Another proof for the rigidity of Clifford minimal hypersurfaces of [S sup n]
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(2) ñ%ó %ôñ°\ó>=õ úöJñ{ñ"õ"RÈñ÷VTñ%û8°Tôoø (x, y) ∈ M n T(x,y) Mkl = (v, w) ∈ Rk+1 × Rl+1 : hx, vi = 0. ÷ "û8"Äñ÷ÈõTù. ν : Mkl → Rn+1. ¤óJþd ô. ν(x, y) =. r. ò. ø. l x, − k. r. k y l. Tô. o hw, yi = 0 .. !. ñó Rwñ÷VTñ ñ÷wþdR ñ"" óJô ó ôõTö<úôóJñÄþdR ñ" V ö Tö ô Mü T M TûOñ÷Èû8 ôk \óJ ôó ôVTöùVR ü \óJ"R ñ°Rwùúñ6Tñó ô (x,y)úóJôkl (v, 0) k ¤óJþdRGúÈñ÷VTñÄóû ñ÷ ô ñ÷w)\ù"R"ó ô û8 A(x,y) (v, 0) = ν (v, 0) ∈ T(x,y) Mkl q ó %Tôy ó d ôþTöJúÈTû q ÷ " 0 û " J ó ñ y ÷ ú öJñóJùöJó óJñ-ø ü ô − kl A(x,y) ÿ k − kl (v, 0) ü+ ñ÷6T Rø X1Tô ÷ ñ÷VTñ q k ó OTô ó d ôþTöJúXTû óJñ÷ úöJñóJùöJó óJñ-ø ÿ ÿ ÿ A(x,y) ÿ l q q <÷VRþd<ñ÷VTñjñ÷O1Tôk úþ¤Tñú" k l ü ÿ-Tÿ H(x, y) = k − kl +l = 0ü l Tö 6 ÷VRþdÄñ÷VTñ 2. kA(x,y) k = k. r !2 r !2 l k +l = l + k = n − 1. − k l. / !#"$" ;45%6$& 5'6 ëàç á¢ç ã à Þ 35 32 687 5 976,819' 76: 7,*;5 "; M -."/ 0 7'91( 3 37:9 ;47,' M = A(Mkl ). ;.5(')#*+#' 5,"-."/ 07'91(&%2/49 ;3 $9 3454 / 5 / 143 5" 86 7 37:9?;7,' / 5,*<1 5=?>"/>@5 ; Mkl k l M /.7:946$& 7A67,*%5,"B' 5'6 9C@D > M ⊂ Sn. A ∈ O(n + 1). ÞFE. ë &-åàäèíîëà á1èìsë jåè\á¢ç ã à &-ãVâõá që ï qèê<ëlãVê<ëoâTè\á1ãVâ ñ÷ þdR ñ"GùVR Tûs\ó@GZ " ôñó ö wñ6Tôd ôoñGþdR ñ" ð ñÈúÈ ôñ" ø ò ò C ∞ (T M ) V ö ô ÷õR¢þTó Tôñ óJþ¤TñóJþdõTû ó Gñ÷õñ" ô" Mü A DA : C ∞ (T M ) × ¤ó J d þ ô ø ÷ " ò DA(V, W ) = DV A(W ) − A(DV W ) ÿ C ∞ (T M ) → C ∞ (T M ) D ó °ñ÷ ð þó*sóJþóJñ6RôôR ñó ô_ô ÷ " R R ô R 1 ¤ þ T ó T o ô G ñ J ó þ T ñ J ó d þ k Tû ó Mü A ¤óJþd ô ø ñ÷Gñ" ô6 2 D A : C ∞ (T M ) × C ∞ (T M ) × C ∞ (T M ) → C ∞ (T M ) ò D 2 (X, Y, Z) = DZ (DA(X, Y )) − DA(DZ X, Y ) − DA(X, DZ Y ). ÷ ð Tùö ó Tô Tû. Tñ. ó sñ÷ÈöJóJô1T%õTù. ¤óJþd ô. ø. ò ∆Ap : Tp M → Tp M Pn−1 A2 p ∈ M ÷ " ó %Tôyñ÷ôõTö "ó ∆Ap (v) = i=1 D Ap (v, ei , ei ) ÿ {e1 , . . . , en−1 } ò Tû Tp M ü ôIH JK.LóJô°ù"1þdR ñ÷VTñ%óû M ⊂ Sn ó ° wóJôóJõTö÷oøù "ú+ûR ñ÷ ô. . ∆A = (n − 1)A − |A|2 A.. ° Rô"ú ôRÄTûñ÷ó %û8 úö . ÿ. G÷VRþd ñ÷VTñ. ∆|A|2 = 2 h∆A, Ai + |DA|2 = 2 |A|2 (n − 1) − |A|2 + |DA|2 ,. Tô ñ÷ "û8". ÿ. Ä 6ñ TóJô ò .
(3) , F @ . . óJô. / 5 ëíîíîè E /46 / & 5 / $6 & 5 6$5 M2 . '=$*;$' 5,"B& %2 /9 ;3 $9 35 4 / #* n $6 & 35,* 17,*+" % 3 >S. |A| = n − 1. ô"! #%$ & ð H*!CKjù"¢þdR . 6ô óJô ÿ ñ÷Èû8öJö ÿ. 56 ;=*7'65* /21435'687:9. >. DA ≡ 0. H KTôGóJô ùZ ô ôoñöJø.*s÷ ô ' óJôõñ÷R" ,+. *XT¾Tô)(° R øo"÷ó ò. ' #*;$' 5," & %2 /9 ;3 9$35 4 / 73 n >0/#3347:9 /1 :/9% 5 - . /46 M / 5 ) S x∈M 5 $ 6 & C / * ' 3 4 ; 6 / 5 < ; 3 ; 4 / 6 7 3 5 . " 0 : 7 9 1 ) ' $* ; $' 5 " & % 2 / 9 ; 3 $ 9 3 5 4 / > kAk2 (x) = n−1 M ÷{ñ÷R" %÷V1þd3+2 úñ ôoñó ôR ó )ô%TûZñ÷%"RúöJñ"O¤"ñûÀ"ú ôoñöJø ú"R "RúöJñ" ÷ ô ÿ õ"÷VT6 ñ" ó ,¢Tñó ôîTû)ñ ÷ *söJó@Z G " ÷oøùZ "ú+û8RRó ¾ôRRR ü .qëãVâTëí. ÿ ÷ "1"ô_ó ñ÷VTñÈñ÷ Rô\óJñó ôÙôhñ÷ ô Tû<ñ÷ ÷VTùZ ù 6Tñ"Èó Äõö ñ 1ó ¾ñ"wþd óû0ø Tôl"ÄöJó d öJø ñ"÷ úù óJô Rwùúñ6Tñó ô ñ÷VTô Tôoø ñ÷ ÿ ù"ùZ ñø ñ÷VTñ{õ1ø 6÷VT6 ñ" ó ,1 ñ÷*söJó@Gb" wóJôóJõTö÷oøù "ú+ûRR ü ô ñ÷ó ¾Rw úôó 1Tñó ô Zÿ ÿ óJöJö ¤óJþdkTôhTöJñ" ôVTñóJþd ù"Tû)Tûñ÷ó °ñ÷R" ü ô_Tû ñ÷ õTóJô \@ó GZ " ôRR ÿ óJñ÷uñ÷_ù" þó ú ù"Tû8 H KkTô H*!CK ú ù"TûR%ôñ%ú"ÈóJôñ"4¤6Tñó ôhTû\ó ñó úñó ô¾ó ü ü óJñ°RXôñ°ú" ý " ôóJú 4 ÷R" ü ÷ó 1 TûXñ÷ó ô ù"Tûò " öJó R ôÙñ÷ û8öJö óJôhö wõ ò ÷¤" ÿ ÿ ÿ ù"Tûó ¾wñ6Tó ¤÷oñ+û8 T" Rwùúñ6Tñó ô ü LR H Kbû0{ ñ6TóJö ü ÿ. . ëíîíîè65 . /#3. ";-587 D /A1. $*9 /496: " / ;%,')' ')#*(n + 1) × (n + 1) +#' 5,"!& %2 /9 ;3 9$35 4 / 7 3 9$345 <4 / >. B "; 5 n M = {x ∈ S : hBx, xi = 0}. -."/ 07'91(')#*+#' ,5 " &%2/9<;<3. E. :}èç0à. /46 984 Sn. 5. '. 5'6 9C7D ,5 *<1 6$&/C* ";5 M. âTëoïCåì á. ôéñ÷ó õ6R ñó ô ÿ ¤óJþd Tô TöJñ" ôVTñóJþdlù"TûÄTû ÷R" ü ÿ óJöJö{ù"¢þd ñ÷ ñ÷R" ø ÷ óJôÅñ÷VTñ óû ó îwóJôóJõTöÄ÷oøù ""ú+û8RhóJô óJñ÷ ò ÿ M Sn ÿ ñ ÷ ô ñ ÷ " ° \ ) ó " ñ s G R ô " 6 ñ T o ô ñ J ó o ô d þ ñ ó ö ø w ñ ó ° õ T ñ > ó ) kAk2 = (n − 1) ò B ú6÷lñ÷VTñ ÷Gû8öJö óJôyö wõ °ù"¢þd ô óJ0ô M ⊂ {x ∈ Sn : hx, B0 xi = 0} ü ÿ ÿ H KTô H*!CK ü ; *7'65,*<1 /21435'687:9 > ëíîíîè<; 8 /46 n / 5=')$*;$' 5" &%2/9<;<3 9#35 4 / &84C& /#3?6$& / 4 7 5:9 5,* 6=M1 /⊂ 9C= S5'6:= : / 73 "; :1 / * 6::4 5,"#"/%?> /9 7,5=?>"/> 3 37:9 5,"#" 56$&A /C* 55'6 /1 :/9% 27,#*6 56$DA & /492/ (V, 5:92/ W/ )D5 =46:"/0% V, W ∈ C ∞ (T M ) p ∈ M 6:+ 7 2 9C#*<4 2 5"B443 91 5'6 3 9 /<; 5,*<1 >A@ 7:9 /27 :/95 6$& /<; / 2%9$*<4 2 5,"B443 91 5'6 3 92/<; 5:9 / 4 7* ;685,* 6+3 3 * 4 6: 7*;5 6$& /C% κ17 1 *7'6 1 κ/ 22/C* 1 7* 6$& / 2 7#* 6 5 5,*<1 >. . p. B 927 73 >. κ1 κ2 = −1. ý ¾Tôø p0 ∈ M LóJôR |A|2 (p0 ) 6= 0 Tô M ó {wóJôóJõTö ñ÷ ô κi (p0 ) 6= û8 6 Tô ü ð ñ hñ6Tôd ôoñ þdR ñ"?V ö ôRuóJô κj (p0 ) i j V, W ò ô ó ¤÷ ÷ Tû ú"÷uñ÷VTñ ò p0 |V (p)| = |W (p)| = 1 Ap (V (p)) = κi V (p) Tô Tô LóJôR ó A (W (p)) = κj W (p) DV W (p0 ) = DW V (p0 ) = 0 ü DA ó ôñó 1pTöJöJø ,1 " %÷VRþd°ñ÷VTñ ÷ " 0 û " ó û ó ) ñ ÷ oÿ K(p0 ) DZ A(U ) = A(DZ U ) ü.
(4) J. "R ñó ôVTö úþTñú"kTû. M. óJô ñ÷Gùö TôÄùVTôôR. κj (p0 )K(p0 ) = = = =. ò. ø. V (p0 ), W (p0 ) . ñ÷ ô. hDW DV V − DV DW V, A(W )i(p0 ) hDW DV A(V ) − DV DW A(V ), W i(p0 ) −hDW DV W − DV DW W, A(V )i(p0 ) κi (p0 )K(p0 ).. ñó RGñ÷VTñ{óJôlñ÷Ä"RRô ñ" ù ÿ Gú"R ñ÷Äøw ñø Tûñ÷Ä÷VTùZÄùZ 6Tñ" T ôlñ÷Gû ñ°ñ÷VTñ þTôó "÷R ü ô ñ÷Äñ÷óJ"l"ñ" ù ú"R ñ÷køw ñó R¾Tû DA ÿ ø @GTú6¾úVTñó ô ñ÷ úþTñú"kñ" ô" ü LóJôR ñ÷ ô d ñ ÿ κi 6= κj K = 0 ü ñ÷VTñ ÷ G ö w õ 0 û J ö ö % 0 û " ñ ÷ ó % ö ° ñ V ú T ñ ó ô ü 2 0 = 1 + κi (p0 )κj (p0 ) ü ÿ. 2 /49 ;3 9$3454 /5 5 *7'9' 5," 3 *; 6 ëíîíîè ) /46 n /?5 &% ν : M → Rn+1 /A4687:9 7 /C"71 ,5 " 7,* 6 M 5⊂5,*<S1 5 . 7 D / 1 / 4 8 6 : 7 9 > 4 / 6 3 ; 1 / 7 % * / 5 M w ∈ Rn+1 lw : M → R % 5,*<1 T 5,*<1 fw : M → R w : M → Rn+1 lw (x) = hx, wi 5,* 1 T > /#3 5,*<1 fw (x) =5+hν(x), 6$& /C* wi . w (x) = w − hx, wix − hν(x), wiν(x). x∈M. v ∈ Tx M. v(lw ) = hw, vi = hw T (x), vi v(fw ) = −hA(wT (x)), vi Dv wT (x) = −lw (x)v + fw (x)Ax (v).. and. B 927 7 3 > ñó Rsñ÷VTñ T ó ñ÷sñ÷:dôVTöù" 2R ñó ô Tûñ÷<þdR ñ" ô w (x) w Tx M T ô ñ÷ "û8"ÄóJñ¾ôR¾ ñ6Tôd ôoñþdR ñ"(V ö ô ð ñ Mü α : (−, ) → M ò w úþd ú"÷ ñ÷VTñ Tô 0 Ä÷VRþdGñ÷VTñ α(0) = x α (0) = v ü ÿ v(lw ) =. ð ó d ÿ. ó ". v(fw ) =. dlw (α(t)) dt. = t=0. dhα(t), wi dt. = hα0 (0), wi = hv, wi. t=0. dfw (α(t)) dt. = t=0. dhν(α(t)),wi dt. = t=0. . dν(α(t)) dt. ,w t=0. . = hdνx (v), wi = −hAx (v), wi = −hAx (v), wT (x)i = −hA wT (x) , vi.. . kTö " ÷VRþd ñ÷VTñ. T. Dv w (x) =. . dwT (α(t)) dt. =. . d (w − lw (α(t)) α(t) − fw (α(t)) ν (α(t))) dt. T. t=0. T. = −lw (x)v − fw (x)dνx (v) = −lw (x)v + fw (x)Ax (v).. ÷ó °ö ñ{úVTñó ô ù"1þdR°ñ÷Gö wõ ü. 2.
(5) , F @ . . kT"Gô ÿ. "1\ø ñ" ù"¢þd. ÷R" . ü. î÷V1þd ñ÷VTñ Tô B 927 7 3 > ÷R" øuö wõ Tô ÿ DA ≡ 0 M óJñ÷ ÷V ) ñöJøÅñ ù óJô óJùVTöG úþ¤Tñú"R Tô óJô þd øéùZóJôoñ Tû κ1 κ2 M ÿ ÿ Ù Tôø ö ñ ú Rôó ñ÷möJóJô1T ñ6Tô+û8õTñó ô κ1 κ2 = −1 ü ý x ∈ M ¤ó J d þ ô ø û8kTôoø T : Rn+1 → Rn+1 v ∈ T M T (x) = −ν(x) ò T (v) = A(v) Tô ñó R ñ÷VTñÈû8G þd xø ÷VRþd T (ν(x)) = −x + (κ1 + κ2 )ν(x) ü x ∈ M Zÿ RRwùZ¤" n+1 {ñ÷ \óJ"R ñ¾ú TûOñ÷Gñ÷"Rkú ùVRR R ∈ R} Tô ñ÷ ô ÷VRþdÄR"ñ6 öJó ÷R ÷ ò ñ"°TxôlM1 6{tx ÷lú : tùV R {tν(x) : t ∈ R} T ÿ ò ÿ ò ñ÷ "û8" ñ÷Xñ6Tô+û8õTñó ô ó Oúôó ú öJø ôR ü ð ñ ñ÷%ùVR T S(n+1) ò Tûqøw ñó õTñó RR üOý %Tôoø óJô ö ñ (n + 1) × (n + 1) x M B(x) ∈ S(n + 1) õñ÷ õTñó>)mú6÷mñ÷VTñ û8 Tôø ñó R ñ÷VTñkóû ò B(x)w = T (w) w ∈ Rn+1 ü ó Èñ÷ 1Tôôó 1Tö "ó GTû n+1 ñ÷ ô e1 = (1, 0, . . . , 0) . . . en+1 = (0, . . . , 0, 1) ò R ÷ " B(x) = {bij } ÿ bij = =. . T (ei ), ej = T eTi + xi x + νi ν , ej A eTi , ej − xi νj + νi (−xj + (κ1 + κ2 )νj ).. " Tô T"Gñ÷Èû0úô ñó ô ¤óJþd ô ø xi : M → R νi : M → R x (x) = hx, ei i Tô Tô T ó wñ÷ ñ÷:dôVTö¾ù" 2+R ñó ò ô Tiû ô ü νi (x) = hν(x), ei i ei ei ñ ó R { ñ V ÷ T ) ñ ñ ÷ { À û ú ô ñ ó ô T ô T " % ñ ÷ { À û ú ô ñ ó ô T ô ô RõóJTôxñM ÷ xi νi l ei f ei ö wõ ü ÷V1þd ñ÷VTñ B : M → S(n + 1) ôRÄ "ñ÷ õTùÙô M ü ÿ óJöJö ù"¢þdÈñ÷VTññ÷ó XõTù ó XRôñ6Tôñ ø ÷ óJôwñ÷VTñXóû ñ÷ ô v(bij ) = 0 ü %1Tô"ú ÿ óJñ÷úñOö ¤"OTûd ò ô 6TöJóJñÿ ø ñ÷VTñ A(v) v=∈κTvx M ð ñ A(v) = κ2 v ü 1 ú 2õñ÷ "ñX1" J ó J ö ö ú " ¾ ñ ÷ w õ Ä J ó y ô ñ ÷ { 8 û J ö ö óJô ð ÿ A(v) = κ1 v ü ÿ ÿ Rwùúñ6Tñó ô ü v A eTi , ej = v A eTi , eTj = A Dv eTi , eTj + A eTi , Dv eTj = hA(−xi v + νi A(v)), ej i + A eTi , −xj v + νj A(v) = hA(−xi v + νi A(v)), ej i + ei , −xj A(v) + νj A2 (v). = −κ1 xi hv, ej i + νi κ21 hv, ej i − κ1 xj hv, ei i + νj κ21 hv, ei i .. ñ÷VTñ. ÷l"RRôéúVTöJóJñø óJôÅñ÷ ù" þó ú Rwùúñ6Tñó ôéû8öJö ûÀ" ÿ DA ≡ 0 ü v(xi νj ) = v(xi )νj + xi v(νj ) = hei , viνj − xi hA(v), ej i = hei , viνj − xi κ1 hv, ej i.. ñ÷ û ñ.
(6) . . (κ1 + κ2 )v(νi νj ) = (κ1 + κ2 )(v(νi )νj + νi v(νj )) = −(κ1 + κ2 ) hei , A(v)iνj + νi hA(v), ej i = −(κ1 + κ2 )κ1 hei , viνj + hv, ej iνi = −κ21 hei , viνj + hv, ej iνi + hei , viνj + hv, ej iνi .. ô_ñ÷wö ñÄúVTöJóJñø ÿ wú"R_ñ÷ û ñGñ÷VTñ κ1 κ2 = −1 ü * ò óJôóJôlñ÷R" úVTñó ô d ññ÷VTñ "óJwóJö TGT2¤ú ôñÄ÷ ñ÷VTñ v(bij ) = 0 ü v(bij ) = 0 ÿ ÿ ÷ ô ÷ " 0 û " û0õTöJö Tô ÿ A(v) = κ2 v ü B(x) = B0 x ∈ M M ⊂ M0 = LóJôR ó TôhóJôoþd ñó ö õTñó>) k÷VRþd ñ÷VTñ B ò ÿ M x ∈ Sn : B x, x = 0 ü ó {k÷oøùZ "ú+û80R ü LóJôR ó XwóJôóJ0õTöjñ÷ ô ó {Tö 6kwóJôóJõTö ü øö wõ)J 0 M M0 ó ° *söJ@ó Gb" wóJôóJõTöq÷øùZ "ú+ûR ü ÷ó {Rwùö ñ"R¾ñ÷Èù"Tû ü M0 2 ó { ëíîèâ ÷Èù"Tû oRþdGTû ÷R" ñúVTöJöJøy÷ %ñ÷VTñ%óû ÷øùZ "ú+ûRÄTû n Àôñ{ôÿ RRR"6TóJöJø wóJôóJõTö Tô ó ñÿ ÷ ô óJñ÷ M S DA ≡ 0 M ú óJöJó 1TöZ ÷ " ó sTôy÷:d ôRúsùZöJøôwó Tö M ⊂ f −1 (0) f : Rn+1 → R Tû<ò 4¤"R ü ÷ó ¾ó ¾ô Tÿ û)ñ÷k\þ¤Tôoñ6dRGTû)ñ÷ ô ù"Tû)óJô Rôoñ6ñ óJñ÷ ÿ ÿ ñ÷Äö lô ü æjàqãõìÀëoäíîëà á. ÷ Túñ÷ úö öJó d ñ"ñ÷VTô *ö ó ô ó ¾û8°óJñ"ôVTô ó TöúùùZñTôlñ÷ ÿ "û8 "RÈû8%÷ó {Rw ôoñ"ô ñ÷ó %ùVTùZ ü ë &-ëoâTëàædëoï. H*!CK)L ü L ; 2+& H K. * ÷ ô ü s ü ' *XT TôIL ü (¾ ò Røo÷ó @ $*;#' 5," ;<3 ' 5,*; 347"@1; 73 5 /9 / ! 6$& ;/ 4 7,*<1 33 * 15,' /C* 685," 347'9' 73 4 7*;685,*6 " /C* 66$& ý úô ñó ôVTö ôVTö ø\ó {Tô ¾ ö Tñ"R ý ó ö O" ü *ô\û ü ü Lñ"ô LùóJôd ! #%$ & ùù ü # $ ü ü Èü ð ÿ "ô 7 4 5 " 9 76,81, :6 % $6 & /27:9 C/ ' ; 347'9 ')$*;$' 5"<& %2 /9 ;3 9$35 4 /<; ôô ü Tû Tñ÷ ü ! ## ùù !%$ ! #%$ ü. H Kk ". ü !#"%$'&)()*+,(.-/ %01&2*3 Rn *3(4&.56+, 7 (981 :+&2*<;-= >-?+,@0ABB;jü õTùùZ1T{óJô ñ÷ 2úôVTö3C ôñ"4¤6 óEDôFC ü. H J,KHG ü LóJô @ $*;$' ,5 "JI5:9C8/46: /; $*LK C/ ' ,5 *+*; 5* ' 5 *; 37,"71; # # ùù ! & ü ! . ôô ü Tû. Tñ÷ ü. . MONPRQ?S?S)NUTVLWQ2XZY1[#\/]1P2^ `_ Ja cb @ 2d=e gf / ih /d ./>jlk9m %n6jloopBooqd%. rp /<s%/%n, ut3v gf /n qvin .
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