(M) andany b anch h i we ha e (6)  áh i + 2hi = 0 , whe e A is he Laplace-Bel ami ope a o on he sphe e 52 . Con e sely, i h is a solu ion o (6) in an open U CS 2 hen he o mula x(n) = h(n)n+g ad h(n) de e mines a smoo h map om U o R3, which is ei he a cons an o a con o mal and minimal imme sion ou side a locally cni e se o isola ed singula i ies (b anch poin s) . P oo . . M is minimal i A(x) = 0 e e ywhe e .  Fo an in e ible ope a o A in 2-space we ha e A -1 =  1, AA so by (5), A(x) = 0 de is equi alen o 0 = (h ¡ (n) E + Hess h¡ (n» = 2hi + Ah i . This p o es he i s s a emen o he heo em . Now Suppose h yields (6) . Deno e F(n) = h(n)n + g ad h . F om he p oo o he Lemma 2 weknow ha F, (n) = h(n)E + Hessh(n) .  In pa icula , i neans ha F, (n) maps T, S 2 o i sel and is symme ic in T n S 2 .  Fu he , by (6), F, (n) = 0 . No e ha any symme ic ope a o wi h he ze o ace in 2-space is ep esen ed by a ma ix  b ba ) in any o hono mal basis and is hus con o mal, so F is con o mal, and o n E U ei he ank F, (n) = 2 o F, (n) = 0 .  Deno e by Z he se o poin s whe e F, = 0 .  As (6) is 4 2  A . G . REZNIKOV ellip ic, F(n) is analy ic along wi h h(n), so i F is non-cons an , hen Z is nowhe e dense . Ou side Z, F(n) ís a con o mal ímme sion and we ha e jus shown ha T F (  ,)F(U) =T  , S 2 , so, o he Gauss map we ha e G(F(n)) = n . Hence he suppo unc ion h o F(U - Z) is h(n) = (h(n)n + g ad h(n), n) = h(n) . F om he i s pa o he heo em we see ha Flu-z is minimal . I ollows ha F is ha monic in U - Z, bu F is analy ic in U and Z is nowhe e dense, hence F is ha monic e e ywhe e in U . Locally in con o mal coo dina es (x, y) we ha e n E Z <=> á  = áF = 0, hence Y~(n) = 0 whe e z = x+ iy and .F is holomo phic and Re .F = F, so Z is locally ini e . Theo em 2 . Suppose N = 4 and M is minimal . Then o any p ope simply-connec ed U C G (M) and any b anch hi, Ahi(n) + 2hi(n) doesn' change sign in U . P oo .. Le Ai (x), 1<_ i <_ 3, be he p incipal cu a u es o M in x . Then I 1 + A2 + 3 = 0 by minimali y condi ion .  Suppose Ah i (n) + 2hi (n) = 0 .  somewhe e in U and le x = Gz 1 (n) .  As A -1 (x) _ a, a2+aLa3+- 2-3 d<,, A  we ob ain by (5) ha A1l 2 + 113 + 1 21 3 = 0 .  This '(x) implies 1 + ; 2 + a3 = ( , 1 + 1 2 + A3) 2 - 2(A1 , 2 + 1A3 + 2 , 3) = 0 in x which is impossible by nondegené acy condi ion . We u n o applica ions o ou esul . Le M1, M2 be wo' minimal su aces in R3 such ha G(M1) n G(M2) has a nonemp y in e io in S 2 . In [5] and [6] hei sum M1 + M2 is de ined by pa ame iza ion x(n) = Gi 1 (n) + G2 1 (n) . Le h 2 ~ 1) and h~ 2 ) be wo b anches o suppo unc ions o M1 and M2 espec i ely . Then by (2) we ha e x(n) _ h(n)n+ g ad h(n) . whe e h = hW + h~ 2 ) . Nex , bo h h ~ 1) and h~ 2 ) yield (6) which is linea , hence h yields (6), oo . Theo em 1 implies hus he minimali y o M1+M2 . Mo eo e , gi en a minimal M and any Killing ec o ield Z in S 2 we can de ine he de i a i e su ace MZ by ( 7 )  h(M') = h' , which is also minimal . 2 . Applica ions MINIMAL SURFACE EQUATION  43 Example . Le M l be a ca enoid de ined in euclidean coo dina es (x, y, z) by he equa ion x 2 + y 2 = (ch z) 2 . Then di ec compu a ions show ha (8)  hl (n) = h(n x , n y , ni) = 1 - nZ a c an h nZ , and G(M1) = S 2 - { p} whe e p = (0,0,1) . Le g i E SO(3), i = 2, . . . , m, gi = id be o a ions such ha he se s { g i p} a e pai wise disjoin . Le h =  hl og i , and V =S 2 - U{ gip} . Then we ha e he i ollowing P oposi ion 1 .  The su ace 0 : n ~--+ h(n)n + g ad h is a comple e minimal su ace in R3 wi h only a ini e numbe o b anch poin s, and i s Gauss image omi s p ecisely 2m-poin se U{ gip} (compa e [5]) . i P oo .: Fi s we no e ha in some neighbou hood o p, V) canno ha e b anch poin s . Indeed, suppose de (h(n)E + Hess h(n)) = 0, hen o some ec o X E T  S?, IIXII = 1, (hi(n)E+ Hesshi(n))X = - E7 .>2 (hi(n)E+ Hess hi(n)) X : We know ha o i >_ 2 hi(n) does no ha e singula i ies nea p, so in some neighbou hood o p and o some cons an C we would ha e 11 (hi(n)E+Hessh i (n))XII < CIIXII . Deno e (hi (n)E+ Hess h i (n)) X=Y, hen 11 (hi (n)E+Hess h i (n)) -1 Y11 > C-1 IIYII . Ac ually (hi (n)E + Hess h, (n» - ' is he second undamen al ope a o o he ca enoid M i , as we saw in Lemma 2, he e o e, i s eigen- alues a e ± -K(n), whe e K(n) is he Gaussian cu a u e a G -1 (n) . As i is well-known (and easy o e i y) ha K is decaying o ze o a in ini y, he abo e inequali y is impossible . O cou se, he same is ue abou all he singula poin s gip, hence, being locally ini e, he se o b anch poin s should be ini e . Nex , as he me ic o ca enoid is comple e, we ha e 7 ~~ (hi(n)E + Hess hi (n)) y( )jj = co o any cu e y : [0, oc) - S 2 such lia  lim y( ) = p .  Hence he same a gumen s ac show ha his is ue o h ins ead o h, and inally, V) is comple e . Now conside he Ennepe su ace [7] e : R2 _ R3 . The composi- ion G o e wi h he Gauss map coincides wi h he in e se s e eog aphic p ojec ion 7 - 1 : R2 _ S 2 - {p}, so K :~ 0 and he suppo unc- ion h is de ined in S 2 _ {p} . S aigh o wa d compu a ions show ha K - -0 on E, hence we can apply he same cons uc ion o ob ain 44  A .G . R ZNIKOV P oposi ion 2 . Fo any gi en ini e se E C S 2 he e exis s a com- ple e minimal su ace in R3 wi h only a, ini e numbe o b anch poin s whose Gauss image omi s p ecisely he se E . The conjec u e o Meeks [4] s a es ha o e e y k > 1 he e exis s an embedded minimal su ace homeomo phic o a compac mani old punc- u ed in k poin s . The p oblem o Osse man [7], sol ed by Fujimo o [4], asks whe he he s a emen o ou P oposi ion 2 holds o some smoo h comple e minimal su ace (wi hou b anch poin s) . Concluding Rema ks . 1 . Ou main equa ion (6) admi s sepa a ion o a iables . Fix xo E S 2 , hen in sphe ical pola coo dina es nea ( , cp) nea xo he sphe ical me ic becomes d 2 + sin 2 dcp 2 and he Laplace- Bel ami ope a o becomes A = + s°ñ + S -, 1 ~, . ~ .  Sub s i u ing ( , cp) =  sin g( , w) we ob ain ha A + 2 = 0 is equi alen o g' . +g (4  + 2 ) + 5  = 0 . By Fou ie me hods one inds whe e C,,( ) sa is y Cm ( )e"I' 1 9 ) _ sin 2 + 4  0 . 2 . A a he su p ising phenomenon ollows om ou desc ip ion . Namely, i Oh + 2h = 0 in an open U C S 2 , hen he "Monge- Ampe e" 0 = de (hE+Hessh) sa is ies some second o de PDE . Indeed, we know om Theo em 1 ha he su ace M pa ame ized by F(n) : n H h(n)n, + g ad h(n) is minimal and i s cu a u e a F(n) is V) -1 (n) . Le 's pull back on S2 he minimal su ace's me - ic . We will ob ain g = e(n)go, because he Gauss nap F -1 (n) is con o mal (he e go is he sphe ical me ic) . To compu e p(n), , , e no e ha F* ds = ±O(n) dso, whe e ds, dso a e he a ca ele- men s on M,S 2 espec i ely . Hence, g = 10(n)Igo . The e o e, he cu a u e o -V)(n)go is 0 -1 (n) (compa e wi h Rice¡-Cu bas o Theo em, [10]) . This is equi alen o some PDE . 3 . Suppose M is a comple e minimal su ace o ini e o al cu a u e . Then by he heo em o Osse man [7] he Gauss mapG _ : M S 2 ex ends o a holomo phic map G o he comple ion M, and MINIMAL SUR ACE EQUATION  45 M - M is ini e, say M - M = {p l , . . . p z} .  Le G C M be he ini e se _ o b anch poin s_  o G, say G={q1 . . . qk} .  Then N = M - ((M - M) U£) (is he ini e co e ing o S 2 - G((M - M) U G) .) Nex , he suppo unc ion h(n) becomes single- alued on N and we see ha e e y comple e minimal su ace o ini e o al cu a u e de e mines a solu ion o he equa ion Oh+ 2h = 0 in a ini e co e ing o he s anda d sphe e punc u ed in a ini e numbe o poin s . 4 . Conside he ia me ic g = dx 2 +dy 2 -dz 2 in R2,1 . I o a su ace M C R 2,1 , g ¡ M is posi i ely de ined, hen he e exis s a co ec ly de ined Gauss map G om M o he hype boloid S : x2 + y 2 - z 2 = -1 . I is well-known ha g 1 S is he s anda d hype bolic me ic . Jus as be o e we can de ine a suppo unc ion h(n) . Fo mula (2) in his case eads G~ 1 (n) = -hi(n) + g adhi(n) . Fo mula 5 becomes A (x) = (-hi(n)E + Hess hi(n))-1 and (6) becomes Ohi(n) - 2hi(n) = 0 o minimal su aces M wi h ime- like no mals . Re e ences 1 .  COSTA C ., Bull . Soc . B as . Ma h . 1 5 (1984), 47-54 . 2 .  Ho MAN D ., MEEKS W . III, Bull . Ame . Ma h . Soc . 1 2 (1985), 134-136 . 3 . HO FMAN D ., MEEKS W . III, J . Di e en ial Geom . 21 (1985), 109-127 . 4 .  FUnMOTo H ., J . Ma h . Soc . Japan 40 (1988), 237-249 . 5 . MEEKS W . III, "P os . In . Con,g . Ma h . Be keley 1986," 1987, pp . 551--558 . 6 . ROSENBERG H . TOUBIANA E ., J . Di e en ial Geom . 28 (1988), 115-132 . 7 .  LANGEVIN R ., ROSENBERG H ., Duke Ma h . J . 57 (1988), 819-828 . 8 .  LANGEVIN R ., LEVITT G ., ROSENBERG H ., Banach cen e publ . 20 (1987) . 9 .  OSSERMAN R ., "A Su ey o Minimal Su aces," Van Nos and Reinhold Company, New Yo k, Cincinna i, To on o, London, Mel- bou ne, 1969 . 46  A .G . REZNIKOV 10 . GROMOLL D ., KIANGCNBGRG W ., MGYGR W ., "Riemannische Ge- ome ie in'G o len," Lec u e No es in Ma h, 55, Sp inge , Be lin, Heidelbe g, New Yo k, 1968 . 11 . LAWSON, H .B ., JR ., "Lec u es on Minimal Submani olds," Publish o Pe ish, 1980 . Pe manen add ess : School o Ma hema ical Sciences Raymond and Be e ly Sackle I'acul y o Bxac Sciences Tel A i Uni e si y Ra na -A i , Tel A i 69978 ISRAEL Cu en add ess : In e na ional Cen e o Theo e ical Physics P .O .B . 586 Mi ama e 34100 T ies e ITA LY P ime a e sió eb ,da el 29 ('Oc ub e de 1990, da e a e sió ebulla el 22 de Maig de 1991