Prescrbg Morse scalar curvatures: subcrtcal blowg-up solutos Adrea Malchod ad Mart Mayer Scuola Normale Superore, Pazza de Cavaler 7, 506 Psa, ITALY adrea.malchod@ss.t, mart.mayer@ss.t arxv:8.0946v [math.ap] Dec 08 December 7, 08 Abstract Prescrbg coformally the scalar curvature of a Remaa mafold as a gve fucto cossts solvg a ellptc PDE volvg the crtcal Sobolev expoet. Oe way of attackg ths problem cosst usg subcrtcal approxmatos for the equato, gag compactess propertes. Together wth the results [30], we completely descrbe the blow-up pheomeo case of uformly bouded eergy ad zero weak lmt postve Yamabe class. I partcular, for dmeso greater or equal to fve, Morse fuctos ad wth o-zero Laplaca at each crtcal pot, we show that subsets of crtcal pots wth egatve Laplaca are oe-to-oe correspodece wth such subcrtcal blowg-up solutos. ey Words: Coformal geometry, sub-crtcal approxmato, blow-up aalyss. Cotets Itroducto Prelmares 4 3 Exstece of subcrtcal solutos 6 4 The secod varato 5 Appedx: some techcal estmates 5. Lst of costats......................................... 6 Itroducto Cosder a compact mafold M, g 0 wth 3 ad a coformal metrc g = u 4 g0, u > 0: wth ths otato the scalar curvature trasforms the followg way see [4] R gu u + 4 = Lg0 u := c g0 u + R g0 u c =, wth g0 the Laplace-Beltram operator of g 0. L g0 s called the coformal Laplaca ad trasforms accordg to the law L g u φ = u + Lg0 φ.
I the 70 s, azda ad Warer cosdered [8] the problem of prescrbg the scalar curvature of mafolds va coformal deformato of the metrc, see also [6], [7]. By the above trasformato law, f oe wshes to prescrbe R g as a gve fucto x the would eed to solve L g0 u = xu + o M, g 0.. There are rather easy obstructos to the solvablty of.: for example, f the sg of s costat, t has to cocde wth that of the frst egevalue of L g0. Depedg o the latter sg, whch s coformally varat, a coformal class of metrcs s sad to be of egatve, zero or postve Yamabe class. We wll dscuss for smplcty the case of fucto wth costat sg, despte the lterature there are may terestg papers dealg wth chagg-sg fuctos. I [8], azda ad Warer proved some exstece results for zero or egatve Yamabe classes usg the sub- ad super-soluto method. For postve Yamabe class stead, they foud a ow well-kow obstructo to exstece o the sphere, amely that f u solves., the oe must have, f gs u dµgs = 0,. S ad hece, for coformal curvatures, the fucto, f gs must chage sg. Later o, some exstece results were foud uder codtos that would mply topologcal rchess of the sub-levels of, cotrary to the above example. I two dmesos, where. s replaced by a equato expoetal form, J. Moser showed that the problem s solvable o the stadard sphere f s atpodally symmetrc. I hgher dmesos, exstece results uder the acto of symmetry groups were prove [0] ad [], []. A geeral dffculty studyg. s the lack of compactess due to the presece of the crtcal expoet. A typcal pheomeo ecoutered here s that of bubblg. Bubbles are solutos of. o S wth : these arse as profles of geeral dvergg solutos ad were classfed [], see also [3], [36]. From the varatoal pot of vew, bubbles geerate dvergg Palas-Smale sequeces for the Euler-Lagrage eergy of., gve by J = J : c u g 0 + R g0 u dµ g0 Ju = M M u dµg0 From a formal expaso of J o a fte sum of bubbles, see e.g. the troducto [30], oe sees a role of the dmeso the stregth of the mutual teracto amog bubbles, whch s weaker as creases: a cosequece of ths fact s that three dmesos oly oe bubble ca form. Explotg ths fact, after some work o S by A. Chag ad P. Yag [6], [7], A. Bahr ad J.M. Coro proved a exstece result [6] o S 3 assumg that s a Morse fucto satsfyg the followg two propertes. { = 0} { = 0} = ;.3 mx,,.4 {x M : x=0, x<0} where mx, stads for the Morse dex of at x, see also [] ad [35] for more geeral related results. The above exstece statemet was exteded to arbtrary dmesos [4] for fuctos satsfyg a sutable flatess codto, ad [8], [], [9] for fuctos close to a postve costat the C -sese. I four dmesos, see [7] ad [5], t was show that eve f multple bubbles ca form, they caot be too close to each-other; such pheomeo s usually refereed to as solated smple blow-up. Results of dfferet kd were also prove [9] for = ad [9] [8], [0]: see also Chapter 6 [4]. Two ma approaches have bee used to uderstad the blow-up pheomeo: sub-crtcal approxmatos or the costructo of pseudo-gradet flows. I ths paper we focus o the former, whle the other oe wll be the subect of [3], where a oe-to-oe correspodece of blowg-up solutos wth bouded eergy ad zero weak lmt ad crtcal pots at fty s show. Cosder the problem c g0 u + R g0 u = x u + τ, 0 < τ,.5
whch, up to a proper dlato, s the Euler-Lagrage equato for the fuctoal c u M g J τ u = 0 + R g0 u dµ g0, u A..6 M up+ dµ g0 p+ Beg ow the expoet lower tha crtcal, solutos ca be easly foud, eve though oe could lose uform estmates as τ teds to zero. I [], [35], [4], the sgle-bubblg behavour for dvergg solutos of.5 was proved. The, by degree- or Morse-theoretcal argumets t was show that uder.4 there must be famles of solutos that stay uformly bouded, therefore covergg to solutos of.. For ths argumet to work, oe crucal step was to completely characterze blowg-up solutos of.5, showg that three dmesos sgle blow-ups occur at ay crtcal pot of wth egatve laplaca ad that they are uque. O four-dmesoal spheres, a smlar property was proved [5] for multple blow-ups see also [7], assumg a sutable codto related to the mult-bubble teractos. For Morse fuctos, f 5 the stuato s more volved, ad blow-ups mght be possbly of fte eergy, see e.g. [3], [4], [5], [37]. I [30] t was however proved that f a sequece of blowg-up solutos has uformly-bouded W, -eergy ad zero weak lmt, the blow-ups are stll solated smple. Although the result s smlar to the case of dmesos three ad four, the pheomeo s somehow opposte sce t s drve by the fucto rather tha from the mutual bubble teractos. Both assumptos, zero weak lmt ad bouded eergy, are deed atural: f the former fals the problem. would have a soluto; the secod oe stead s usually foud whe usg m-max or Morse-theoretcal argumets, as t wll be doe [3]. However, dfferetly from = 3, 4, [30] o restrcto s prove o the umber or locato of blow-up pots, provded they occur at crtcal pots of wth egatve Laplaca. The goal of ths paper s to show that the characterzato of the above blow-ups [30] s sharp, amely that they ca occur at arbtrary subsets of { = 0} { < 0}. Furthermore, we prove uqueess of such solutos, ther o-degeeracy ad determe ther Morse dex. Our ma result s the followg oe, that follows from Proposto 3., Corollary 4. ad Theorem [30]. Theorem. Let M, g be a compact mafold of dmeso 5 wth postve Yamabe class, ad let : M R be a postve Morse fucto satsfyg.3. Let x,..., x q be dstct crtcal pots of wth egatve Laplaca. The, as τ 0, there exsts a uque soluto u τ,x,...,x q developg a smple bubble at each pot x ad covergg weakly to zero W, M, g as τ 0. Moreover, up to scalg by costats, u τ,x,...,x q s o-degeerate for J τ ad mj τ, u τ,x,...,x q = q + q = m, x. Furthermore, all blow-ups wth uformly bouded eergy ad zero weak lmt are of the above type. As t wll be show [3], for 5 there caot be a drect couterpart of.4, whch s a dexcoutg codto. However, exstece results of dfferet type wll be derved there. Remark.. A more precse expresso for u τ,x,...,x q s gve by the followg formula α,m = u m q α,m δ,m,a,m W 0 as m,, = M,g 0 Θ x 4 + o, a,m x ad,m τm = τ m. Here the multplcatve costat Θ depeds o the blowg-up solutos but t s depedet of. For ths ad more precse formulas we refer to Secto 3 ad Theorem the Appedx. If = 4, the same coclusos hold replacg a < 0 for all wth v of Theorem [30]. Eve though upo scalg the above solutos u τ,x,...,x q are o-degeerate, they Hessa of J τ there has q = m, x egevalues approachg zero as τ 0, see Secto 4. Theorem gves a oe-to-oe correspodece of zero weak lmt subcrtcal blow-up solutos to subsets of crtcal pots of wth egatve Laplaca, whle [3] ths correspodece wll be show wth zero weak lmt,.e. pure crtcal pots at fty, accordg to the termology [5], see also [33] 3
The proof of Theorem reles o the estmates [30] ad a fte-dmesoal reducto, see e.g. [], wth a careful asymptotc aalyss. I dmeso four, ths approach was used Secto of [5]: here we show that hgher dmesos blow-up mght occur at arbtrary crtcal pots of wth egatve Laplaca, whch affects the global structure of the solutos of problem.. Va careful expasos, we also determe the Hessa of the Euler-Lagrage fuctoal ad the Morse dex of these solutos, whch we prove to be o-degeerate. The solutos we cosder here le a set V q, ε the fuctoal space W, M, g 0 whch cotas a mafold of approxmate solutos for.5, q = α ϕ a,, whch s trasversally o-degeerate see Secto for the otato used here. Ths allows to solve.5 orthogoally to ths mafold va a proper trasversal correcto to the approxmate solutos, see Defto 3. ad Lemma 3., ad reduce to the study of the taget compoet. By Theorem from [30] we ca reduce ourselves to a smaller set V q, ε, see 3., where more precse estmates hold for the gradet of J τ. These allow us to use a orthogoal correcto v small sze, solve also for the taget compoet ad to estmate the secod dfferetal of J τ at q = α ϕ a, + v, see Secto 4. Fally, ths allows tur to compute the Morse dex of the solutos u τ,x,...,x q ad to prove ther uqueess. I ths step we show that, eve though the correcto v s of the same order of the small egevalues of the Hessa of J τ, some cacellato occurs the estmate of the Morse dex. The pla of the paper s the followg: Secto we collect some prelmary materal cocerg approxmate solutos ad the fte-dmesoal reducto of the problem, whch s the worked-out detal Secto 3. I Secto 4 we study the Hessa of the Euler-Lagrage fuctoal J τ V q, ε, fdg a proper base wth respect to whch the Hessa early dagoalzes. Fally, we collect a Appedx some useful ad techcal estmates from [30] ad a table of costats. Ackowledgmets. A.M. has bee supported by the proect Geometrc Varatoal Problems ad Fazameto a supporto della rcerca d base from Scuola Normale Superore ad by MIUR Bado PRIN 05 05B9WPT 00. He s also member of GNAMPA as part of INdAM. Prelmares I ths secto we collect some backgroud ad prelmary materal, cocerg the varatoal propertes of the problem ad some estmates o hghly-cocetrated approxmate solutos of bubble type. We cosder a smooth, closed remaa mafold M = M, g 0 wth volume measure µ g0 ad scalar curvature R g0. Lettg A = {u W, M, g 0 u 0, u 0, } the Yamabe varat s defed as Y M, g 0 = f A c u g 0 + R g0 u dµ g0 u dµg0 ; c = 4, ad t turs out to deped oly o the coformal class of g 0. We wll assume from ow o that the varat s postve, amely that M, g 0 s of postve Yamabe class. As a cosequece, the coformal Laplaca L g0 = c g0 + R g0 s a postve ad self-adot operator. Wthout loss of geeralty we assume R g0 > 0 ad deote by G g0 : M M \ R + the Gree s fucto of L g0. Cosderg a coformal metrc g = g u = u 4 g0, there holds dµ gu = u dµg0 ad R = R gu = u + c g0 u + R g0 u = u + Lg0 u. Note that c u W, M,g 0 u L g0 u dµ g0 = c u g 0 + R g0 u dµ g0 C u W, M,g 0. I partcular we may defe u = u L g0 := u L g0 u dµ g0 4
ad use as a equvalet orm o W, M, g 0. Settg R = R u for g = g u = u 4 g0, we have r = r u = Rdµ gu = ul g0 udµ g0,. ad hece J τ u = r k p+ τ wth k τ = u p+ dµ g0.. The frst- ad secod-order dervatves of the fuctoal J τ are gve by J τ uv = [ L k g0 uvdµ g0 r u p ] vdµ g0 ;.3 p+ k τ τ J τ uvw = k p+ τ [ L g0 vwdµ g0 p r k τ 4 [ L g0 uvdµ g0 k p+ + τ k p+ + τ u p vwdµ g0 ] u p wdµ g0 + p + 3r + u p vdµ g0 u p wdµ g0. L g0 uwdµ g0 u p vdµ g0 ] I partcular, J τ s of class C,α loc A ad, for ε > 0, uformly Hölder cotuous o each set of the form U ε = {u A ε < u, J τ u ε }. To uderstad the blow-up pheomeo, t s coveet to cosder some hghly cocetrated approxmate solutos to.. Let us frst recall the costructo of coformal ormal coordates from [3]: gve a M, these are defed as geodesc ormal coordates for a sutable coformal metrc g a [g 0 ]. Let r a be the geodesc dstace from a wth respect to the metrc g a : wth ths choce, the expresso of the Gree s fucto G ga for the coformal Laplaca L ga wth pole at a M, deoted by G a = G ga a,, smplfes cosderably. I Secto 6 of [3] oe ca fd the expaso G a = Here H r,a C,α loc.4 ra + H a, r a = d ga a,, H a = H r,a + H s,a for g a = u 4 a g 0..5 4 ω, whle the sgular error term s of the type: r a for = 5 H s,a = O l r a for = 6. ra 6 for 7 The leadg term H s,a for = 6 s Wa 88c ϕ a, = u a + γ G a We otce that the costat γ s chose so that l r, wth W the Weyl tesor. For > 0 large defe, G a = G ga a,, γ = 4 ω..6 γ G a x = d g a a, x + od g a a, x as x a. Such fuctos are approxmate solutos of., see Lemma 5., ad for sutable values of depedg o τ these are also approxmate solutos of.5, see Lemma 5.7 for a mult-bubble verso. Notato. For p, L p g 0 wll stad for the famly of fuctos of class L p wth respect to the measure dµ g0. Recall also that for u W, M, g 0 we have set r u = ul g0 udµ g0, whle for a M we deote by r a the geodesc dstace from a wth respect to the coformal metrc g a troduced before. For a fte set of pots {a } of M we wll deote by,, W, the quattes a, a, Wa, etc.. For k, l =,, 3 ad > 0, a M, =,..., q, let 5
ϕ = ϕ a, ad d,, d,, d 3, =,, a ; φ, = ϕ, φ, = ϕ, φ 3, = a ϕ, so φ k, = d k, ϕ. Wth these deftos, the φ k, s are uformly bouded W, M, g 0 for every value of the s. We ext recall a stadard fte-dmesoal reducto for fuctos that are close W, to a fte sum of bubbles. It s useful to defe the followg quatty ε, := + + γ G g 0 a, a..7 Gve ε > 0, q N, u W, M, g 0 ad α,, a R q +, R q +, M q, we set 4 A u q, ε = {α,, a,, ε,, rα a 4 k τ, u α ϕ a, < ε, τ < + ε}; V q, ε = {u W, M, g 0 A u q, ε }, see.,. ad.6. For A u q, ε to be o-empty, we wll always assume that τ ε. Uder the above codtos o the parameters α, a ad, the fuctos q = α ϕ a, costtute a smooth mafold W, M, g 0, whch mples the followg well kow result see e.g. [5]. Proposto.. Gve ε 0 > 0 there exsts ε > 0 such that for u V q, ε wth ε < ε, the problem u α ϕã, L g0 u α ϕã, dµ g0 f α,ã, A uq,ε 0 admts a uque mmzer α, a, A u q, ε 0 ad we set Moreover, α, a, depeds smoothly o u. ϕ = ϕ a,, v = u α ϕ, = a..8 The term v = u α ϕ s orthogoal to all ϕ, ϕ, a ϕ, wth respect to the product Fally, for u V q, ε let, Lg0 = L g0, L g0. H u = H u q, ε = ϕ, ϕ, a ϕ Lg 0..9 3 Exstece of subcrtcal solutos Theorem, from [30], descrbes detal the behavour as τ 0 of blowg-up solutos to.5 wth uformly bouded eergy ad zero weak lmt V q, ε, provdg postve lower bouds o J τ a sutable subset of the fuctoal space. I vew of ths, we ca restrct our atteto to ceters a,..., a q close to dstct crtcal pots x,..., x q of wth egatve Laplaca: more precsely, for 6 we ca assume the followg codtos for = 5 they are slghtly modfed: see the above-metoed statemet α Θ p θ a < ɛ ; 3 ā + c x x + c x x τ ɛ, 3 ɛ 3 ; 6
for = τ ad some x { = 0} { < 0} wth x x,. Here, Θ > 0 uformly bouded ad bouded away from zero depeds o the fucto V q, ε, determed Remark 6. of [30]. We ext defe the followg refed egbourhood of potetal subcrtcal blowg-up solutos as V q, ε = {u V q, ε, ad above hold true.} 3. From Lemmata 5.4, 5.5 ad 5.6 t follows that recallg. there exsts ɛ > 0, tedg to zero as ε 0, such that J τ u ɛ 3 for u V q, ε \ V q, ε wth k τ =, so ths ustfes to look for solutos V q, ε oly. For α ϕ V q, ε wth c < α < C, we have the expaso J τ α ϕ + v = J τ α ϕ + J τ α ϕ v + J τ α ϕ v + O v 3. 3. Recall the uform postvty of J τ α ϕ o H u q, ε see.9 ad [5], whch ustfes the followg Defto 3.. For α ϕ V q, ε we defe v as the uque soluto of the mmzato problem J τ α ϕ + v = m J τ α ϕ + v. 3.3 v H α ϕ, v <ε Lemma 3.. Let v be as the above defto. The oe has the followg propertes for α ϕ V q, ε there holds v τ; f u V q, ε s such that J τ u = 0, the α ϕ V q, ε ad u = α ϕ + v. Moreover, for α ϕ V q, ε oe has that J τ α ϕ + v = O ɛ, where ɛ 0 as ε 0. 3.4 3 Proof. Let us deote by Π Hα ϕ the proecto oto H α ϕ : we eed to solve Π Hα ϕ J τ α ϕ + v = 0. Sce J τ s vertble o ths subspace, we ca wrte Π Hα ϕ J τ α ϕ + v = 0 as v = H α ϕ J τ α ϕ [ J τ α ϕ + J τ α ϕ + v J τ α ϕ J τ α ϕ v ]. We kow from Lemma 5.7 that for α ϕ V q, ε oe has J τ α ϕ. Sce by Hölder s cotuty the quatty wth roud brackets the last formula s of order o v, we ca use a cotracto argumet a ball of sze to get the exstece of a soluto to Π Hα ϕ J τ α ϕ + v = 0, wth the estmate. By the defto of v ad the above cotracto argumet we have that J τ α ϕ v = J τ α ϕ + o o φ k, Lg 0. 3.5 Testg thus J τ α ϕ o φ k,, we fd from Lemmata 5.4, 5.5 ad 5.6, aga for α ϕ V q, ε J τ α ϕ φ k, ɛ 3. It s easy to see from.4 ad Lemma 5. that J τ φ k, = o, ad sce v we have that J τ α ϕ vφ k, = o, 3.6 3 7
More geeral, oe fds also that Jα ϕ + θ v vφ k, = o 3 for ay θ 0,. To see ths, sce v φ k, Lg 0, recallg.4 t s suffcet to show that α ϕ + θ v p vϕ dµ g0 α ϕ p vϕ dµ g0 = O 3. Ths, tur, ca be verfed by dvdg the doma of tegrato to { v α ϕ } ad ts complemetary set, usg Hölder s equalty ad the fact that v. Cosequetly J τ α ϕ + v = J τ α ϕ + v φk, = J τ α ϕ φk, +o ɛ = O 3 3, where ɛ teds to zero as ε does. Fally, f a soluto J τ u = 0 exsts o V q, ε, the we may wrte But the u = α ϕ + v + ṽ wth ṽ Lg0 φ k,. 0 = J τ α ϕ + v + ṽṽ = J τ α ϕ + vṽ + J τ α ϕ + vṽṽ + o ṽ, whece ecessarly ṽ = 0 by uform postvty of J τ α ϕ o φ k, Lg 0. Thus J τ u = 0 wth u V q, ε = u = α ϕ + v where v = v α,a, s the uque soluto to 3.3, for whch α ϕ + v V q, ε. Remark 3.. For α ϕ V q, ε ad ν W, M, g 0 wth ν = t ca be show that k τ p+ α ϕ 8 J τ α ϕ ν = α τ + α τ α B εa B εa B εa ϕ + l + r c ϕ + + c c ϕ νdµ g0 c c r c ϕ + c c ϕ + + c ϕ νdµ g0 c c c k,l x k x l r ϕ + νdµ g0 + o, referrg to the table at the ed of the paper for the defto of the costats. As a cosequece of these formulas, oe ca prove that v s deed of order ad ot smaller, as well as determe the leadg order ts expaso. Ayway, due to some cacellato propertes, ths wll ot substatally affect the egevalues of the Hessa of J τ at α ϕ + v, estmated the ext secto. Let us ow set d,, d,, d 3, =,, a, for =,..., q. Lemma 3.. For u = α ϕ + v V q, ε there holds v, d l, v = O. Proof. The boud o v follows from Lemma 3.. Dfferetatg φ k,, v Lg0 = 0 we obta φ k,, d l, v Lg0 = d l, φ k,, v Lg0 = O v, 8
whece deotg by Π φk, the orthogoal proecto oto Π φk, we have Π φk, v due to v. Moreover, sce J τ α ϕ + vv = 0 for every smoothly-varyg vector feld v φ k, Lg 0 of ut orm we have 0 =d l, J τ α ϕ + vv = J τ α ϕ + vd l, α ϕ + vv + J τ α ϕ + vd l, v ad we ca estmate the last summad above as J τ α ϕ + vd l, v = J τ α ϕ + vπ φk, d l, v = O J τ α ϕ + v v, sce φ k,, d l, v = d l, φ k,, v = O v. Thece, J τ α ϕ + v = O mples J τ α ϕ + vvd l, v = J τ α ϕ + vvd l, α ϕ + O. The the clam would follow from Π φk, d l, v, whch we had see before, ad the uform postvty of J τ α ϕ o φ k, Lg 0, provded we show J τ α ϕ + vφ l, v =O, 3.7 cf. 4. ad 4.7 for weaker statemets. Let us prove 3.7 for l =. We ext clam that J τ α ϕ + vϕ v = J τ α ϕ ϕ v + O. From.4, sce v φ k, Lg 0, t s suffcet to show that we must show see the proof of Lemma 3. α ϕ + v p vϕ dµ g0 α ϕ p vϕ dµ g0 = O. Aga, ths ca be see cosderg the set { v α ϕ } ad ts complemetary, usg Hölder s equalty ad v. Thus, from the above clam ad.4 we fd, due to the orthogoaltes φ k,, v Lg0 = 0, J τ α p r α ϕ ϕ v = ϕ α ϕ k τ p ϕ vdµ g0 p+ k τ α ϕ α ϕ 4 L k τ p+ + g0 α ϕ ϕ dµ g0 α ϕ p vdµ g0 α ϕ + p + 3r α ϕ α ϕ k τ p+ + p ϕ dµ g0 α ϕ p vdµ g0. By defto of V q, ε we have τ α ϕ J τ α ϕ ϕ v 4 + L c 0 α g0 α ϕ ϕ dµ g0,τ + 4 + + 3α c 0 α,τ ad recallg 5. ad 5.5 we may smplfy ths to α α,τ α ϕ 4 ϕ vdµ g0 α ϕ + vdµg0 α ϕ + ϕ dµ g0 up to error O. Moreover, from 5.3 ad 5.4 we have L g0 α ϕ ϕ dµ g0 = 4 c 0 α + O α ϕ + vdµg0 9
ad sce da, a, we fd by expadg ad usg Lemma 5. α ϕ 4 ϕ vdµ g0 = α 4 ϕ + vdµ g0 ; α ϕ + vdµg0 = α + ϕ + vdµ g0 ; α ϕ + ϕ dµ g0 = α + ϕ dµ g0 ; α ϕ + vdµg0 = α + ϕ + vdµ g0, up to errors of order O. Therefore, sce J τ α ϕ ϕ v 4 + 8 α α,τ = O due to 3., we obta α α,τ α + + 4 + + 3α α,τ α 4 + α ϕ + ϕ + vdµ g0 vdµ g0 α + ϕ + vdµ g0 up to a error O. Therefore usg aga 3. we have J τ α ϕ ϕ v + ϕ + vdµ g0 α α α ϕ + vdµ g0 + + + 3 α α α ϕ + vdµ g0 up to the same error. Thus, J τ α ϕ ϕ v = O usg 5.6, obtag 3.7 for l =. For l =, 3 oe ca reaso aalogously. Theorem follows from the ext proposto, based o the aalyss of Secto 4, ad Corollary 4.. Proposto 3.. Let 5 ad let : M R be a postve Morse fucto satsfyg.3. The, for every subset {x,..., x q } of { = 0} { < 0}, as τ 0 there exsts a uque u = α ϕ a, + v V q, ε wth u L g0 =, da, x = o ad J τ u = 0. Proof. Due to 3.4, we have J ɛ 3 o V q, ε ad J ˆɛ 3 o V q, ε as log as c < α < C. Thus, by Lemma 3., t s suffcet to look for crtcal pots the set C := {ũα,, a := α ϕ + vα,, a V q, ε ũ L g0 = }, whch s a smooth 3 + -dmesoal mafold W, M, g 0. Vce-versa, we clam that a crtcal pot of J τ C s deed a crtcal pot of J τ. I fact, by Lagrage multpler s rule, the gradet of J τ at a costraed crtcal pot ũ 0 must be orthogoal to C. Sce J τ s dlato-varat, ts gradet o C must be taget to the ut sphere the Lg0 orm. O the other had, by costructo of v, the gradet of J τ at ũ 0 s taget to C := {α ϕ V q, ε u L g0 = } at the pot u 0 such that ũ 0 = u 0 + v 0 wth obvous otato. By the estmate o the dervatves of v Lemma 3., Tũ0 C s early parallel to Tu0 C, whch mples that J τ ũ 0 = 0, as desred. It remas to prove exstece ad uqueess of crtcal pots of J τ C. For the exstece part, oe ca use the expasos Lemmas 5.4, 5.5 ad 5.6, together wth the defto of V q, ε to show that J τ s o-vashg o the boudary of C. For example see the defto of V q, ε, suppose = c x x τ + ε ; = τ. 0
From Lemma 5.5 oe deduces that there exsts ɛ > 0, tedg to zero as ε 0, such that From Lemmas 3. ad 3. oe has also that J τ α ϕ > ɛ 3. J τ uα,, a > ɛ 3, wth a smlar reversed equalty, wth opposte sg, f = c x x ε τ. Aalogous estmates ca be derved for the α ad a dervatves, yeldg that the degree of J τ o C s well-defed ad o-zero. Ths shows the exstece of a crtcal pot for J τ C, whch s freely crtcal for J τ by the above dscusso. Sce by costructo the egatve part of the above solutos s small W, orm, t s possble to show from Sobolev s equalty that t has to vash detcally, so full postvty follows the from the maxmum prcple. Uqueess follows from Lemma 3. ad Proposto 4., mplyg the strct covexty or cocavty of J τ C wth respect to all parameters α s, s ad the coordates of the pots a, provded they are chose so that x s dagoal. 4 The secod varato Let V q, ε be the ope set defed 3.: the am of ths secto s to fd there a early dagoal form of the secod dfferetal of J τ. Let us recall our otato from Secto, ad partcular that of the orthogoal space H u.9. Proposto 4.. For α ϕ + v V q, ε, cosder the decomposto W, M, g 0 = H α ϕ ϕ q ϕ q a ϕ q =: V X α X X a. The there exsts a bass B of W, M, g 0, wth elemets the subspaces of the above decomposto, such that the coeffcets of the the secod dfferetal of J τ wth respect to B have the form [ J τ α k ϕ k + v] B = V + 0 0 0 0 A q,0 0 0 0 0 Λ + 0 0 0 + o, where: V + represets the coeffcets of a symmetrc, postve-defte operator o V wth egevalues uformly bouded away from zero; A q,0 has q egatve egevalues uformly bouded away from zero ad oe-dmesoal kerel; Λ + s postve-defte, wth egevalues uformly bouded away from zero; v stads for the dagoal matrx =,...,q. Remark 4.. The bass elemets B correspodg to the frst two blocks have orms of order, whle the oes correspodg to the last two blocks have orm of order. We made ths choce to guaratee the off-dagoal terms the above matrx to be of order o. Proof of Proposto 4.. We wsh to aalyse.4 for u = α ϕ + v V q, ε. Recall that W, M, g 0 = φ k, k, H α ϕ,
see Secto. We the choose a bass {ν 0, ν, ν,...} for H α ϕ whch s orthoormal wth respect to, Lg 0 ad for... q τ defe B = { φ k,, ν } := { ϕ, ϕ, a ϕ, ν }; k =,, 3, =,..., q. It s ot hard to see that, wth ths choce, the coeffcets [ J τ α k ϕ k + v] B are all of order O, ad our goal s to make ther estmates more precse, cosderg dfferet matrx blocks. Frst block. The fact that J τ α ϕ s uformly postve-defte o H α ϕ s well-kow, see e.g. [5]. The postvty of J τ α ϕ + ε v o the same subspace follows from the Hölder cotuty of the secod dfferetal ad the fact that v = O. Frst two blocks. Testg the secod dfferetal wth ν ad φ, = ϕ we get J τ α ϕ + v ν φ, = o 4. usg the orthogoalty ν, φ, Lg0 = 0, Lemma 5. ad the fact that v. Moreover, from.4 ad the fact that φ, s of order, we fd J τ α k ϕ k + v φ 6 c, φ, = 0 α,τ up to a error of order o. Let us compare the above expresso to δ k,l + α kα l α = A dx,; c 0 = R + r, 4. fα = α α ; α := wth frst- ad secod-order dervatves gve by q α, = α := q = α, α fα = α α fα = δ, α α α α α = α + α α α + + α α α α α 4 + + α 4 + α 4 α + α α α α α α α + α + α + α +. α 4 ; α 4 α 4 The fucto f s scalg varat ad restrcted to {α = } attas ts maxmum at α satsfyg where we have α α α 4 = for all =,..., q, 4 α α fα = α δ, + α α α. 4.3 Comparg 4. ad 4.3 we coclude, wth obvous otato V + 0 J τ ν φ J τ ν φ 3 [ J τ α k ϕ k + v] B = 0 A q,0 J τ φ φ J τ φ φ3 J τ φ ν J τ φ φ J τ φ φ J τ φ φ3 + o. J τ φ3 ν J τ φ3 φ J τ φ3, φ J τ φ3 φ3
Terms off x blocks. Let us cosder ext the teracto of ν wth φ k, = φ k, for k =, 3. Sce v = O, ν = O, ϕ k, φ k, Lg0 = O ad ν, φ k, Lg0 = 0 we smply fd for.4 J τ α l ϕ l + v ν φ,k = J τ α l ϕ l ν φ,k = pr α ϕ k p+ + τ α l ϕ l p ν φ,k dµ g0, 4.4 up to a error of order o. Ideed, by.4, the crucal estmates eeded to verfy 4.4 are α l ϕ l p ν dµ g0 = o = α l ϕ l p φk, dµ g0. 4.5 These however follow easly by expaso ad teracto estmates usg ϕ l, φ k, Lg0 = O, ν, φ k, Lg0 = 0, L g0 ϕ l = 4 ϕ + l + o W, M, g 0 ad Lemma 5.3. For the remag tegral 4.4, we the have α l ϕ l p ν φ,k dµ g0 = α l ϕ l p ν φ,k dµ g0 + o = {ϕ > l αl ϕ l }α l ϕ l p ν φ,k dµ g0 + O ϕ p l ϕ p+ + o L p l = α p ϕ p ν φ,k dµ g0 + O ϕ p l ϕ + ϕ l ϕ p p+ + o L p {ϕ > l αl ϕ l } l 4.6 ad therefore, usg Lemma 5. wth p = + τ α l ϕ l p ν φ,k dµ g0 = α p ϕ p ν φ,k dµ g0 + o. The, sce ν = O, τ = O ad ε r,s = O, we fd α l ϕ l p ν φ,k dµ g0 = α 4 ν φ,k dµ g0 + o = o, where the last equalty follows from Lemma 5. ad φ k,, ν Lg0 = 0. Thus J τ α l ϕ l + v ν φk, = o for k =, 3. 4.7 By exactly the same argumets wth φ, = O as for 4.5 there holds J τ α l ϕ l + v φ, φk, = J τ α l ϕ l + v φ, φ k, = J τ α l ϕ l ϕ φ k, = o for k =, 3. Thus we arrve at V + 0 0 0 [ J τ α l ϕ l + v] B = 0 A q,0 0 0 0 0 J τ φ φ J τ φ φ3 + o 0 0 J τ φ3 φ J τ φ3, φ. 3 3
Last x block. We are left wth the estmate of J τ α k ϕ k + v φ k, φl, = J τ α k ϕ k + vφ k, φ l, for k, l =, 3. Usg the fact that φ k, L g0 α m ϕ m + vdµ g0 = o = φ k, α m ϕ m + v p dµ g0 for k =, 3, whch follows from v = O, Lemma 5. ad Lemma 5., we fd for.4 J τ α m ϕ m + vφ k, φ l, [ = φ k τ k, L g0 φ l, p r ] α m ϕ m + v p φ k, φ l, dµ g0 p+ k τ =: I =: I k τ p+ k τ I = I p+ c 0 α I + o,τ. 4.8 I the latter formula, recallg. ad the defto of V q, ε, we have used the fact that k τ p+ = c 0 α,τ + o ad that both I, I are of order. Let us frst compute I, for whch we clearly have I = p r α m ϕ m p r αm ϕ φ k, φ l, dµ g0 + pp m+ v α m ϕ m p φ k, φ l, vdµ g0 k τ k τ up to a error o, as v = O, ad therefore stll up to a error o I =p r α m ϕ m p φ k, φ l, dµ g0 k τ + 4 + 4 α α m ϕ m 6 φk, φ l, vdµ g0. α,τ As due to da, a for, the teractos terms ε,.7 are of order = o, we fd r δ, α p I =p k τ + 4 + 4 ϕ p φ k, φ l, dµ g0 α α,τ δ, α 6 ϕ 6 φ k, φ l, vdµ g0 up to a error o. Usg 3., up to the same error we may smplfy ths to I =p r δ, α p k τ + 4 + 4 δ,α ϕ p φ k, φ l, dµ g0 + 4 + δ, ϕ 6 φ k, φ l, vdµ g0 for some ε > 0 small ad fxed. Moreover, by orthogoalty ad 5. B εa x φ k, φ l, dµ g0 r α ϕ + v k τ α ϕ + v = r α ϕ k τ α ϕ = 4 α α p+,θ c c 0 c c c c 0 τ + o, 4
whece by 3. ad the fact that p = + τ we arrve at I =4 + [ + + c c c τ] θ δ, c 0 c c 0 + 4 + δ, x φ k, φ l, dµ g0 B εa + 4 + 4 δ,α ϕ 6 φ k, φ l, vdµ g0. ϕ p φ k, φ l, dµ g0 Let us compute the last tegral above, whch s of order O, as t s v. There holds 4 ϕ 6 φ k, φ l, vdµ g0 = =d k, d k, φ l, vdµ g0 φ l, vdµ g0 d k, φ l, vdµ g0 Due to orthogoalty, the frst tegral above s of order o ad deotg by φ l, d k, vdµ g0. ŵ = Π φk, Lg 0 w for w W, M, g 0 4.9 the orthogoal proecto oto φ k, Lg 0 we have, up to a error o d k, φ l, vdµ g0 = d k, φ l, vdµ g0 due to the orthogoaltes v, φ k, Lg0 = 0 ad the fact that v = O. Hece, usg the same otato as 4.9, we arrve at I =4 + [ + + c c c τ] θ δ, ϕ p φ k, φ l, dµ g0 c 0 c c 0 + 4 + δ, x φ k, φ l, dµ g0 B ca 4 + δ,α d k, φ l, vdµ g0 + φ l, d k, vdµ g0. Due to the fact that v = O we have, stll up to a o J τ α m 8 L g0 ϕ m v = c 0 α p+,τ 4 v + m m ad we recall from 3.5 that J τ α m ϕ m v = J τ α m ϕ m +o o φ l, Lg 0. From ths we deduce, aga by smalless of teractos terms ε, v, + d k, φ l, vdµ g0 = c 0α p+,τ 8 J τ α m ϕ m d k, φ l, + v, d k, φ l, Lg0 4 ad, by orthogoalty ad Lemma 5., there holds up to a error o v, d k, φ l, Lg0 = d k, v, φ l, Lg0 = 4 d k, vd l, ϕ + dµg0 = 4 + d k, vφ l, dµ g0. 5
We therefore coclude that, up to a error o, I =4 + [ + + c c c τ] θ δ, c 0 c c 0 + 4 + δ, x φ k, φ l, dµ g0 B εa 4 δ, α c 0 α p+,τ 8 J τ α m ϕ m d k, φ l,, ϕ p φ k, φ l, dµ g0 at whch pot v has bee elmated from the ma terms the expaso. By Lemma 3. we the have J τ α m ϕ m φk, = o, so we may pass from d k, φ l, to d k, φ l, the above formulas ad, as J τ α m ϕ m = O, we obta c 0 α p+,τ 8 J τ α m ϕ m d k, φ l, = α m τ ϕ + m l + mr c ϕ + m + c + α m τ α m B εa m B εa m B εa m c m m m ϕ m d k, φ l, dµ g0 c c mr c ϕ + m c c ϕ + m + c m m m ϕ m d k, φ l, dµ g0 c c c m x m r ϕ + m d k, φ l, dµ g0. m m Stll by the fact that ε, = o we therefore arrve at I =4 + [ + + c c c τ] θ δ, c 0 c c 0 + 4 + δ, x φ k, φ l, dµ g0 4 δ, τ + τ B εa B εa B εa B εa ϕ p φ k, φ l, dµ g0 ϕ + l + r c ϕ + + c c ϕ d k, φ l, dµ g0 c c r c ϕ + c c ϕ + + c ϕ d k, φ l, dµ g0 c c c x r ϕ + d k, φ l, dµ g0, 6
up to some o. By oddess, we may smplfy ths to I =4 + [ + + c c c τ] θ δ, δ k,l c 0 c c 0 + 4 + δ,δ k,l x φ k, φ k, dµ g0 4 δ, δ k,l τ + τ B εa B εa B εa B εa ϕ p φ k, φ k, dµ g0 ϕ + l + r c ϕ + + c c ϕ d k, φ k, dµ g0 c c r c ϕ + c c ϕ + + c ϕ d k, φ k, dµ g0 c c c x r ϕ + d k, φ k, dµ g0 By Lemma 5. t follows that, up to some o, for k =, 3 4 + ϕ d k, φ k, dµ g0 = L g0 ϕ d k, φ k, dµ g0 = ϕ, d k, ϕ Lg0 = d k, ϕ, d k, ϕ Lg0 d k, ϕ, d k, ϕ Lg0 =d k, φ,, φ k, Lg0 φ k, Lg0 = o, as φ,, φ k, Lg0 ad φ k, L g0 are almost costat a ad. So we smplfy to I 4 = + + + δ,δ k,l + τ B εa B εa [ + + c c c τ] θ δ, δ k,l c 0 c c 0 x φ k, φ k, dµ g0 δ, δ k,l τ c r c c c c c ϕ + d k, φ k, dµ g0 ϕ p φ k, φ k, dµ g0 l + r c B εa B εa Next, for the frst summad above we fd that, up to a error o θ ϕ p φ k, φ k, dµ g0 = φ k, φ k, dµ g0 + θ ϕ τ = + d k, ϕ + φ k, dµ g0 + B εa = 4 + φ k,, φ k, Lg0 + θ B εa B εa c ϕ + d k, φ k, dµ g0 x r ϕ + d k, φ k, dµ g0. + r θ φ k, φ k, dµ g0 l + r φ k, φ k, dµ g0 φ k, φ k, dµ g0 7
usg Lemma 5. ad properly expadg. Recallg 4.8, we thus coclude k τ p+ 8 J τ α m ϕ m + vφ k, φ l, = =δ, δ k,l + + c c 0 c c c c 0 τ τ B εa B εa l + r c L g0 4 φ k,φ l, dµ g0 φ k, φ k, dµ g0 + τ c ϕ + B εa d k, φ k, dµ g0 + τ x r ϕ + d k, φ k, dµ g0 + I 4 l + r φk, φ k, dµ g0 B εa B εa ad partcular for =,..., q, ad =,..., we have, up to a error o c r c c c c c V + 0 0 0 [ J τ α k 0 ϕ k + v] B = A q,0 0 0 0 0 J τ ϕ ϕ 0 0 0 0 J τ a ϕ + x φ k, φ k, dµ g0 ϕ a ϕ. d k, φ k, dµ g0 4.0 Last dagoal terms. Cocerg -dervatves, we frst otce that mxed dervatves dfferet s are of order, whch s a o sce 5. Therefore t s suffcet to compute secod dervatves wth respect to the same. Ths correspods to k τ p+ 8 J τ α m ϕ m + v ϕ = + + c c c τ c 0 c c 0 + τ τ B εa B εa B εa l + r ϕ dµ g0 l + r c c ϕ + ϕ dµ g0 + τ x r ϕ + ϕ dµ g0 + B εa B εa φ k, φ k, dµ g0 c r c c c c c ϕ + x ϕ dµ g0. ϕ dµ g0 The secod-last summad vashes ad ϕ p φ k, φ k, dµ g0 = c k + o, cf. Lemma 5., whece k τ p+ 8 J τ α m ϕ m + v ϕ = c + + c c c τ c 0 c c 0 + τ + τ B εa B εa l + r ϕ dµ g0 τ c r c c c c c ϕ + B εa ϕ dµ g0 + l + r c B εa c ϕ + r ϕ dµ g0. ϕ dµ g0 Moreover, ϕ + ϕ dµ g0 = ϕ + ϕ dµ g0 + ϕ dµ g0 = + c + o, 8
ad + B εa r ϕ dµ g0 = = B εa B εa r ϕ ϕ + dµ g0 r ϕ ϕ + B εa dµ g0 r ϕ + ϕ dµ g0. Thus, recallg 3., partcular c τ + c = o, we arrve at k τ p+ 8 J τ α m ϕ m + v ϕ = c τ + τ τ B εa l + r B εa + ϕ ϕ dµ g0 + c c B εa τ 3 l + r ϕ dµ g0 B εa ad for the last tegral above we fd passg to tegrato over R r ϕ ϕ + dµ g0 = r δ 0, δ + 0, dx R = up to some error of order o. Cosequetly, r ϕ ϕ + dµ g0, r δ 0, dx = R r + r dx = 8 k τ p+ 8 J τ α m ϕ m + v ϕ = c + c c 6 τ c c + τ B εa l + r ϕ dµ g0 τ B εa Fally, we calculate passg to tegrato over R ad up to a o + l + r ϕ dµ g0 B εa l + r + ϕ c ϕ dµ g0. = l + r δ 0, δ + 0, dx = l + r δ 0, δ + R R R r + r δ 0, δ + 0, dx l + r δ 0, δ + 0, dx, R where the frst summad above vashes by rescalg, ad we are reduced to 0, dx k τ p+ 8 J τ α m ϕ m + v ϕ = c + c c R 6 τ + τ r c c + r δ 0, δ + 0, dx, where, up to some o, r + r δ 0, δ + 0, dx = R R r r + r dx = + ĉ 3, ĉ 3 = R r r + r dx. 4. + By a explct computato all the above costats ca be explctly evaluated, we coclude that up to a error o k τ p+ 8 J τ α m ϕ m + v ϕ = c + c c 6 c c 9 ĉ 3 τ = Γ 8Γ + τ.
Thece we arrve at wth =,..., q ad =,..., V + 0 0 0 [ J τ α k 0 ϕ k + v] B = A q,0 0 0 0 0 Λ + 0 0 0 J a τ ϕ a ϕ up to o, where Λ + > 0 s as the statemet. We are left wth the computato of the terms for stace we cosder k τ p+ 8 J τ α m ϕ m + v a ϕ = + + c c 0 c c c c 0 τ τ B εa B εa l + r c k τ p+ 8 J τ α m ϕ m + v a a ϕ dµ g0 + τ c a ϕ dµ g0 + τ ϕ + x r ϕ + a ϕ dµ g0 + At ths pot some smplfcatos occur. From the relato c τ + c we obta cacellato of the terms volvg ad c a ϕ dµ g0 = c 3 + o; = o r c B εa ϕ a ϕ, B εa B εa l + r a c r c c c c c ϕ + ϕ dµ g0 a ϕ dµ g0 x a ϕ dµ g0.. Usg as well the relatos ϕ + a ϕ dµ g0 = + c 3 + o together wth c c 0 c c c c0 c3 = c c c c c c c, due to the fact that c 0 = c ad c = c 3, to obta k τ p+ 8 J τ α m ϕ m + v a ϕ = c 3 τ + τ B εa B εa l + r a x ϕ + a ϕ dµ g0 + ϕ dµ g0 τ B εa B εa l + r x a ϕ dµ g0. a ϕ dµ g0 + ϕ Moreover we have, passg to tegrato over R, up to a error o + l + r a ϕ dµ g0 = l + r a δ + a 0, R δ 0, dx B εa + x a = R + δ 0, r δ 0, dx l + r + δ 0, a δ 0, dx R 0
ad fd for the frst summad x R + We therefore are left wth + δ r 0, a k τ p+ 8 J τ α m ϕ m + v a ϕ = x ϕ + a ϕ dµ g0 + B εa δ 0, dx = δ 4 0, a δ 0, dx = c 3 R. B εa Fally, passg to tegrato over R, up to some o there holds + x l a ϕ a dµ g0 = δ + a 0, δ 0, dx B εa = δ,l R x δ + 0, a R x l x a ϕ dµ g0. δ 0, dx x a l δ + 0, R a δ 0, dx, ad smlarly for =,...,. Dagoalzg the Hessa we have x = l= l x l ad x δ + a 0, R δ 0, dx = δ x R r 0, R + dx = r + r dx, + ad smlarly for =,...,, so we coclude that k τ p+ 8 J τ α m ϕ m + v a Smlarly, oe ca show aalogous formula for ay couple of dces The proof s thereby complete. k τ p+ 8 J τ α m ϕ m + v a k ϕ a l ϕ = c. ϕ = c k,l. From Proposto 4. we deduce that the kerel of J τ s exactly oe-dmesoal. The presece of a kerel s uavodable due to the scalg varace of J τ, but ths degeeracy turs out to be mmal. We ca therefore restrct ourselves to some homogeeous costrat. Corollary 4.. of.5 V q, ε. The Let I τ = J τ [ Lg0 =] or I τ = J τ [ kτ =], ad let ũ be ormalzato of a soluto u mi τ, ũ = q + q m, a. 5 Appedx: some techcal estmates I ths appedx, recallg our otato, we collect some useful statemets ad formulas proved [30]. Lemma 5.. There holds L g0 ϕ a, = Oϕ + a,. More precsely o a geodesc ball B αa for α > 0 small L g0 ϕ a, = 4 ϕ + a, c ra H a + r a ra H a ϕ + a, = + R g a u a ϕ a, + or a ϕ + where r a = d ga a,. Sce R ga = Ora coformal ormal coordates, cf. [3], we obta a,,
L g0 ϕ a, = 4 [ c r a H a a + H a ax]ϕ + a, + O ϕ a, for = 5; L g0 ϕ a, = 4 ϕ + a, = 4 [ + c L g0 ϕ a, = 4 ϕ + a, = O ϕ a, for 7, W a l r]ϕ + a, + O ϕ a, for = 6; where W a = Wa. The expasos stated above persst upo takg ad a dervatves. Lemma 5.. Let θ = τ ad k, l =,, 3 ad, =,..., q. The, for ε, as.7, there holds uformly as 0 τ 0 φ k,, φ k,, a φ k, Cϕ ; θ ϕ 4 τ φ k, φ k, dµ g0 = c k d + Oτ + +θ, c k > 0; for up to some error of order Oτ + + ε + 4, θ ϕ + τ φ k, dµ g0 = b k d k, ε, = ϕ τ d k, ϕ + dµ g0 ; v θ 4 ϕ τ φ k, φ l, dµ g0 = O for k l ad for k =, 3 θ for = 5 φ k, dµ g0 = O τ + l for = 6 ; 4 4 for 7 ϕ + τ v v θ ϕ α τ ϕ β dµ g 0 = Oε β, for, α + β =, α τ > > β ; ϕ ϕ dµ g0 = Oε, l ε,, ; v,, a ε, = Oε,,. wth costats b k = R c = R dx +r + dx + r, c = for k =,, 3 ad 4 R r dx + r +, c 3 = Lemma 5.3. For u V q, ε wth k τ =, cf..,ad ν H u q, ε there holds [ J τ α ϕ ν = O r τ θ r + r r +θ r + r +θ + r r s ε + r,s θ r R r dx + r +. ] ν. Lemma 5.4. For u V q, ε ad ε > 0 suffcetly small the three quattes J τ uφ,, J τ α ϕ φ,, α J τ α ϕ ca be wrtte as α α,τ `c 0 α α p+,τ + `b k l θ α p `c α k α l α ε k,l α ε, α k k k k `d αk α H 3 W l 4 α k H k k α for = 5 3 k α k W k l k k α for = 6 4 k 0 for 7
up to a error of order O τ + r r s + r 4 r + ε + r,s + J τ u, wth postve costats `c 0, `c, `b, `d 8 + 8 8 `b = b c, `c = c 0 c, `d = d 0 c, `c 0 = 8 c 0. 5. 0 I partcular for all α α p+,τ θ α p = + O τ + r s r + ε r,s + J τ u. Lemma 5.5. For u V q, ε ad ε > 0 suffcetly small the three quattes J τ uφ,, J τ α ϕ φ, ad α J τ α ϕ ca be wrtte as α α,τ c τ + c b α ε, + α d wth postve costats c, c, d, b up to some error O τ + r s H 3 W l 4 r r for = 5 for = 6, 0 for 7 + 4 r + ε + r,s + J τ u. Lemma 5.6. For u V q, ε ad ε > 0 suffcetly small the three quattes J τ uφ 3,, J τ α ϕ φ 3, ad a α J τ α ϕ ca be wrtte as α α,τ č 3 + č 4 3 + ˇb 3 wth postve costats č 3, č 4, ˇb 3 up to some error O τ + r s Lemma 5.7. For every u V q, ε there holds α r r α a + 4 r ε,, + ε + r,s + J τ u. J τ u τ + r s r r + r + α α p+,τ r θ α p r r + ε + r,s + v. Theorem. Suppose that 5, : M R s postve, Morse ad satsfes.3. The for ε > 0 suffcetly small there exsts c > 0 such that for ay u V q, ε wth k τ = there holds Ju c τ + r s r r + r + α α p+,τ r θ r αr p + εr,s, uless there s a volato of at least oe of the four codtos τ > 0; there exsts x x { = 0} { < 0} ad da, x = O ; v α = Θ θ p + o ; c τ = c k k + o k k where Θ s a postve costat, uformly bouded ad bouded away from zero, that depeds o u see Remark 6. [30]. I the latter case there holds... q = τ ad settg a = exp gx ā, 3
we stll have up to a error o the lower boud 3 Ju τ + x 9 x + 5 9π [Hx 3 + x G g0 x, x ] x γ 3 + ā + č4 x x č 3 3 + α Θ p θ a x 90 x + 86 Hx π 3 k x k x k k k + 86 π x k Hx k x k 3 k case = 5 ad Ju τ + c x c x + ā + č4 x x θ + α Θ p č 3 a 3 case 6. The costats appearg above are defed by c 0 = R dx +r, r c = c + r + l + r dx, c r r = 0 c + r R 0 R 4 R r č 3 = R + r dx, č 4 = + r dx.; 4 dx 4 b = ; d = r + r c R 0 + r + c R + r dx. + 0 + dx; From the proof of Proposto 5. ad Sectos 4,5 ad 6 [30] we wll eed the followg estmates up to a error of order O τ + r + 4 r r s ε + r,s there holds b = b α ϕ p+ dµ g0 = + d θ α c 0 θ α p+ H 3 W l 4 0 + c θ + b α α + τ + c +θ α α θ ε, ; d = R r dx + r + ; 5. recallg.7, oe has ϕ L g0 ϕ dµ g0 = b ε, + O r s 4 r + ε + r,s, b = 4 b ; 5.3 up to a error Oτ +, there holds 4 v up to a error of order O τ + r 4 r α α ϕ L g0 ϕ 4 dµ g 0 = c 0 ; 5.4 + r s ε + r,s. oe has ϕ L g0 ϕ dµ g0 =4 c 0 α + b α α ε,. 5.5 4
v If ϕ s as.6, the ϕ + νdµ g0 v L g0 ϕ 4 ϕ + v up to a error Oτ + oe has 4 ϕ p+ dµ g0 = c 0 τ θ + c θ + c +θ L + g 0 + d 3 for = 5 = O l 3 0 3 for = 6 v ; 5.6 4 for 7 H 3+θ W l 4+θ 0, c = R r dx + r ; 5.7 v up to a error or order Oτ + r s J τ α ϕ = c α α ϕ L g0 ϕ dµ g0 α ϕ p+ p+ +θ α α,τ d r r + 4 r + ε + r,s there holds = α α ϕ L g0 ϕ dµ g0 θ c 0 α p+ θ p+ H 3 W l 4 0 α α,τ b c θ α + α α θ,τ α α,τ τ ε,.; 5.8 v f ε, s as.7, the ε, = ε, + O 4 + ε +, case < or d g0 a, a o. 5.9 Fally, we derve oe last techcal estmate. Recallg., from 5.5 we have, up to a error o, r α ϕ =α α L g0 ϕ ϕ dµ g0 = 4 c 0 α = 4 c 0 α 5.0 dx wth c 0 = R +r. From 5. stead, stll up to a error o, we get α ϕ p+ dµ g0 = c 0 θ α p+ + c θ α τ + c +θ α = c 0 α p+,θ + α θ c τ + c wth costats gve by Therefore c = r α ϕ k τ α ϕ = 4 α α p+,θ ad we coclude aga from 3. that l + r R + r dx, ad c = 4 r α ϕ k τ α ϕ = 4 α α α p+,θ α p+,θ α θ R r + r dx. 5. c τ + c c 0 c 0 + o c c c τ + o. 5. c 0 c c 0 5
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