LOCAL MONOTONICITY AND MEAN VALUE FORMULAS FOR EVOLVING RIEMANNIAN MANIFOLDS


 Marcia Cameron
 2 years ago
 Views:
Transcription
1 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS FOR EVOLVING RIEMANNIAN MANIFOLDS KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Abstract. We erive ietities for geeral ows of Riemaia metrics that may be regare as local meavalue, mootoicity, or Lyapuov formulae. These geeralize previous work of the rst author for mea curvature ow a other oliear i usios. Our results apply i particular to Ricci ow, where they yiel a local mootoe quatity irectly aalogous to Perelma s reuce volume ~ V a a local ietity relate to Perelma s average eergy F.. Itrouctio To motivate the local formulas we erive i this paper, cosier the followig simple but quite geeral strategy for ig mootoe quatities i geometric ows, whose core iea is simply itegratio by parts. Let (M ; g(t)) be a smooth oeparameter family of complete Riemaia maifols evolvig for t [a; b] by (.) t g = h: Observe that the formal cojugate of the timeepeet heat operator t o the evolvig maifol (M ; g(t)) is ( t + + tr gh). If '; : M [a; b]! R are smooth fuctios for which the ivergece theorem is vali (e.g. if M is compact or if ' a a their erivatives ecay rapily eough at i ity), oe has (.) ' = f [( )'] + '[( t M M t t + + tr gh) ]g : If ' solves the heat equatio a solves the ajoit heat equatio, it follows that the itegral R ' is iepeet of time. More geerally, if [( M t )'] a '[( t + + tr gh) ] both have the same sig, the R ' will be mootoe i M t. If the prouct ' is geometrically meaigful, this ca yiel useful results. Here are but a few examples. Example. The simplest example uses the heat equatio o Eucliea space. Let (.3) (x; t) = [4(s t)] e jy xj = 4(s t) (x R ; t < s) eote the backwar heat kerel with sigularity at (y; s) R R. If ' solves the heat equatio a either it or its erivatives grow too fast at i ity, the '(y; s) = lim '(x; t) (x; t) x: t%s R R Because t '(x; t) (x; t) x = 0, oe has '(y; s) = R '(x; t) (x; t) x for all R R y R a t < s, which illustrates the averagig property of the heat operator. Here a throughout this paper, eotes the volume form associate to g(t).
2 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Example. Let F t : M,! M t R + be a oeparameter family of hypersurfaces evolvig by mea curvature ow, t F t = H, where H is the mea curvature a the outwar uit ormal of the hypersurface M t. This correspos to h = HA i (.), where A is the seco fuametal form. De e by formula (.3) applie to x R + a t < s. Usig tr g h = H, oe calculates that ( t + H ) = (x y)? H (s t) : Hece by (.), ' = [( )'] (x y)? t t H (s t) ' : M t M t This is establishe for ' by Huiske [0, Theorem 3.] a geeralize by Huiske a the rst author [8, ] to ay smooth ' for which the itegrals are ite a itegratio by parts is permissible. Hece R is mootoe oicreasig i time a is costat precisely M t o homothetically shrikig solutios. The mootoicity implies that the esity MCF O := lim t%0 of the limit poit O = (0; 0) is well e e. Aother RM t cosequece is that sup M b ' sup M a ' if ( t )' 0 for t [a; b]. Example 3. A compact Riemaia maifol (M ; g(t)) evolvig by Ricci ow correspos to h = Rc i (.), so that tr g h = R. If ' a = (f jrfj + R) + f (4) = e f ; the Perelma s etropy may be writte as W (g(t); f(t); (t)) = R M '. If =t = a ( t + )f = jrfj R, the ( t + R) = j Rc +rrf gj (4) = e f : I this case, (.) becomes W(g(t); f(t); (t)) = j Rc +rrf t M gj (4) = e f ; which is formula (3.4) of [3]. I particular, W is mootoe icreasig a is costat precisely o compact shrikig graiet solitos. Example 4. Agai for (M ; g(t)) evolvig smoothly by Ricci ow for t [a; b], let ` eote Perelma s reuce istace [3] from a origi (y; b). Take ' a choose v to be the reucevolume esity 3 v(x; t) = M t [4(b t)] = e `(x;b t) (x M ; t < b): The Perelma s reuce volume is give by V ~ (t) = R '. By [3, 7], M ( t + R)v 0 hols i the barrier sese, hece i the istributioal sese.4 Throughout this paper, r represets the spatial covariat erivative, a = trgrr. 3 The formula use here a throughout this paper i ers from Perelma s by the costat factor (4) =. This ormalizatio is more coveiet for our applicatios. 4 It is a staar fact that a suitable barrier iequality implies a istributioal iequality. See [4] for relevat e itios a a proof. A irect proof for v is fou i [38, Lemma.].
3 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 3 Thus oe obtais mootoicity of the reuce volume if M is compact or if its Ricci curvature is boue. More geerally, oe gets mootoicity of R ' for ay oegative supersolutio ' of the heat equatio. I particular, takig '(x; t) = R(x; t) R mi (0) M o a compact maifol a otig that ( t )' 0 hols poitwise, oe veri es that R [R R M mi (0)]v is oecreasig i time. I [], Felma, Ilmae, a the thir author itrouce a expaig etropy a a forwar reuce volume for compact maifols evolvig by Ricci ow. Mootoicity of these quatities may also be erive from (.) with '. Similar ieas play importat roles i Perelma s proofs of i eretial Harack estimates [3, 9] a pseuolocality [3, 0]. The strategy of itegratio by parts ca be aapte to yiel local mootoe quatities for geometric ows. We shall preset a rigorous erivatio i Sectio whe we prove our mai result, Theorem 7. Before oig so, however, we will explai the uerlyig motivatios by a purely formal argumet. Suppose for the purposes of this argumet that = S atb t is a smooth, precompact subset of M [a; b]. Assume that t is smooth with outwar uit ormal, a let eote the measure o t iuce by g(t). If the prouct ' vaishes o, the (.4) [( f )'] + '[( t t + + tr gh) ]g t b = ' t + (' hr ; i hr'; i) t: a t t This formula may be regare as a spacetime aalog of Gree s seco ietity. I the special case that is the superlevel set f(x; t) : (x; t) > 0g a both a a b are empty, the = jr j r, whece (.4) reuces to (.5) [( f >0gf )']+'[( t t ++tr gh) ]g t+ 'jr j t = 0: f =0g Formula (.5) eables a strategy for the costructio of local mootoe quatities. Here is the strategy, agai presete as a purely formal argumet. Let ' a > 0 be give. De e = log, a for r > 0, let (r) = log(r ). Notice that r (r) = r for all r > 0. Take to be the set e e for r > 0 by (.6) := f(x; t) : (x; t) > r g = f(x; t) : (r) > 0g: (Whe is a fuametal solutio 5 of a backwar heat equatio, the set is ofte calle a heatball.) Assume for the sake of this formal argumet that the outwar uit ormal to the time slice (t) := \(M ftg) is = jr j r. Observe that (.7) ( t + ) = ( t + + tr gh) jr j tr g h: Applyig the coarea formula to each time slice (t), followe by a itegratio i t, oe obtais (.8) jr j ' t = jr j' t: r r 5 See Sectio 5 below.
4 4 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Similarly, oe has (.9a) (tr g h) (r) ' t = (tr g h)[ r r log(r )]' t = (.9b) (tr g h)' t; r because the bouary itegral vaishes i this case. Now by rearragig (.5) a usig (.7) (.9), oe gets f (r) [( )'] + [( t t + + tr gh) ]'g t = jr j (tr g h) (r) ' t 'jr j t + (tr g h)' t E r = jr j r (tr g h) (r) ' t jr j (tr g h) (r) ' t: r De ig (.0) P '; (r) := jr log j (tr g h) log(r ) ' t a applyig a itegratig factor, oe obtais the followig formal ietity. Sice log(r ) = (r) > 0 i, this ietity prouces a local mootoe quatity wheever ( t )' a '( t + + tr gh) have the same sig. Prototheorem. Wheever the steps above ca be rigorously justi e a all itegrals i sight make sese, the ietity (.) r P'; (r) = r r + log(r will hol i a appropriate sese. )( t )' + [( t + + tr gh) ]' t I spirit, (.) is a parabolic aalogue of the formula! r r '(x) = jx yj<r r + (r jx yj )' ; jx yj<r which for harmoic ' (i.e. ' = 0) leas to the classical local meavalue represetatio formulae '(y) =! r '(x) = jx yj<r! r '(x) : jx yj=r The mai result of this paper, Theorem 7, is a rigorous versio of the motivatioal prototheorem above. We establish Theorem 7 i a su cietly robust framework to provie ew proofs of some classical meavalue formulae (Examples 5 8), to geerate several ew results (Corollaries 3, 5, 8, 9, 3, 6, 7) a to permit geeralizatios for future applicatios. Our immeiate origial results are orgaize as follows: i Sectio 4, we stuy Perelma s reuce volume for maifols evolvig by Ricci ow; i Sectio 5, we iscuss heat kerels o evolvig Riemaia maifols (icluig xe maifols as a iterestig special case); a i Sectio 6, we cosier Perelma s average eergy for maifols evolvig by Ricci ow. I [7], the thir author applies some of these results to obtai local regularity theorems
5 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 5 for Ricci ow. Potetial future geeralizatios that we have i mi cocer varifol (Brakke) solutios of mea curvature ow, solutios of Ricci ow with surgery, a fuametal solutios i the cotext of weak (Bakry Émery) Ricci curvature, e.g. [3]. As ote above, Theorem 7 allows ew proofs of several previously kow local mootoicity formulae, all of which shoul be compare with (.). To wit: Example 5. Cosier the Eucliea metric o M = R with h = 0. If is the backwars heat kerel (.3) cetere at (y; s) a the heatball (y; s) is e e by (.6), the (.0) becomes P '; (r) = '(x; t) jy xj t: 4(s t) Thus (.) reuces to P'; (r) (.) r r = jy xj (y;s) r + log(r (y;s) )( t )' t: Sice R (y;s) 4(s t) t =, this implies the mea value ietity (.3) '(y; s) = r '(x; t) jy xj t 4(s t) (y;s) for all ' satisfyig ( t )' = 0. This localizes Example. To our kowlege, Pii [8, 9, 30] was the rst to prove (.3) i the case =. This was later geeralize to > by Watso [35]. The geeral formula (.) appears i Evas Gariepy [9]. There are may similar meavalue represetatio formulae for more geeral parabolic operators. For example, see Fabes Garofalo [0] a Garofalo Lacoelli [4]. (Also see Corollaries 3 a 6, below.) Example 6. Surface itegrals over heatballs rst appear i the work of Fulks [3], who proves that a cotiuous fuctio ' o R (a; b) satis es '(y; s) = jy xj r '(x; t) p 4jy xj (s t) + [jy xj (s t)] for all su cietly small r > 0 if a oly if ' is a solutio of the heat equatio. (Compare to Corollary 5 below.) Example 7. Previous results of the rst author [5] localize Example for mea curvature ow. O R + ( ; 0), e e (x; t) := ( 4t) = e jxj =4t. Substitute ( t + + tr gh) = jr? + Hj a tr g h = H ito (.0) a (.). If the spacetime track M = S t<0 M t of a solutio to mea curvature ow is well e e i the cylier B(0; p r =) ( r =4; 0), the [5] proves that formula (.), with the itegrals take over \ M, hols i the istributioal sese for ay r (0; r) a ay ' for which all itegral expressios are ite. I particular, P ; (r)=r is mootoe icreasig i r. The esity MCF O := lim t%0 (x; t) of the limit poit O = (0; 0) ca thus be calculate RM t locally by MCF P ; (r) O = lim r&0 r :
6 6 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING (Compare to Corollary 8 below.) Relate work of the rst author for other oliear i usios is fou i [7]. Example 8. Perelma s scale etropy W a the forwar reuce volume + are localize by the thir author [5, Propositios 5., 5.3, 5.4]. Although oly state there for Kähler Ricci ow, these localizatios remai vali for Ricci ow i geeral. They are motivate by the rst author s work o mea curvature ow [6] a arise from (.) by takig ' to be a suitable cuto fuctio e e with respect to ` a `+, respectively. The remaier of this paper is orgaize as follows. I Sectio, we rigorously erive Theorem 7: the geeral local mootoicity formula motivate by formula (.) above. I Sectio 3, we erive a local graiet estimate for solutios of the cojugate heat equatio. I Sectios 4 5, we apply this machiery to obtai ew results i some special cases where our assumptios ca be checke a i which (.) simpli es a becomes more familiar. The Appeix (Sectio 7) reviews some relevat properties of Perelma s reuce istace a volume. Ackowlegmets. K.E. was partially supporte by SFB 647. D.K. was partially supporte by NSF grats DMS0584, DMS , a a Uiversity of Texas Summer Research Assigmet. L.N. was partially supporte by NSF grats a a Alfre P. Sloa Fellowship. P.T. was partially supporte by a EPSRC Avace Research Fellowship. L.N. thaks both Professor Beett Chow a Professor Peter Ebefelt for brigig Watso s meavalue equality to his attetio. This motivate him to stuy heatball costructios a i particular [0] a [5]. He also thaks Professor Peter Li for may helpful iscussios.. The rigorous erivatio Let < a < b <, a let (M ; g(t)) be a smooth oeparameter family of complete Riemaia maifols evolvig by (.) for t [a; b]. As ote above, the formal cojugate of the heat operator t o (M ; g(t)) is ( t + + tr gh). For R, we aopt the staar otatio [] + := maxf; 0g. Let be a give positive fuctio o M [a; b). As i Sectio, it is coveiet to work with (.) := log a the fuctio e e for each r > 0 by (.) (r) := + log r: For r > 0, we e e the spacetime superlevel set ( heatball ) (.3a) (.3b) = f(x; t) M [a; b) : > r g = f(x; t) M [a; b) : (r) > 0g: We woul like to allow to blow up as we approach time t = b; i particular, we have i mi various fuctios which have a sigularity that agrees asymptotically with a (backwars) heat kerel cetere at some poit i M at time t = b. (See Sectios 6 5.) I this cotext, we make, for the momet, the followig three assumptios about.
7 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 7 Assumptio. The fuctio is locally Lipschitz o M [a; s] for ay s (a; b). Assumptio. There exists a compact subset M such that outsie [a; b). Assumptio 3. There exists r > 0 such that! j j = 0 a lim s%b \(M fsg) jr j t < : is boue Remark 4. By the cotiuity of from Assumptio a its boueess from Assumptio, we ca be sure, after reucig r > 0 if ecessary, that r outsie some compact subset of M (a; b]. I particular, we the kow that the superlevel sets lie isie this compact subset for r (0; r]. Remark 5. By Assumptio 3 a compactess of, oe has R j j t <. Remark 6. We make o irect assumptios about the regularity of the sets themselves. Let ' be a arbitrary smooth fuctio o M (a; b]. By Assumptio 3, the quatity (.4) P '; (r) := [jr j (r)(tr g h)]' t is ite for r (0; r]. Our mai result is as follows: Theorem 7. Suppose that (M ; g(t)) is a smooth oeparameter family of complete Riemaia maifols evolvig by (.) for t [a; b], that : M [a; b)! (0; ) satis es Assumptios 3, that r > 0 is chose accorig to Assumptio 3 a Remark 4, a that 0 < r 0 < r r. If is smooth a the fuctio belogs to L ( ), the (.5) P '; (r ) r r r 0 r + If, istea, iequality P '; (r 0 ) r 0 [ ( t + + tr gh) = ( t + + jr j + tr g h)' t + + jr j + tr g h ( + log r)( ' t ')] t r: is merely locally Lipschitz i the sese of Assumptio, a the ( t + + tr gh) 0 hols i the istributioal sese, a ' 0, the P '; (r ) P '; (r 0 ) r (.6) r r0 r + ( + log r)( ' t r 0 ') t r:
8 8 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Remark 8. If ' solves the heat equatio a solves the cojugate heat equatio, the (.5) implies that P '; (r)=r is iepeet of r. See Example 5 (above) a Corollary 3 (below). Proof of Theorem 7. We begi by assumig that is smooth. I the proof, we write P () P '; (). For most of the proof, we will work with a moi e fuctio, amely (.7) P (r; s) := [jr j \(M [a;s]) (r)(tr g h)]' t; arisig from restrictio to the time iterval [a; s], for some s (a; b). As a result, we will oly be workig o omais o which a its erivatives are boue, a the covergece of itegrals will ot be i oubt. A limit s % b will be take at the e. Let : R! [0; ] be a smooth fuctio with the properties that (y) = 0 for y 0 a 0 (y) 0. Let : R! [0; ) eote the primitive of e e by (y) = R (x) x. Oe shoul keep i mi that ca be mae very close to y the Heavisie fuctio, i which case (y) will lie a little below [y] +. For r (0; r] a s (a; b), we e e (.8) Q(r; s) := [jr j ( (r) ) M [a;s] ( (r) )(tr g h)]' t; which shoul be regare as a perturbatio of P (r; s), a will relieve us of some techical problems arisig from the fact that we have o cotrol o the regularity of. Note that ( (r) ) a ( (r) ) have support i. Therefore, the covergece of the itegrals is guaratee. I the followig computatios, we suppress the epeece of Q o s a assume that each itegral is over the spacetime regio M [a; s] uless otherwise state. Oe has (.9) r + r Q(r) r = r Q0 (r) Q(r) = [jr j 0 ( (r) ) 0 ( (r) )(tr g h)]' t Q(r) = [jr j 0 ( (r) )]' t [( (r) )(tr g h)]' t [jr j ( (r) )]' t + [( (r) )(tr g h)]' t: The rst itegral a the last itegral i the last equality o the rightha sie require further attetio.
9 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 9 For the rst of these, we keep i mi that r = r (r) a compute (.0) [jr j 0 ( (r) )]' t r = ; r(( (r) )) ' t = [( )( (r) )' + r (r) ; r' ( (r) )] t = [ ( )( (r) )' + (') (r) ( (r) ) + hr ; r'i (r) 0 ( (r) )] t; the calculatio beig vali o each time slice. For the fourth itegral, we compute that at each time t (a; b), oe has ( (r) )' = [ 0 ( (r) ) t t '+( (r)) ' t +( (r))'(tr g h)] ; M ftg M ftg the al term comig from i eretiatio of the volume form. Itegratig over the time iterval [a; s] a usig the facts that 0 = a that ( (r) ) = 0 at t = a (which hols because (r) 0 at t = a by Remark 4), we that (.) ( (r) )(tr g h)' t = [( (r) ) t ' + ( (r)) ' ] t t + ( (r) )' ; M fsg where the itegrals are still over M [a; s] uless otherwise iicate. We ow combie (.9) with (.0) a (.) to obtai r + Q(r) (.) r r = ( t + + jr j + tr g h)( (r) )' t ' + (') (r) ( (r) ) t t ( (r)) t + hr ; r'i (r) 0 ( (r) ) t + ( (r) )' : M fsg The etire ietity may ow be multiplie by =r + a itegrate with respect to r betwee r 0 a r, where 0 < r 0 < r r, to get a ietity for the quatity Q(r ; s)=r Q(r 0 ; s)=r0. We may simplify the resultig expressio by pickig a appropriate sequece of vali fuctios a passig to the limit. Precisely, we pick a smooth : R! [0; ] with the properties that (y) = 0 for y =, (y) = for y, a (y) 0 0. The we e e a sequece k : R! [0; ] by k (y) = ( k y). As k icreases, this sequece icreases poitwise to the characteristic fuctio of (0; ). The correspoig k coverge uiformly to the fuctio y 7! [y] +. Crucially, we also ca make use of the facts that k ( (r) ) coverges to the characteristic fuctio of i L (M [a; b]) a that (r) k 0 ( (r)) is a boue sequece of fuctios o M [a; b) with isjoit supports for each k. Iee, the support of k 0 lies withi the iterval ( k ; k ).
10 0 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING For each r (0; r], we have Q(r; s)! P (r; s) as k!. Usig the omiate covergece theorem, our expressio becomes (.3) P (r ; s) r = + P (r 0 ; s) r0 r r 0 r r 0 r r 0 r + r + r + \(M [a;s]) \(M [a;s]) M fsg ( t + + jr j + tr g h)' t r (r)( ' t [ (r) ] + ' r: ') t r Now we may take the limit as s % b. By Assumptio 3, the al term coverges to zero, a we e up with (.5) as esire. Next we tur to the case that is merely Lipschitz, i the sese of Assumptio. Give s (a; b) a fuctios a as above, there exists a sequece of smooth fuctios j o M [a; b] such that j! i both W ; a C 0 o the set \ (M [a; s]). By hypothesis o our Lipschitz, we have I := M [a;s] ( t + + jr j + tr g h)( (r) )' t 0; where we make sese of the Laplacia term via itegratio by parts, amely ( )( (r) )' t := [hr ; r'i ( (r) ) + jr j 0 ( (r) )'] t: By e itio of j, we have lim j! M [a;s] ( j t + j + jr j j + tr g h)(( j ) (r) )' t = I 0; uiformly i r (0; r]. Cosequetly, we may carry out the same calculatios that we i i the rst part of the proof to obtai a iequality for the quatity Q(r ; s)=r Q(r 0 ; s)=r0, with j i place of. We the pass to the limit as j! to obtai the iequality Q(r ; s) Q(r 0 ; s) r (.4) r r0 r + (') (r) ( (r) ) t r M [a;s] + + r 0 r r 0 r r 0 r r 0 r + r + r + M [a;s] M [a;s] M fsg ' t ( (r)) t r hr ; r'i (r) 0 ( (r) ) t r ( (r) )' r for our Lipschitz. Fially, we replace with the same sequece of cuto fuctios k that we use before (thus approximatig the Heavisie fuctio), take the limit as k!, a the take the limit as s % b. This gives the iequality (.6).
11 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS The argumet above may be compare to proofs of earlier results, especially the proof [5] of the local mootoicity formula for mea curvature ow. There is a alterative formula for (.4) that we useful i the sequel: Lemma 9. Suppose that (M ; g(t)) is a smooth oeparameter family of complete Riemaia maifols evolvig by (.) for t [a; b], that : M [a; b)! (0; ) satis es Assumptios 3, that r > 0 is etermie by Assumptio 3 a Remark 4, a that 0 < r 0 < r r. If ' a t + jr j L ( ), the for all r (0; r], oe has P '; (r) = ( t + jr j ) t: Proof. I the case that ', substitutig formula (.) ito formula (.8) yiels Q(r; s) = ( t + jr j )( (r) ) t ( (r) ) : M [a;s] M fsg Although (.) was erive assumig smoothess of, oe ca verify that it hols for locally Lipschitz satisfyig Assumptio by approximatig by a sequece of smooth j (as i the proof of Theorem 7) a the passig to the limit as j!. The if t + jr j L ( ), oe may (agai as i the proof of Theorem 7) choose a sequece k alog which Q(r; s)! P (r; s) as k! a the let s % b to obtai the state formula. 3. A local graiet estimate I orer to apply Theorem 7 to a fuametal solutio of the heat equatio of a evolvig maifol i Sectio 5, we ee a local graiet estimate. Oe approach woul be to aapt existig theory of local heat kerel asymptotics. Istea, we prove a more geeral result which may be of iepeet iterest. Compare [7], [], [4], the recet [33], [34], a [36, 37]. Let (M ; g(t)) be a smooth oeparameter family of complete Riemaia maifols evolvig by (.) for t [0; t]. We shall abuse otatio by writig g() to mea g((t)), where (t) := t t: I the remaier of this sectio, we state our results solely i terms of. particular, g() satis es g = h o M [0; t]. Give x M a > 0, e e (3.) () := [ B g() (x; ) fg M [0; t]: 0t We ow prove a local a priori estimate for boue positive solutios of the cojugate heat equatio (3.) ( We will apply this i Sectio 5. tr g h)v = 0: I
12 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Theorem 0. Let (M ; g()) be a smooth oeparameter family of complete Riemaia maifols evolvig by g = h for 0 t. Assume there exist k ; k ; k 3 0 such that h k g; Rc k g; a jr(tr g h)j k 3 i the spacetime regio () give by (3.). Assume further that v() solves (3.) a satis es 0 < v A i (). The there exist a costat C epeig oly o a a absolute costat C such that at all (x; ) (), oe has jrvj v ( + log A v ) + C k + k + k 3 + p k 3 + C Proof. By scalig, we may assume that A =. We e e 6 computig that f := log v a w := jr log( f)j ; p k coth( p k ) + C ( )f = jrfj + (tr g h): The usig Bocher Weitzeböck, we calculate that ( )jrfj = h(rf; rf) Rc(rf; rf) jrrfj + r(tr g h + jrfj ); rf a ( )w = ( f) [h(rf; rf) Rc(rf; rf) + hr(tr gh); rfi] " rjrfj ( f) jrrfj ; rf # + + jrfj4 f ( f) 4 jrfj4 ( f) 4 + (tr gh)jrfj + jrfj 4 rjrfj ; rf ( f) 3 f ( f) 3 : By rewritig the last term above as rjrfj ; rf f ( f) 3 = f jrfj4 hrw; rfi + 4 f ( f) 4 4 jrfj4 ( f) 3 a cacellig terms, we obtai ( )w = ( f) [h(rf; rf) Rc(rf; rf) + hr(tr gh); rfi] " rjrfj ( f) jrrfj ; rf # + + jrfj4 f ( f) + (tr gh)jrfj jrfj 4 f ( f) 3 + hrw; rfi : f Now let (s) be a smooth oegative cuto fuctio such that (s) = whe s a (s) = 0 whe s, with 0 0, j 0 j C, ( 0 ) C, a 00 C. De e g() (x; x) u(x; ) := : 6 Note that w is use by Souplet hag [33, Theorem.] i geeralizig Hamilto s result [7]. A similar fuctio is employe by Yau [36]. Also see relate work of the thir author [6]. :
13 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 3 Observe that at each xe, u is smooth i space o of the g() cut locus of x. However, for our purposes of applyig the maximum priciple, Calabi s trick allows us to procee as though u were smooth everywhere. Thus, we calculate that a a jruj Now let G := uw a compute that ( )(G) = G + u ( u C u C k u C p k coth( p k ) + C : )w + w ( )u hru; rwi : For ay > 0, cosier G o M [0; ]. At ay poit (x 0 ; 0 ) where G attais its maximum o M [0; ], we have 0 ( )( G) a ( )(G) G hru; rwi h(rf; rf) Rc(rf; rf) + hr(trg h); rfi + u ( f) + (tr gh)jrfj ( f) 3 f + u f hrw; rfi jrfj 4 ( f) 3 + w C k + C p k coth( p k ) + C : Usig the fact that rg(x 0 ; t 0 ) = 0, we ca replace urw by wru above. The multiplyig both sies of the iequality by u [0; ] a usig =( f), we obtai p o 0 G + [( + )k + k ]G + k 3 G f + Gjrujjrfj ( f)g f + G C k + C p k coth( p k ) + C : p Noticig that k 3 G k3 (G + ) a that f Gjrujjrf j G f jrfj f u + jruj u f f ( f)g + G C f f ; we estimate at (x 0 ; t 0 ) that 0 k 3 + G + C k + k + k 3 + C p k coth( p k ) + C + G C f f ( f)g :
14 4 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Diviig both sies by ( f) while otig that =( f) a f=( f), we get 0 k 3 + G + C k + k + k 3 + C p k coth( p k ) + C G ; from which we ca coclue that G + C k + k + k 3 + p k 3 + C p k coth( p k ) + C at (x 0 ; 0 ). Hece W ( ) := sup xbg() (x;) w(x; ) may be estimate by W ( ) 0 G(x 0 ; 0 ) + 0 C k + k + k 3 + p k 3 + C p k coth( p k ) + C + C k + k + k 3 + p k 3 + C Sice > 0 was arbitrary, the result follows. p k coth( p k ) + C : Remark. I the special case that h 0, we have jrvj v ( + log A v ) + k + C p k coth( p k ) + C at (x; ), for all times [0; t] a poits x B g() (x; ), which slightly improves a result of [33]. 4. Reuce volume for Ricci flow Our rst applicatio of Theorem 7 is to Ricci ow. Let (M ; g(t)) be a complete solutio of Ricci ow that remais smooth for 0 t t. This correspos to h = Rc a tr g h = R i (.). 4.. Localizig Perelma s reuce volume. Perelma [3, 7] has iscovere a remarkable quatity that may be regare as a ki of parabolic istace for Ricci ow. De e (t) := t t, otig that g((t)) the satis es g = Rc for 0 t. Fix x M a regar (x; 0) (i (x; ) cooriates) as a spacetime origi. The spacetime actio of a smooth path with (0) = (x; 0) a () = (x; ) is p L() := j (4.a) 0 j + R p (4.b) = j s j + s R s (s = p ): 0 Takig the i mum over all such paths, Perelma e es the reuce istace from (x; 0) to (x; ) as (4.) `(x; ) = `(x;0) (x; ) := p if L(); a observes that (4.3) v(x; ) = v (x;0) (x; ) := `(x;) e (4) =
15 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 5 is a subsolutio of the cojugate heat equatio u = u Ru i the barrier sese [3], hece i the istributioal sese. 7 It follows that the reuce volume (essetially a Gaussia weighte volume) (4.4) V ~ (t) = V(x;0) ~ (t) := v(x; ) M is a mootoically icreasig fuctio of t which is costat precisely o shrikig graiet solitos. (Compare to Example 4 above.) The iterpretatios of ` as parabolic istace a V ~ as Gaussia weighte volume are eluciate by the followig examples. Example 9. Let (M ; g) be a Riemaia maifol of oegative Ricci curvature, a let q be ay smooth superharmoic fuctio (q 0). I their semial paper [], Li a Yau e e (x; ) = if 4 0 j j + 0 q(()) ; where the i mum is take over all smooth paths from a origi (x; 0). As a special case of their more geeral results [, Theorem 4.3], they observe that (4) = e (x;) is a istributioal subsolutio of the liear parabolic equatio ( + q)u = 0. Example 0. Let (R ; g) eote Eucliea space with its staar at metric. Give R, e e X = gra( 4 jxj ). The oe has 0 = Rc = g L X g. Hece there is a Ricci solito structure (i.e. a i itesimal Ricci solito) o Eucliea space, calle the Gaussia solito. It is otrivial wheever 6= 0. Take = to give (R ; g) the structure of a graiet shrikig solito. The () = p = x is a Lgeoesic from (0; 0) to (x; ). Thus the reuce istace is `(0;0) (x; ) = jxj =4 a the reuce volume itegra is exactly the heat kerel v (0;0) (x; ) = (4) = e jxj =4. Hece V ~ (0;0) (t). (Compare [, 5].) Example. Let Sr() eote the rou sphere of raius r() = p ( ). This is a positive Eistei maifol, hece a homothetically shrikig (i t) solutio of Ricci ow. Alog ay sequece (x k ; k ) of smooth origis approachig the sigularity O at = 0, oe gets a smooth fuctio `O(x; ) := lim k! `(xk ; k )(x; ) = measurig the reuce istace from O. Hece V ~ O (t) [( )=(e)] = Vol(S ) for all t < 0. (See [4, 7.].) Our rst applicatio of Theorem 7 is where is Perelma s reucevolume esity v (4.3). Let ` eote the reuce istace (4.) from a smooth origi (x; t) a assume there exists k (0; ) such that Rc kg o M [0; t]. I what follows, we will freely use results from the Appeix (Sectio 7, below). Lemma 39 guaratees that ` is locally Lipschitz, hece that Assumptio is satis e. (Also see [38] or [4].) The estimate i Part () of Lemma 8 esures that Assumptio is satis e. Assumptio 3 follows from combiig that estimate, Corollary 3, a Lemma 40. Here we may take ay r > 0 satisfyig r < mift=c; 4g, where c = e 4kt=3 =(4). So for r (0; r], cosier P ';v (r) := [jr`j r + R( log p `)]' t: 4 7 See [38] for a irect proof of the istributioal iequality.
16 6 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING Notice that jr`j replaces the term jx xj 4 i the heatball formulas for Eucliea space a solutios of mea curvature ow. See Examples 5 a 7, respectively. Remark. For r (0; r], oe may write P ;v (r) i either alterative form (4.5a) P ;v (r) = ( + ` + jr`j ) t = ( (4.5b) 3= K) t: Here K(x; ) = R 0 3= H(=) is compute alog a miimizig Lgeoesic, where H(X) = Rc(X; X) (R + hrr; Xi+R=) is Hamilto s trace i eretial Harack expressio. If R 0 a ' 0 o, the for all r (0; r], oe has (4.6) P ';v (r) = [jr`j + R (r) ]' t jr`j ' t 0: If (M ; g(0)) has oegative curvature operator a r < 4t( some C >, the for all r (0; r], = + C` (4.7) P ;v (r) t: =C) for Proof. By Part () of Lemma 39, the argumets of Lemma 40 apply to show that t + jr j = + ` + jr`j L ( ). Hece Lemma 9 a ietities (7.5) a (7.6) of [3] imply formulae (4.5). Sice (r) > 0 i, the iequalities i (4.6) are clear. If (M ; g(0)) has oegative curvature operator, Hamilto s trace i eretial Harack iequality [8] implies that alog a miimizig Lgeoesic. Hece H( ) R( + t ) = t R t 3= t K 3= t 0 p (R + j j ) = By Lemma 3, oe has < r =4, which gives estimate (4.7). t t ` : Our mai result i this sectio is as follows. Recall that (r) := log( p 4 ) `. Corollary 3. Let (M ; g(t)) be a complete solutio of Ricci ow that remais smooth for 0 t t a satis es Rc kg. Let ' be ay smooth oegative fuctio of (x; t) a let c = e 4kt=3 =(4). The wheever 0 < r 0 < r < mif p t=c; p g, oe has (4.8) Furthermore, P ';v (r ) r P ';v (r 0 ) r 0 r r 0 r + (4.9) '(x; t) = lim r&0 P ';v (r) r : (r)( ' t ') t r: r
17 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 7 I particular, (4.0) '(x; t) P ';v(r ) r r + 0 r + (r)[( t )'] t r: Proof. The quatity = v satis es v t + v Rv 0 as a istributio. (This is implie by Perelma s barrier iequality [3, (7.3)]; see [38, Lemma.] for a irect proof.) Hece we may apply Theorem 7 i the form (.6) to obtai (4.8). Formula (7.6) of Perelma [3] implies that P ';v (r) = [ ` + R (r) R 3= K]' t: By Corollary 3, there is a precompact eighborhoo U of x with U [0; cr ] for all r > 0 uer cosieratio. By Lemma 35, there exists a precompact set V such that the images of all miimizig Lgeoesics from (x; 0) to poits i U [0; cr ] are cotaie i the set V [0; cr ], i which oe has uiform bous o all curvatures a their erivatives. So by Lemma 8, oe has ` 4 + O( ) a r R (r) = R( log p 4 `) = O( ) as & 0. By Corollary 37, 3= K is also O( ) as & 0. Aaptig the argumets i the proof of Lemma 40, oe coclues that P ';v (r) lim r&0 r = lim 0(x; x) r&0 r 4 ' t = '(x; t); exactly as i the calculatio for Eucliea space. (Also see Corollary 3, below.) = 0 (x;x) A example of how this result may be applie is the followig local Harack iequality, which follows irectly from (4.0). Remark 4. Assume the hypotheses of Corollary 3 hol. If R 0 o, the R(x; t) r r [jr`j + R (r) ]R t + 0 r + (r) jrcj t r: The iequality (4.8) is sharp i the followig sese. Corollary 5. Let (M ; g(t)) be a complete solutio of Ricci ow that is smooth for 0 t t, with Rc kg. If equality hols i (4.8) for ', the ( ; g(t)) is isometric to a shrikig graiet solito for all r < mif p t=c; p g. Proof. From the proof of Theorem 7, it is easy to see that P;v (r) ( r r = r Er t + + tr gh)v + t v for almost all r < mif p t=c; p g. Therefore, equality i (4.8) implies that v is a istributioal solutio of the parabolic equatio ( + R)v = 0 i for almost all small r. By parabolic regularity, v is actually smooth. This implies that oe has equality i the chai of iequalities ` jr`j + R ` = ( R + 3= K) + ` ` + ` that follow from equatios (7.3), (7.5), a (7.0) of [3]. Hece oe has u := (` jr`j + R) + ` = 0:
18 8 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING By equatio (9.) of [3] (where the roles of u a v are reverse), this implies that 0 = ( + R)(uv) = j Rc +rr` gj v: This is possible oly if ( ; g(t)) has the structure of a shrikig graiet solito with potetial fuctio `. Remark 6. For applicatios of Corollary 5 to regularity theorems for Ricci ow, see [7] by the thir author. 4.. Comparig global a local quatities. Corollaries 3 a 5 suggest a atural questio: how oes the purely local mootoe quatity P ;v (r)=r compare to Perelma s global mootoe quatity V ~ (t) = R v? A path to a partial M aswer begis with a observatio that geeralizes Example above. Cao, Hamilto, a Ilmae [] prove that ay complete graiet shrikig solito (M ; g(t)) that exists up to a maximal time T < a satis es certai ocollapsig a curvature ecay hypotheses coverges as t % T to a icomplete (possibly empty) metric coe (C; ), which is smooth except at the parabolic vertex O. The covergece is smooth except o a compact set (possibly all of M ) that vaishes ito the vertex. 8 Furthermore, they prove that alog a sequece (x k ; k ) approachig O, a limit `O(x; ) := lim `(xk ; k )(x; ) exists for all x M a (t) > 0. They show that the cetral esity fuctio RF O (t) := V ~ O (t) = lim ~V (xk ; k )(t) k! of the parabolic vertex O is iepeet of time a satis es RF O (t) e, where is the costat etropy of the solito (M ; g()). O a compact solito, there is a poitwise versio of the Cao Hamilto Ilmae result, ue to Beett Chow a the thir author: Lemma 7. If (M ; g()) is a compact shrikig (ecessarily graiet) solito, the the limit `O(x; ) exists for all x M a (t) > 0. This limit agrees up to a costat with the solito potetial fuctio f(x; ). See [4] for a proof. Recall that the etropy of a compact Riemaia maifol (M ; g) is (M ; g) := if W (g; f; ) : f C0 ; > 0; (4) = e f = ; M where (4.) W (g; f; ) : = (jrfj + R) + f (4) = e f : M (Compare to Example 3.) Uer the couple system (4.a) (4.b) (4.c) t g = Rc ( t + )f = jrfj R + t = ; 8 See [] for examples where (C; ) = lim&0 (M ; g()) is oempty.
19 LOCAL MONOTONICITY AND MEAN VALUE FORMULAS 9 the fuctioal W(g(t); f(t); (t)) is mootoe icreasig i time a is costat precisely o a compact shrikig graiet solito with potetial fuctio f, where (after possible ormalizatio) oe has (4.3) Rc +rrf g $ 0: Here a i the remaier of this sectio, the symbol $ eotes a ietity that hols o a shrikig graiet solito. We are ow reay to aswer the questio we pose above regarig the relatioship betwee P ;v (r)=r a V ~ (t). (Compare to Example 7.) Corollary 8. Let (M ; g(t)) be a compact shrikig Ricci solito that vaishes ito a parabolic vertex O at time T. The for all t < T a r > 0, oe has where P ;v (r) = R [jr`j + R( log RF O (t) := ~ V O (t) = P ;v(r) r ; r p 4 `)] t is compute with ` = `O. Proof. It will be easiest to regar everythig as a fuctio of (t) := T t > 0. Because (M ; g()) is a compact shrikig solito, there exist a timeiepeet metric g a fuctio f o M such that Rc(g) + r r f g = 0. The solutio of Ricci ow is the g() = (g), where f g >0 is a oeparameter family of i eomorphisms such that = i a (x) = gra g f(x). The solito potetial fuctio satis es f(x; ) = f(x) a f = jrfj. (Notice that (4.3) implies that system (4.) hols.) Let = (4) = e `(x;), where ` is the reuce istace from the parabolic vertex O. By Lemma 7, ` = f + C. So Assumptios a are clearly satis e. Because j j = O[ = log( = )] a jr j = O( = ) M fg M fg as & 0, Assumptio 3 is satis e as well. Because = jrfj, Lemma 9 implies that P ;v (r) = ( + ` + jr`j ) t $ t: (Compare Remark.) Computig V ~ () = V ~ O (), oe s that ~V () = (4) = e `(x;) (g()) M = $ = = E $ P ;v (): Vol g() fx : (4) = e `(x;) zg z h Vol g() Vol g()fy : `(y; ) < log t i fx : (4) = e `(x;) g (z = = ) p 4 g
20 0 KLAUS ECKER, DAN KNOPF, LEI NI, AND PETER TOPPING But o a shrikig graiet solito, P ;v (r)=r is iepeet of r > 0, while ~V () is iepeet of > 0. Sice they agree at r = a =, they agree everywhere. Sice the reuce istace a reuce volume are ivariat uer parabolic rescalig, similar cosieratios apply to solutios whose rescale limits are shrikig graiet solitos Localizig forwar reuce volume. I [], Felma, Ilmae, a the thir author itrouce a forwar reuce istace t `+(x; t) := if p p s j t s j + R s: 0 Here the i mum is take over smooth paths from a origi (x; 0) to (x; t). De e u(x; t) = (4t) = e `+(x;t) a = log u. I [5], it is prove that ( t R)u 0 hols i the istributioal sese if (M ; g(t)) is a complete solutio of Ricci ow with boue oegative curvature operator for 0 t T. Followig the same argumets as i the proof of Corollary 3 the leas to the followig result for P ';u (r) = [jr`+j R( log r p 4t `+)]' t: Corollary 9. Let (M ; g(t)) be a complete solutio of Ricci ow with boue oegative curvature operator for 0 t T. Let ' be ay smooth oegative fuctio. The wheever 0 < r 0 < r < p 4T, oe has (4.4) P ';u (r ) r P ';u (r 0 ) r 0 r r 0 r + ( + log r)( ' t I irect aalogy with Corollary 5, oe also has the followig. + ') t r: Corollary 0. Let (M ; g(t)) be a complete solutio of Ricci ow with boue oegative curvature operator for 0 t T. If equality hols i (4.4) with ', the ( ; g(t)) is isometric to a expaig graiet solito for all r < p 4T. 5. Meavalue theorems for heat kerels I this sectio, we apply Theorem 7 to heat kerels of evolvig Riemaia maifols, especially those evolvig by Ricci ow, with statioary (i.e. timeiepeet) maifols appearig as a iterestig special case. Let (M ; g(t)) be a smooth family of Riemaia maifols evolvig by (.) for t [0; t]. We will agai abuse otatio by regarig certai evolvig quatities, where coveiet, as fuctios of x M a (t) := t t. A smooth fuctio : (M [0; t])(x; 0)! R + is calle a fuametal solutio of the cojugate heat equatio (5.) ( tr g h) = 0 with sigularity at (x; 0) if satis es (5.) at all (x; ) M (0; t], with lim &0 (; ) = x i the sese of istributios. We call a miimal fuametal solutio of (5.) a heat kerel.
In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationSection 6.1. x n n! = 1 + x + x2. n=0
Differece Equatios to Differetial Equatios Sectio 6.1 The Expoetial Fuctio At this poit we have see all the major cocepts of calculus: erivatives, itegrals, a power series. For the rest of the book we
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationThe Field of Complex Numbers
The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationMeasurable Functions
Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these
More information8.5 Alternating infinite series
65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationif A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,
Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σalgebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More informationMATH 361 Homework 9. Royden Royden Royden
MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,
More informationDistributions of Order Statistics
Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More information8.3 POLAR FORM AND DEMOIVRE S THEOREM
SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More information3.2 Introduction to Infinite Series
3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are
More informationU.C. Berkeley CS270: Algorithms Lecture 9 Professor Vazirani and Professor Rao Last revised. Lecture 9
U.C. Berkeley CS270: Algorithms Lecture 9 Professor Vazirai a Professor Rao Scribe: Aupam Last revise Lecture 9 1 Sparse cuts a Cheeger s iequality Cosier the problem of partitioig a give graph G(V, E
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More informationA Gentle Introduction to Algorithms: Part II
A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The BigO, BigΘ, BigΩ otatios: asymptotic bouds
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationSumming The Curious Series of Kempner and Irwin
. INTRODUCTION. Summig The Curious Series of Kemper a Irwi Robert Baillie I 94, Kemper prove [3] that if we elete from the harmoic series all terms whose eomiators have the igit 9 that is, /9, /9, /9,...,
More informationSection 9.2 Series and Convergence
Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives
More informationTAYLOR SERIES, POWER SERIES
TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationON THE DENSE TRAJECTORY OF LASOTA EQUATION
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON THE DENSE TRAJECTORY OF LASOTA EQUATION by Atoi Leo Dawidowicz ad Najemedi Haribash Abstract. I preseted paper the dese trajectory
More informationTaylor Series and Polynomials
Taylor Series ad Polyomials Motivatios The purpose of Taylor series is to approimate a fuctio with a polyomial; ot oly we wat to be able to approimate, but we also wat to kow how good the approimatio is.
More informationA CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationπ d i (b i z) (n 1)π )... sin(θ + )
SOME TRIGONOMETRIC IDENTITIES RELATED TO EXACT COVERS Joh Beebee Uiversity of Alaska, Achorage Jauary 18, 1990 Sherma K Stei proves that if si π = k si π b where i the b i are itegers, the are positive
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationMath 475, Problem Set #6: Solutions
Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b oegative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),
More informationEconomics 140A Confidence Intervals and Hypothesis Testing
Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter is
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationNotes on exponential generating functions and structures.
Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a elemet set, (2) to fid for each the
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More information5. SEQUENCES AND SERIES
5. SEQUENCES AND SERIES 5.. Limits of Sequeces Let N = {0,,,... } be the set of atural umbers ad let R be the set of real umbers. A ifiite real sequece u 0, u, u, is a fuctio from N to R, where we write
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationLecture 7: Borel Sets and Lebesgue Measure
EE50: Probability Foudatios for Electrical Egieers JulyNovember 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationDivide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016
CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito
More informationPartial Di erential Equations
Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More information7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b
Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5
More informationDivide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015
CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationSection IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationApproximating the Sum of a Convergent Series
Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationAdvanced Probability Theory
Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σalgebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio
More information1. a n = 2. a n = 3. a n = 4. a n = 5. a n = 6. a n =
Versio PREVIEW Homework Berg (5860 This pritout should have 9 questios. Multiplechoice questios may cotiue o the ext colum or page fid all choices before aswerig. CalCb0b 00 0.0 poits Rewrite the fiite
More informationLecture Notes CMSC 251
We have this messy summatio to solve though First observe that the value remais costat throughout the sum, ad so we ca pull it out frot Also ote that we ca write 3 i / i ad (3/) i T () = log 3 (log ) 1
More information1 Set Theory and Functions
Set Theory ad Fuctios. Basic De itios ad Notatio A set A is a collectio of objects of ay kid. We write a A to idicate that a is a elemet of A: We express this as a is cotaied i A. We write A B if every
More informationAn example of nonquenched convergence in the conditional central limit theorem for partial sums of a linear process
A example of oqueched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which
More informationAN ASYMPTOTIC ROBIN INEQUALITY. Patrick Solé CNRS/LAGA, Université Paris 8, SaintDenis, France.
#A8 INTEGERS 6 (206) AN ASYMPTOTIC ROBIN INEQUALITY Patrick Solé CNRS/LAGA, Uiversité Paris 8, SaitDeis, Frace. sole@est.fr Yuyag Zhu Departmet of Math ad Physics, Hefei Uiversity, Hefei, Chia zhuyy@hfuu.edu.c
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationAnalysis Notes (only a draft, and the first one!)
Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationContents. 7 Sequences and Series. 7.1 Sequences and Convergence. Calculus II (part 3): Sequences and Series (by Evan Dummit, 2015, v. 2.
Calculus II (part 3): Sequeces ad Series (by Eva Dummit, 05, v..00) Cotets 7 Sequeces ad Series 7. Sequeces ad Covergece......................................... 7. Iite Series.................................................
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationAll solutions of the CMCequation in H n R invariant by parabolic screw motion
All solutios of the CMCequatio i H R ivariat by parabolic screw motio Maria Ferada Elbert ad Ricardo Sa Earp March 21, 2012 Abstract I this paper, we give all solutios of the costat mea curvature equatio
More information4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then
SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or pseries (the Compariso Test), but of
More informationB1. Fourier Analysis of Discrete Time Signals
B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 007 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 4 diagram sheets ad a page formula sheet. Please tur over Mathematics/P DoE/Exemplar
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationA sharp TrudingerMoser type inequality for unbounded domains in R n
A sharp TrudigerMoser type iequality for ubouded domais i R Yuxiag Li ad Berhard Ruf Abstract The TrudigerMoser iequality states that for fuctios u H, 0 (Ω) (Ω R a bouded domai) with Ω u dx oe has Ω
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationMath Background: Review & Beyond. Design & Analysis of Algorithms COMP 482 / ELEC 420. Solving for Closed Forms. Obtaining Recurrences
Math Backgroud: Review & Beyod. Asymptotic otatio Desig & Aalysis of Algorithms COMP 48 / ELEC 40 Joh Greier. Math used i asymptotics 3. Recurreces 4. Probabilistic aalysis To do: [CLRS] 4 # T() = O()
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a dregular
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
More informationNUMBERS COMMON TO TWO POLYGONAL SEQUENCES
NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively
More information