Minimal Surfaces and Gauss Curvature of Conoid in Finsler Spaces with (α, β)-metrics *

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1 Advace Pre Mathematc -5 Plhed Ole Jly ( Mmal Srface ad Ga Crvatre of Cood Fler Space wth (α β)-metrc Dghe Xe Q He Departmet of Mathematc Togj Uverty Shagha Cha Emal: _dghe877@6com Receved Decemer 6 ; receved March 8 ; accepted March 6 ABSTRACT I th paper mmal mafold Fler pace wth (α β)-metrc are tded Epecally helcod are alo mmal (α β)-mkowk pace The the mmal rface of cood Fler pace wth (α β)-metrc are gve Lat the Ga crvatre of the cood the -dmeo Rader-Mkowk pace tded Keyword: Iometrcal Immero; Mmal Smafold; (α β)-metrc; Cood Srface; Ga Crvatre Itrodcto I recet decade geometry of mafold Fler geometry ha ee rapdly developed By g the Bema-Hadorff volme form Z She [] trodced the oto of mea crvatre ad ormal crvatre for Fler mafold Beg aed o t Berte type theorem of mmal rotated rface Rader-Mkowk pace wa codered [] Later Q He ad Y B She ed aother mportat volme form e Holme-Thompo volme form to trodce oto of aother mea crvatre ad the ecod fdametal form [] Th Q He ad Y B She cotrcted the correpodg Berte type theorem a geeral Mkowk pace [] The theory of mmal rface Ecldea pace ha developed to a rch rach of dfferetal geometry A lot of mmal rface have ee fod Ecldea pace Mkowk pace a aaloge of Ecldea pace Fler geometry A atral prolem to tdy mmal rface wth Bema-Hadorff or Holme- Thompo volme form M Soza ad K Teelat frt tded the mmal rface of rotato Rader- Mkowk pace ad ed a ODE to characterze the BH-mmal rotated rface [5] Later the otrval HT-mmal rotated hyperrface qadratc (α β)- Mkowk pace are tded [6] N C ad Y B She ed aother method to gve mmal rotatoal hyperrface qadratc Mkowk (α β)-pace [7] However thee eample oly coder the pecal (α β)- metrc ether Rader or qadratc Therefore what the cae wth the geeral (α β)-metrc? Project pported y NNSFC (o 979 o 776) ad the Natral Scece Fodato of Shagha (o 9ZR) The ma prpoe of th paper to tdy the cood (α β)-pace It clde mmal mafold Fler pace wth geeral (α β)-metrc ( F ) ad the Ca crvatre Rader-Mkowk -pace We preet the eqato that characterze the mmal hyperrface geeral (α β)-mkowk pace We prove that the cood Mkowk -pace wth metrc F mmal f ad oly f t a helcod or a plae der ome codto Fally mlar to [7] we gve the Ga crvatre of cood Rader-Mkowk -pace ad pot ot that the Ga crvatre ot alway opotve o mmal rface Prelmare Let M e a -dmeoal mooth mafold A Fler metrc o M a fcto F : TM atfyg the followg properte: ) F mooth o TM \ ; ) F y F y for all ; ) The dced qadratc form g potvely defte j g : gj y d d () gj : F j yy Here ad from ow o F F mea F j y y F ad we hall e the followg coveto of j y y de rage le otherwe tated: j ; m Copyrght ScRe

2 D H XIE Q HE The projecto π :TM M gve re to the pll- ack dle π TM ad t dal π TM whch t over TM \ We hall work o TM \ ad rgdly e oly oject that are varat der potve recalg y o that oe may vew them a oject o the projectve phere dle SM g homogeeo coordate I π TM there a gloal ecto F d y called the Hlert form whoe dal l l l y F called the dtghed feldthe volme elemet dvsm of SM wth repect to the Remaa metrc g ˆ the pll-ack of the Saak metrc o TM \ ca e epreed a dv d d () SM g j : det d F d d () d : y dy dy dy () The volme form of a Fler -mafold (M F) defed y d V : d : d (5) c M ( )-phere c SM deote the volme of the t Ecldea S SM= y ytm Let (M F) ad M F e Fler mafold ad f : M M e a mmero If F y F fdfy for all y TM\ the f called a ometrc mmero It clear that g y g y f f (6) j for the ometrc mmero f y f y f e the orthogoal complemet of wth repect to g ad et j f : M F M F Let f π TM π TM π f TM j k h fj fk G G h (7) h g h h F f k fj G ad G j are the geodec coeffcet of F ad F repectvely We ca ee that h π TM (ee () []) whch called the ormal crvatre Recall that for a ometrc mmero f : M F M F we have (ee formlae () ad () of Chapter V [8]) k k j G fj G (8) k lk fl g g From (7) t follow that j h p fj G (9) p : f Set h d d S M c () F whch called the mea crvatre form of f A ometrc mmero f : M F M F called a mmal mmero f ay compact doma of M the crtcal pot of t volme fctoal wth repect to ay varato vector feld The f mmal f ad oly f Mmal Hyperrface of (α β)-space Here ad from ow o we coder geeral (α β)-metrc Let F a potve C fcto o j a y j j a j If the F a Rader metrc If a Ecldea metrc ad parallel wth repect to F a locally Mkowk metrc ad (M F) called a (α β)-mkowk metrc By [9] F a Fler metrc atfe f ad oly f ( ) Let g Adet aj g det gj F It have ee proved ([9]) that g H A H () () () I the followg part we wll dc mmal hyperrface Mkowk pace wth (α β)-metrc Let f : M F M F e a ometrc mmero F Copyrght ScRe

3 D H XIE Q HE a y Sce f a ometrc mmero we get F f F j a a a f f j j y f Note t rrface M F let e e the t ormal vector feld of f M wth repect to ad e e the t ormal vector feld of M wth repect to g repectvely That hat (M F) a hype of f g f a g g There et a fcto y j o SM ch that g a a g The g a ( ) From aove we kow that f mmal f ad oly f From () ad () ad a mlar way a [5] we ca get h gh g fj G g j fj G a a h d F (5) S M j A A HA g g HA A A The (5) eqvalet to a S M j fj G d (6) If F a (α β)-mkowk metrc the G I Mkowk-(α β) pace f mmal f ad oly f f j SM j yy d (7) Theorem Let (M F) e a hyperrface of M F ad F e a (α β)-mkowk metrc The f : M F M F a mmal mmero f ad oly f f j j S ( ) ( ) d (8) S a phere ch that Now we coder the cood -dmeoal (α β)- Mkowk pace parallelg to -a Set F y y y y ad a cotat Let f co v v hv h v a kow fcto The v v v co v h co f y y y co v v v co v h y co vy v y vy co v y h Ame that y co y π the h y y y y h y Note that the ormal vector of the rface h v hcov h h h ad Set W f f f v v f = cov v h j j S co d (9) Copyrght ScRe

4 D H XIE Q HE The (8) eqvalet to f W f W () Sce S ymmetrc wth rep ect to y ad a fct o oly depe dg o y W S However d Therefore () ecome to hw W mpole Recall th W y S at g d ad y ot detcally vahg we ca ota W The h h cv d c d are artrary cotat Theorem Let V F e a (α β)-mkowk pace F y ad f co v v hv e a cood The f mmal f ad oly f f a helcod or a plae Remark From theorem we ca affrm that a helcod mm al o t oly Ecldea (α β) k y pace t alo Mkow pace Th a teretg relt for mmal rf ace Bt whether the relt hold f the codto y ot atfed? Now we coder the followg codto: y y y cov v y co v v y are ot all zero To mplfy the com- ptato we oly dc qadratc (α β)-metrc: F k Set B co v v B co v v The (8) ecome a eqato repect to : C v C v 5 5 C v C v C v C v 5 C5 Bh 8 5 C B h B hh () C k B B k BB kbb k BB h Bh B k Bh k πb B 8 π kb π kb k h C B h h 9 k B Bh k B hh 5 C h h 8 B k B kb k B k h k B kπb k π B 8 9 kb k h h C k B B h Sce () vald for ay we ca ota C 5 v If or the B or B ch that hv Therefore whe are ot all zero hv cot That to ay a mmal cood hyperrface a plae wth repect to the gve metrc aove Theorem Let V F e a (α β)-mkowk pace F k atfyg y y y ( are ot all zero) The a mmal cood hyperrface V F a plae Ga Crvatre of Cood Rader -Space A we all kow the Ga crvatre of a mmal rface opotve every Ecldea pace The a atral prolem are: whether th fact hold for mmal rface Mkowk-Rader -pace? I th ecto we tdy the Ga crvatre of cood Mkowk-Rader -pace arod -a the drecto # # that y Coder the cood f v co v vhv ad v S Let e df e df v The y e e gve a atral coordate v o t taget dle I th ecto we hall e the coveto that j ad Bede the otato : : v ad y : y : are alo ed Note that the dced -form f o the rface cloed The the Rcc crvatre teor of F f F gve y ([] Page 8) Copyrght ScRe

5 D H XIE Q HE Rc Rc F r Fr Rc deote the Rcc crvatre teor of the j dced Remaa metrc f r j ad j deote the coeffcet of the covaret dervatve of wth repect to The the Ga crvatre of the rface gve y K y f v wth repect to f Deote z Rc y F K F r Fr () () K deote the Ga crvatre ad z f The j j cov v z v co v hv z z z v z co v co v Notg that the Ga crvatre compted Ecldea pace a follow: LN M K EG F v h L z M z N z E z z F z z G z z By drect comptato we ca ota K h h Meawhle the coeffcet of a j h a kl a kl () f are gve y h aj z zj It eay to verfy that k kl a z z By a drect comptato we have j l j j Sce j j From Bede h j z hh h h j h h hh h h h jk k j jk j k we have h h h h h h h h h h h h h h h h h j r j y h r y h j k jk h h h h h y h h The from () ad () we ota the followg theorem Theorem Let V F e a Rader-Mkow pace wth y k the Ga crvatre of the cood f v co v vhv at f v drecto of y e e gve y K y 8F F 5 6 F Copyrght ScRe

6 D H XIE Q HE 5 h h h h h h hh h h h h 5 h 6 h h h h h Note that a helcod mmal f ad oly f t a cood wth repect to (α β)-metrc ( y ) Let hv cv d (c a cotat) the the Ga crvatre of th rface gve y K y F F c c c (5) c c c However for a gve pot f v TS K y y? drecto of K ) If ) If the whch K y K y < for ay c ; Sce j j F a y c c Eqato () ecome K y c c c c c If c let the K y we ca alo make c c c c c c c c c K y ; Otherwe let the the c K y c c c c we ca make c c c c the K y I m the Ga crvatre ot opotve ay REFERENCES [] Z She O Fler Geometry of Smafold Mathematche Aale Vol No 998 pp do:7/8 5 [] M Soz a J Sprck ad K Teelat A Berte Type theorem o a Rader Space Mathematche Aale Vol 9 No pp 9-5 do:7/8--5- [] Q He ad Y B She O the Mea Crvatre of Fler Smafold Chee Joral of Cotemporary Mathematc Vol 7C 6 pp - [] Q He ad Y B She O Berte Type Theorem Fler Space wth the Volme form Idced from the Projectve Sphere Bdle Proceedg of the Amerca Mathematcal Socety Vol No 6 pp do:9 /S [5] M Soza ad K Teelat Mmal Srface of Rotato a Fler Space wth a Rader Metrc Mathematche Aale Vol 5 No pp 65-6 do:7/ [6] Q He ad W Yag The Volme Form ad Mmal Srface of Rotato Fler Space wth (α β)-metrc Iteratoal Joral of Mathematc Vol No pp - do:/s967x68 [7] N C ad Y B She Mmal Rotated Hyperrface Mkowk (α β)-space Geometrae Dedcata Vol 5 No pp 7-9 do:7/ [8] H Rd The Dfferetal Geometry of Fler Space Sprger-Verlag Berl 959 [9] Z She Laderg Crvatre S-crvatre ad Rema Crvatre I: Z She Ed A Sampler of Fler Geometry MSRI Sere Camrdge Uverty Pre Camrdge [] Z She Dffertal Geometry of Spray ad Fler Space Klwer Academc Plher Berl Copyrght ScRe