Manifolds with Bakry-Emery Ricci Curvature Bounded Below

Transcription

1 Advaces Pue Maemacs, 6, 6, ://wwwscog/joual/am ISSN Ole: ISSN P: Maolds w Baky-Emey Rcc Cuvaue Bouded Below Issa Allassae Kaboye, Bazaaé Maama Faculé de Sceces e Tecques, Uvesé de Zde, Zde, Nge Déaeme de Maémaques e Iomaque, Uvesé Abdou Moumou, Namey, Nge ow o ce s ae: Kaboye, IA ad Maama, B (6 Maolds w Baky-Emey Rcc Cuvaue Bouded Below Advaces Pue Maemacs, 6, ://dxdoog/436/am666 Receved: Augus 4, 6 Acceed: Ocobe 4, 6 Publsed: Ocobe 7, 6 Coyg 6 by auos ad Scec Reseac Publsg Ic Ts wok s lcesed ude e Ceave Commos Abuo Ieaoal Lcese (CC BY 4 ://ceavecommosog/lceses/by/4/ Oe Access Absac I s ae we sow a, ude some codos, M s a maold w Baky- Émey Rcc cuvaue bouded below ad w bouded oeal uco e M s comac We also esabls a volume comaso eoem o maolds w oegave Baky-Émey Rcc cuvaue wc allows us o ove a oolologcal gdy eoem o suc maolds Keywods Baky Émey Rcc Cuvaue, Myes Teoem, Volume Comaso Teoem, Toologcal Rgdy Teoem Ioduco Le ( M, g be a comlee Remaa maold ad : M R a smoo uco A Baky-Émey Rcc cuvaue s deed by Rc = Rcg + ess, wee Rc sads g e Rcc cuvaue o ( M, g ad ess deoes e essa o Te uco s called e oeal uco Fo smlcy, deoe Rc g by Rc Te Baky-Émey eso occus may dee subjecs, suc as duso ocesses ad Rcc low We s a cosa uco, e Baky-Émey Rcc eso becomes e Rcc eso so s aual o vesgae wc geomec ad oologcal esuls o e Rcc eso exed o e Baky-Émey Rcc eso As a exeso o Rcc cuvaue, may classcal esuls Remaa geomey asseed ems o Rcc cuvaue ave bee exeded o e aalogous oes o Baky-Émey Rcc cuvaue codo I [] G We ad W Wyle oved some comaso eoems o smoo mec DOI: 436/am666 Ocobe 7, 6

2 I A Kaboye, B Maama measue saces w Baky-Émey Rcc eso bouded below I s ae we esabls a Myes ye eoem o maolds bouded below by a egave cosa Teeoe we ove a s a geealzao o e eoem o M Lmocu [] o Tadao [3] I e secod a o s ae we esabls a codo o ocomac maold w oegave Baky-Émey Rcc cuvaue o be deomoc o e eucldea sace Mas Resuls Te ollowg eoem s a smla eoem oved [4] ad [5] ad s a geealzao o Myes eoem Teoem Le (,,e M g dvolg be a mec sace suc a Rc ( k Suose a M coas a ball B ( x, o cee x ad adus suc a e mea cuvaue m o e geodesc see S ( x, w esec e wad og omal veco vees m <( k I ee exss a cosa c suc a c e M s comac ad ( k ( + k l dam( M + ( k m( x wee = su (, x S I s well kow a ee exs ocomac maolds w oegave Rcc cuvaue wc ae o e oologcal ye Recall a a maold M s sad o ave e oologcal ye ee s a comac doma Ω wose bouday Ω s a oologcal maold suc a M \ Ω s omeomoc o Ω [, + A moa esul abou oologcal eess o a comlee Remaa maold M s due o Abesc ad Gomoll (See [6] Le be a oeal uco o M sasyg ( x cd ( x, o some oegave cosa c ad a xed o Vol ( B (, Se ( x = ( x + 3 d( x, ; le α = lm ad α ( M = α ω I s ae we sow a oologcal gdy eoem o ocomac maolds w oegave Baky-Émey Rcc cuvaue as ollow: Teoem Le ( M, g,e dvolg be a mec sace suc a Rc ( x Suose α M > ad K k o a o M ad ( x cd (, x I o all > Vol ( B ( ω e M s deomoc o 3 Poos, < + l 8k + e k α ( M Poo o eoem Te ecques used e oo o s eoem ae based o ( 755

3 I A Kaboye, B Maama [4] ad [5] Fs, le cosuc a comaso model sace Le m ad ake a eal ad R { + } soluo o e deeal equao m S be e u see so a < < R Le φ be e ( k φ( φ = (3 w al values φ = a ad φ = a < Suose φ ( o all [ R, ece m O S [ R, wee m g ca a a k ( a k ( φ ( = + k e + k e k a a we dee a Remaa mec eso by s e sadad mec o m ( u, ca (4 g = φ g (5 m S m Tus e Remaa comlee maold S [ R, cosa equal o ( m k m Fo all s, e yesuace S { s} m S [ R, veco w ouwad og veco e w og osve s s w Rcc cuvaue ( s ( s w mea cuvaue φ ( us, = ( m (6 φ Now le ove, ude e yoeses o eoem, a M s comac Le y be a abay o M\ S(, ; ee exss a o x S(, a d( ys, (, = d( xy, Le γ be a mmal geodesc jog x o y; γ ( = exx ( u w u TS x (, ad u = Le ( γ, e,, e be a aallel ooomal ame alog γ ad se d Y ( = ex e ( s ece x ds s= Y s a (, suc S -Jacob eld alog γ Te geodesc γ ca be exed o a mmal geodesc γ sag a : ( u T exv Y s a (, γ = ex v w = x (see [4], Pooso 3 ad S -Jacob eld alog γ ad oly Y ca be exeded o a Jacob eld alog γ, ull a I e geodesc ola coodaes e volume eleme ca be we as: wee dθ ( θ θ d vol = A, d d (7 s e volume om o e u see (, θ = ece A (, θ ( ( θ d A (, θ g S ad A Y Y e d volg = A, θ d dθ = e A, θ d dθ We ave d l,, = A = = Y Y = m (8 (, θ ( θ d A l ( A (, θ = = m ( (9 d A, To ove e eoem we use e ollowg eoem oved by G We ad W 756

4 I A Kaboye, B Maama Wyle [] Teoem 3 (Mea Cuvaue Comaso Le be a o M Assume I a ( a Rc, ( alog a mmal geodesc segme om (we π assume e m m + a ( alog a mmal geodesc segme om Equaly olds ad oly e adal secoal cuvaues ae equal o ad ( = ( a o all < I c alog a mmal geodesc segme om ad < o > ad π 4 e m + m ( alog a mmal geodesc segme om 3 I c alog a mmal geodesc segme om ad > ad π π, 4 e m + m + 4 c (3 I acula we = we ave + m (4 wee m + s e mea cuvaue o e geodesc see M + e smly coeced model sace o dmeso + w cosa cuvaue ad m s e mea cuvaue o e model sace o dmeso π π I ac [] G We ad W Wyle saed a,, 4 ( ( s d ( c s (5 wee s ( s e soluo o equao y ( + y ( = Fom eoem 3 above ad Equaos ((8 ad (9 o all s ( s, θ ( θ s + + (, θ d m (, θ d A ( s, θ k k + ( θ, we ave: A s m = e e = (6 A, A, + wee 4 c A ( s, θ k ad cosa Rcc cuvaue ( k + + A ( s, θ = ( φ( s k φ I = < k e ( s φ deoes e volume eleme e sace o dmeso + Fom e assumo we ave: φ we k ( k ( + s R = + l k k 757

5 I A Kaboye, B Maama ece ee exss R R so a A ( R, θ = wc meas a ee exss γ Teeoe we coclude a so a e S(, -Jacob eld Y vases a ( R γ ( R s a cojugae o o e cee o e see (, ( k be mmal, a s d( y, R= + l ad k ( + k Dam M ( k ( + + l k k S ece γ ceases o I [] M Lmocu geealzed a classcal Myes eoem by usg e Baky-Émey Rcc cuvaue eso o comlee ad coeced Remaa maolds ( M, g Ts eoem ca be vewed as a coollay o eoem Coollay 3 Le (M, g be a comlee ad coeced Remaa maold o dmeso I ee exss a smoo uco : M sasyg e equales ad c e M s comac Poo o Coollay Rc + ess k > (7 To ove s coollay suces o sow a ee exs a osve eal w < k ad a geodesc see S(, wc mea cuvaue vees m <( Le x be a o M ad le [ ] ad ( e ( Se Y ( = φ ( e ( wee φ ( γ :, M be a mmal geodesc jog o x be a aallel ooomal veco elds alog γ ooomal o γ π = s We ave (, φ φ ( γ x I Y Y Rc = ( ( φ φ φ k + ess d π π π = ( cos k s d + φ d φ φ 4 π π π ( cos k s d ( d + + c d φ φ 4 Teeoe π k π π π π ( + + c cos c cos 8 ( k π c π wc allows a m ( < ( k (8 cπ m = π + (9 8 π 8c > + k π ( By Comacess o S(,, ee exss a osve cosa so a, o ay 758

6 I A Kaboye, B Maama m < >, e cocluso ollows om eoem Coollay 33 (E Calab geodesc γ emaag om we ave Sce Rc ( k ( Le ( M, g be a comlee ad coeced Remaa maold o dmeso Suose ee exss a smoo uco : M so a c ad Rc I M s ocomac e ee exss a geodesc γ M so a lm Rc ( γ ( Poo I s clea a, o a geodesc γ ssug om ee exs wo osve eals k ad so a Rc ( γ ( ( k o all e adms a cojugae o alog γ ece, M s ocomac, o all M, ee exss a geodesc γ ssug om so a o ay wo osve eal k ad ee exss so a Rc ( γ ( < ( k I acula k = = we ake ( Rc γ < ad e cocluso ollows Coollay 34 (Ambose Le ( M, g be a comlee ad coeced Remaa maold o dmeso Suose ee exss a uco o M so a Rc I ee exss a o M so a, o ay geodesc γ emaag om, aamezed by s ac-leg we ave e M s comac Poo + ( γ I M s ocomac, om coollay 33, ee exss o Teeoe, Rc d = + ( > so a Rc ( γ ( < Rc ( γ ( d Rc ( γ ( d + d < + ( + + Poo o eoem Le Vol ( (, e B s = dvol B( s, g deoes e weged volume o e geodesc ball m o cee ad adus s M ad vol ( s e volume o geodesc ball o adus s m e model sace M w cosa cuvaue ad dmeso m I Deeal Geomey, e volume comaso eoy lays a moa ule May moa esuls s oc ca o be obaed wou volume comaso esuls as oologcal gdy esuls Fo comlee smoo mec measue sace w Rc e ollowg lemma moved e volume comaso eoem oved by G We ad W Wyle I []: Lemma 35 Le ( M, g,e dvolg be comlee smoo mec measue sace w Rc Fx M x cd, x e o R > Poo ; ee exss c so a Vol ( B (, R 3c R e Vol B (, ( 759

7 I A Kaboye, B Maama Le x be a o M ad le γ :[,] M be a mmal geodesc jog o x ad ( e ( be a aallel ooomal veco elds alog γ ooomal o γ Se Y ( = e ( By e secod vaao omula we ave: = (, m I X X = = ( γ γ = X R X,, X d + Rc d + ( γ γ, d ( γ d ( ( γ d ( γ d = + d d = + ( x + ( γ d c ( ( γ ess ece m = + 3 c Fom (9 ad e above elao, we ave d A (, θ m ( = l ( A (, θ = + 3 c d A, θ Fo all osve eals ad s, egag s elao we ave: (, θ (, θ s 3c m ( d = e A Teeoe we ave ( θ ( θ wc mles R (3 A s s (4 A s, e A, s ece 3c R 3c R A ( s, θ dθd e A (, θ s dθd (5 S S 3c R 3c A ( s, θ dθ e s A (, θ dθd = e s vol ( B (, R (6 S ad egag om o R w esec o s we oba e cocluso Se ( x ( x 3 cd( x, ece we ave = + Te S m = = = 3 c (7 ( (, ( B (, Vol B R R Vol (8 76

8 I A Kaboye, B Maama Fom e elao (8 we deduce a e uco vol s o- ω ceasg Le α ( B (, Vol( B (, Vol( B (, = lm ad α ( M = α ω We ave α ( M ( B (, Vol ω ω We say a M s o lage weged volume gow α ( M > Le R, be e se o e u al age vecos o e geodescs sag om c wc ae mmzed a leas o ad R, s comlemeay se Se B, = x B,, γ :, s M mmal, γ =, γ U (9 Le R, { } [ ] Σ a subse o e u see U TM Se { } γ [ ] γ γ B, = x B,, :, s M mmal, =, Σ (3 Σ Lemma 36 I ( x cd( x, Vol B (, e uco o ay >, 3 (, x = x + cd x ad Rc e ( Σ s oceasg ad ω ( (,, vol B R ω α wee s deed by: Poo By Equao (7 we ave d d l A(, θ = m( = l ; (3 d d A (, θ ece we deduce a e uco s deceasg By lemma 3 [7] we ave: R R, dd A dd θ θ θ Σ Σ R ωr = = A (, dd dd θ θ θ ω Σ Σ Teeoe m [ R, cuθ ] R Vol (, dd (, dd B R A θ θ A θ θ Σ Σ Σ = m [, cuθ ] Vol B, A, θ dd θ A, θ dd θ ( (, ( Σ Σ Σ Fo Σ Σ ad by a ( o e lemma 36 we ave: volb (, vol,, B vol B Σ Σ Σ (34 vol B, vol B, vol B, < we ave ad e a ( ca be oved as e lemma 3 [8] (3 (33 76

9 I A Kaboye, B Maama Lemma 37 Le ( M, g be a comlee ocomace Remaa maold ad a oeal uco o M w ( x cd (, x ad Rc I M s o lage weged volume e ad Poo We ave ( [ (, Σ Vol B ω ( M ( α, > (35 ( Vol B, = Vol B, \ B, + Vol B, (36 Σ Σ Σ Σ (, = (, dd θ θ (37 Vol B A Σ \ Σ Σ \ R Σ Σ dd θ Vol \ \ R Σ (38 Sce Σ we ave ( > Vol Σ Σ α M, lm Σ \ Σ = ; ece ( Vol B Vol B lm = lm (39 ω ω Lemma 38 Le ( M, g be a comlee ocomace Remaa maold ad a oeal uco o M w ( x cd (, x ad Rc I M s o lage weged volume e o ay x B(, we ave VolB, α ω d( x, α ( M ( M (4 Te oo o s lemma s se by se smla o e oe [9] (lemma 4 Le q, be wo os M Te excess uco s deed as: e x = d x, + d qx, d q, (4 q By agle equaly e excess uco s oegave ad s lscz Le γ be a ay om ad se s( x d( x, γ ( γ γ = ece, o ay we ave: e x = d x, + d, x, (4 Te uco e, γ ( s oceasg o ad e, ( x γ Se e, γ ( x = lm e, ( x γ e γ s oceasg o, we ave By e ac a, ( e, γ x e x, >, γ Alyg e Tooogov s eoem ad e deo o ccal o we ave: Lemma 39 Le M be a comlee ocomace Remaa maold suc a m K k o some k ad M Suose a x s a ccal o o d Te o ay ay γ ssug om, we ave e γ Recall a a o x s a ccal o o l k + e ( x kd (, x d o ay veco x (43 u TM ee exss 76

10 I A Kaboye, B Maama π a mmal geodesc γ om x o so a ( v, γ ( Fom e equaly (8 ad usg e agumes o e oo o e Pooso 3 [6], we deduce e ollowg excess esmae o comlee smoo mec measue sace w Rc ad oeal uco bouded by cd ( x, Teoem 3 Le ( M, g be a comlee ocomace Remaa maold ad a oeal uco o M w cd (, x, o some xed o, Rc ad s x < m d x,, d qx, e { } e q ( x s 8 By e same agumes as [] ad usg sead o, oe ca ove e above lemma To ove e eoem, suces o sow a M coas o ccal o o d oe a Fo s, le x be a o M ad x ad se = d( x, Fom e lemma 38 ad e equaly ( we ave: ( (44 < k (45 d x, l 8k + e ece, ee exss a ay γ ssug om veyg s = d( x, γ < l k 8k + e < Fom e a- γ = o all, wc meas Le q be a o o γ so a d( x, q = d( x, γ e d( xq, gle equaly we ave: m ( d( x,, d(, x s q γ ([, ] Suc om e elaos (44 ad (45 we oba (46 ( s e, γ ( x e, ( ( x 8 l γ < e k k + Te equales (43 ad (47 sow a x s o a ccal o o sooy lemma M s deomoc o Reeeces (47 d ece, by [] We, G ad Wyle, W (9 Comaso Geomey o e Baky-Emey Rcc Teso Joual o Deeal Geomey, 83, [] Lmocu, M ( Te Baky-Émey Rcc Teso ad Is Alcaos o Some Comacess Teoems Maemasce Zesc, 7, 75-7 ://dxdoog/7/s [3] Tadao, (6 Remak o a Damee Boud o Comlee Remaa Maolds w Posve Baky-Émey Rcc Cuvaue Deeal Geomey ad Is Alcaos, 44, ://dxdoog/6/jdgeo5 [4] Maama, B ( U éoème de Myes ou les vaéés à coubue de Rcc moée a ue cosae egave Aka Maemaka,,

11 I A Kaboye, B Maama [5] Iokawa, Y (99 Dsace See ad Myes-Tye Teoems o Maolds w Lowe Bouds o, e Rcc Cuvaue Illos Joual o Maemacs, 34, [6] Abesc, U ad Gomoll, D (99 O Comlee Maolds w Noegave Rcc Cuvaue Joual o e Ameca Maemacal Socey, 3, ://dxdoog/9/s [7] Maama, B ( A Volume Comaso Teoem ad Numbe o Eds o Maolds w Asymocally Noegave Rcc Cuvaue Revsa Maemáca Comluese, 3, [8] Maama, B (5 Oe Maolds w Asymocally Noegave Cuvaue Illos Joual o Maemacs, 49, [9] Xa, C (999 Oe Maolds w Noegave Rcc Cuvaue ad Lage Volume Gow Commea Maemac elvec, 74, ://dxdoog/7/s4599 [] Se, Z (996 Comlee Molds w Noegave Rcc Cuvaue ad Lage Volume Gow Iveoes Maemacae, 5, ://dxdoog/7/s58 Subm o ecommed ex mausc o SCIRP ad we wll ovde bes sevce o you: Acceg e-submsso ques oug Emal, Facebook, LkedI, Twe, ec A wde seleco o jouals (clusve o 9 subjecs, moe a jouals Povdg 4-ou g-qualy sevce Use-edly ole submsso sysem Fa ad sw ee-evew sysem Ece yeseg ad ooeadg ocedue Dslay o e esul o dowloads ad vss, as well as e umbe o ced acles Maxmum dssemao o you eseac wok Subm you mausc a: ://aesubmssoscog/ O coac am@scog 764