NEUMANN ISOPERIMETRIC CONSTANT ESTIMATE FOR CONVEX DOMAINS

Transcription

1 NEUMANN ISOPERIMETRIC CONSTANT ESTIMATE FOR CONVEX DOMAINS XIANZHE DAI, GUOFANG WEI, AND ZHENLEI ZHANG Abstract We preset a geometrc ad elemetary proof of the local Neuma sopermetrc equalty o covex domas of a Remaa mafold wth Rcc curvature bouded below 1 Itroducto Isopermetrc ad Sobolev equaltes are equvalet equaltes (see eg Theorem 13 below) whch play mportat role geometrc aalyss o mafolds Ideed, dog aalyss o mafolds usually depeds o the estmate of the Sobolev costat whch could the be obtaed va the sopermetrc costat There are extesve work o sopermetrc costat estmates A mportat method poeered by Gromov reles o the geometrc measure theory ad ts regularty theory, whch works for closed mafolds or covex domas wth smooth boudary, see eg the survey artcle [9] ad the recet paper [14] Oe may also obta a estmate through L- Yau gradet estmate for heat kerel [12] ad the equvalece of heat kerel bouds, Sobolev equalty, sopermetrc equalty, see [15, Page 448], whch aga requres smooth ad covex boudary Aother method usg eedle decomposto from covex geometry has also bee very successful ad, very recetly, bee combed wth optmal trasport ad exteded to o-smooth case, see [3] ad the refereces there For star-shaped domas a mafold wth Rcc curvature bouded from below, Buser [2] gave a elemetary proof for a Neuma sopermetrc costat (the Cheeger costat) estmate usg comparso geometry, but the estmate depeds o the ad out radus of the doma, whch does ot gve a uform estmate for covex doma as the -radus mght be small We would lke to pot out that for covex domas wth o-smooth boudares, the estmate for sopermetrc costat s oly obtaed the very recet paper [3] metoed above I ths short ote we gve a very geometrc ad elemetary proof of a Neuma sopermetrc equalty, albet wth o-optmal costat, for covex domas whose boudares eed ot be smooth Frst we recall some deftos 1991 Mathematcs Subject Classfcato 53C2 XD s partally supported by the Smos Foudato, NSF ad NSFC, GW partally supported by the Smos Foudato ad NSF DMS , ZZ partally supported by CNSF

2 Defto 11 Whe M s compact (wth or wthout boudary), the Neuma α-sopermetrc costat of M s defed by IN α (M) = sup H m{vol(m 1 ), vol(m 2 )} 1 1 α vol(h) where H vares over compact ( 1)-dm submafold of M whch dvdes M to two dsjot ope submafolds M 1, M 2 (wth or wthout boudary) Defto 12 The Neuma α-sobolev costat of M s defed by SN α (M) = f a R f a α α 1 sup f C (M) f 1 The sopermetrc costat ad Sobolev costat are equvalet Theorem 13 ([4], see also [11]) For all α, IN α (M) SN α (M) 1 2 IN α(m) For coveece we cosder the ormalzed Neuma α-sopermetrc ad α- Sobolev costat: IN α(m) = IN α (M) vol(m) 1/α, SN α(m) = SN α (M) vol(m) 1/α Usg comparso geometry ad Vtal coverg we gve a estmate o the ormalzed Neuma sopermetrc costat for covex doma terms of the Rcc curvature lower boud ad the dameter of the doma Theorem 14 Let (M, g) be a complete Remaa mafold of dmeso, wth Rc ( 1)K for some K Let Ω be a bouded covex doma The (11) IN (Ω) 4 e 11( 1) Kd d where d s the dameter of the doma Ω I partcular, f M s closed wth dameter d, the (12) IN (M) 4 e 11( 1) Kd d Corollary 15 Let (M, g) be a complete Remaa mafold of dmeso, wth oegatve Rcc curvature Let Ω be a bouded covex doma The (13) IN (Ω) 4 d where d s the dameter of the doma Ω I partcular, f M s closed wth dameter d, the (14) IN (M) 4 d Remark 16 The case whe Ω equals the whole mafold s well-kow The referece we metoed earler for covex doma the lterature deals wth domas wth (smooth) covex boudary whch s a stroger codto 2,

3 Remark 17 For balls we ca obta both Drchlet ad Neuma sopermetrc costat estmates eve uder the much weaker tegral Rcc lower boud assumpto [7, 18] O the other had t s ot clear f that wll rema true for covex domas Remark 18 Usg the mea curvature estmate from [16] oe gets smlar estmate whe the Bakry-Emery Rcc curvature s bouded from below ad oscllato of the potetal fucto s bouded 2 Proof of Theorem 14 The proof goes by a coverg argumet of Aderso [1], combed wth a observato of Gromov [1] See [1] or [7] for a smlar argumet of estmatg the local Drchlet sopermetrc costat Frst of all we recall a lemma whose proof s a slght modfcato of Gromov s observato [1, 5(C)] Lemma 21 Let M be a complete Remaa mafold Let Ω be a covex doma of M ad H be ay hypersurface dvdg Ω to two parts Ω 1, Ω 2 For ay Borel subsets W Ω, there exsts x 1 oe of W, say W 1, ad a subset W aother oe, W 2, such that (21) vol(w ) 1 2 vol(w 2) ad ay x 2 W has a uque mmal geodesc coectg to x 1 whch tersects H at some z such that (22) dst(x 1, z) dst(x 2, z) The covexty assumpto of Ω s essetal It mples that ay mmal geodesc wth edpots dfferet parts must tersects H The Bshop-Gromov relatve volume comparso theorem gves Lemma 22 Let H, W ad x 1 be as the lemma above The (23) vol(w ) 2 1 De ( 1) KD vol(h ) where D = sup x W dst(x 1, x) ad H s the set of tersecto pots wth H of geodescs γ x1,x for all x W Proof Let Γ S x1 be the set of ut vectors such that γ v = γ x1,x 2 for some x 2 W We compute the volume the polar coordate at x 1 Wrte dv = A(θ, t)dθ dt the polar coordate (θ, t) S x1 R + For ay θ Γ, let r(θ) be the radus such that exp x1 (rθ) H The W {exp x1 (rθ) θ Γ, r(θ) r 2r(θ)} So, by 3

4 relatve volume comparso, vol(w ) ˆ Γ ˆ 2r(θ) r(θ) A(θ, t)dtdθ sh 1 (2 ˆ KD) sh 1 ( KD) D sh 1 (2 KD) sh 1 ( KD) vol(h ) Γ r(θ)a(θ, r(θ))dθ The requred estmate follows from sh(2t) sh t = 2 cosh t e t wheever t Corollary 23 Let H be ay hypersurface dvdg a covex doma Ω to two parts Ω 1, Ω 2 For ay ball B = B r (x) we have (24) m ( vol(b Ω 1 ), vol(b Ω 2 ) ) 2 +1 re ( 1) Kd vol(h B 2r (x)) where d = dam(ω) I partcular, f B Ω s dvded equally by H, we have (25) vol(b r (x) Ω) 2 +2 re ( 1) Kd vol(h B 2r (x)) Proof Put W = B Ω the above lemma ad otce that D 2r ad H H B 2r (x) Now we are ready to prove our ma theorem Proof of Theorem 14 We may assume that vol(ω 1 ) vol(ω 2 ) For ay x Ω 1, let r x be the smallest radus such that vol(b rx (x) Ω 1 ) = vol(b rx (x) Ω 2 ) = 1 2 vol(b r x (x) Ω) Let d = dam(ω) By above corollary, (26) vol(b rx (x) Ω) 2 +2 r x e ( 1) Kd vol(h B 2r (x)) The doma Ω 1 has a coverg Ω 1 x Ω 1 B 2rx (x) By Vtal Coverg Lemma, cf [13, Secto 13], we ca choose a coutable famly of dsjot balls B = B 2rx (x ) such that B 1rx (x ) Ω 1 Applyg the relatve 4

5 volume comparso theorem ad the covexty of Ω we have vol(ω 1 ) 1rx sh 1 ( Kt)dt rx sh 1 ( Kt)dt vol ( ) 1 sh 1 (1 Kr x ) sh 1 ( Kr x ) vol ( ) 1 sh 1 (1 Kd) sh 1 ( vol ( ) Kd) 1 e 9( 1) Kd vol ( ) = e 9( 1) Kd vol ( B rx (x ) Ω ) Moreover, sce the balls B are dsjot, (26) gves, vol(h) vol(b H) 2 2 e ( 1) Kd rx 1 vol(b rx (x ) Ω) These two estmates lead to vol(ω 1 ) 1 vol(h) 2 2 e 1( 1) Kd 4 e 1( 1) Kd 4 e 1( 1) Kd sup = 4 e 1( 1) Kd sup ( vol(b r x (x ) Ω) ) 1 x vol(b rx (x ) Ω) r 1 vol(b r x (x ) Ω) 1 r 1 x vol(b rx (x ) Ω) vol(b rx (x ) Ω) 1 rx 1 vol(b rx (x ) Ω) ( rx vol(b rx (x ) Ω) ) 1 O the other had, sce vol(ω 1 ) vol(ω 2 ), we have r x d for ay x Ω 1 Thus, by the relatve volume comparso ad covexty of Ω aga, we have d vol(ω) sh 1 ( Kt)dt rx sh 1 ( Kt)dt vol(b r x (x) Ω) Therefore, ( vol(ω) 1 vol(ω 1 ) 1 4 1( 1) Kd r d e sup sh 1 ( ) 1 Kt)dt vol(h) r <rd sh 1 ( Kt)dt The last term o the rght had sde has the estmate r d sh 1 ( Kt)dt r sh 1 ( r d Kt)dt r sh 1 ( Kd) sh 1 ( Kr) d sh 1 ( Kd) ( d e ( 1) Kd) 1 5 Kd

6 The requred ormalzed Neuma sopermetrc costat estmate ow follows Refereces [1] M T Aderso, The L 2 structure of modul spaces of Este metrcs o 4-mafolds, GAFA, 2 (1992), [2] P Buser, A ote o the sopermetrc costat, A Sc École Norm Sup (4) 15 (1982), o 2, [3] F Cavallett & A Modo Sharp ad rgd sopermetrc equaltes metrc-measure spaces wth lower Rcc curvature bouds, arxv: [mathmg], to appear Ivetoes [4] J Cheeger, A lower boud for the smallest egevalue of the Laplaca, Problems Aalyss, R Gug ed, Prceto Uversty Press, 197 [5] J Cheeger ad ST Yau, A lower boud for the heat kerel, Comm Pure Appl Math, 34 (1981), [6] S Y Cheg & S T Yau, Dfferetal equatos o Remaa mafolds ad ther geometrc applcatos, Comm Pure Appl Math 28 (1975), [7] X Da, G We ad Z Zhag, Local Sobolev Costat Estmate for Itegral Rcc Curvature Bouds, arxv: [8] S Gallot, Isopermetrc equaltes based o tegral orms of Rcc curvature, Soc Math de Frace, Astersque (1988), [9] S Gallot, Iegaltes sopermetrques et aalytques sur les varetes remaees (Frech) [Isopermetrc ad aalytc equaltes o Remaa mafolds] O the geometry of dfferetable mafolds (Rome, 1986) Astersque No (1988), [1] M Gromov, Paul Levy s sopermetrc equalty, Publcatos IHES, 198 [11] P L, Geometrc Aalyss, Cambrdge studes advaced mathematcs 134, Cambrdge Uversty Press, 212 [12] P L ad ST Yau, O the parabolc kerel of the Schrodger operator, Acta Math 156 (1986) [13] FH L ad XP Yag, Geometrc Measure Theorey-A Itroducto, Scece Press, Bejg/New York ad Iter Press, Bosto, 22 [14] E Mlma, Sharp sopermetrc equaltes ad model spaces for the curvature-dmesodameter codto, J Eur Math Soc (JEMS) 17 (215), o 5, [15] L Saloff-Coste,Uformly ellptc operators o Remaa mafolds Joural of Dff Geom 36 (1992), [16] G We ad W Wyle, Comparso Geometry for the Bakry-Emery Rcc Tesor, Joural of Dff Geom 83, o 2 (29), [17] D Yag, Covergece of Remaa mafolds wth tegral bouds o curvature I, A Scet Èc Norm Sup, 25 (1992), [18] Z Zhag, Notes o the sopermetrc costat estmate, preprt (Xazhe Da) Departmet of Mathematcs, ECNU, Shagha ad UCSB, Sata Barbara CA 9316, emal:da@mathucsbedu (Guofag We) Departmet of Mathematcs, Uversty of Calfora, Sata Barbara CA 9316, emal: we@mathucsbedu (Zhele Zhag) Departmet of Mathematcs, Captal Normal Uversty, Cha, emal: zhlego@alyucom 6