Construction of Manifolds of Positive Ricci Curvature with Big Volume and Large Betti Numbers

Transcription

1 Comparison Geometry MSRI Pulications Volume 30, 1997 Construction of Manifolds of Positive Ricci Curvature with Big Volume and Large Betti Numers G. PERELMAN Astract. It is shown that a connected sum of an aritrary numer of complex projective planes carries a metric of positive Ricci curvature with diameter one and, in contrast with the earlier examples of Sha Yang and Anderson, with volume ounded away from zero. The key step is to construct complete metrics of positive Ricci curvature on the punctured complex projective plane, which have uniform euclidean volume growth and almost contain a line, thus showing topological instaility of the splitting theorem of Cheeger Gromoll, even in the presence of the lower volume ound. In the asence of such a ound, the topological instaility was earlier shown y Anderson; metric staility holds, even without the volume ound, y the recent work of Colding Cheeger. 1. Outline We start from a singular space of positive curvature, namely the doule spherical suspension of a small round two-sphere. The size of that sphere can e estimated explicitly and is fixed in our construction. The singular points of our space fill a circle of length 2π. We smooth our space near the singular circle in a symmetric way. Then we remove a collection of disjoint small metric alls centered at the former singular points, and glue in our uilding locks instead. A uilding lock is a metric on C P 2 \ all, having positive Ricci curvature and strictly convex oundary. We arrange that the oundary of the uilding lock is isometric to the oundary of the removed all and is more convex. This allows us to smooth the resulting space to get a manifold of positive Ricci curvature. Its volume is close to the volume of the doule suspension we started from, whereas its second Betti numer equals the numer of uilding locks it contains. This numer can e made aritrarily large y localizing the initial This paper was prepared while the author was a Miller fellow at the University of California at Berkeley. 157

2 158 G. PERELMAN smoothing of the doule suspension to smaller neighorhoods of the singular circle, and y removing a larger numer of smaller alls. The main difficulty is to construct uilding locks. This is carried out in Section 2. In Section 3 we construct the amient space, i.e., we make precise the relation etween the numer and the size of alls to e removed and the smoothing of our doule suspension. In Section 4 we explain how to smooth the result of gluing two manifolds with oundaries, so as to retain positive Ricci curvature. Notation. KX Y ) denotes the sectional curvature in the plane spanned y X, Y. We areviate RicX, Y )/ X Y as RiX, Y ). 2. Construction of the Building Block Our uilding lock is glued from two pieces: the core and the neck. The core is a metric on C P 2 \ all with positive Ricci curvature and strictly convex oundary; moreover, the oundary is isometric to a round sphere. The neck is ametricons 3 [0, 1] with positive Ricci curvature, such that the oundary component S 3 {0} is concave and is isometric to a round sphere, whereas the oundary component S 3 {1} is convex and looks like a lemon, that is, like a smoothed spherical suspension of a small round two-sphere. There are some additional conditions on the normal curvatures of the oundary of the neck; they will e made clear elow. Construction of the core. It is well known that the canonical metric of C P 2 in a neighorhood of C P 1 can e expressed as ds 2 = dt 2 + A 2 t) dx 2 + B 2 t) dy 2 + C 2 t) dz 2, where t is the distance from C P 1 and X, Y, Z is the standard invariant framing of S 3, satisfying [X, Y ]=2Z, [Y, Z] =2X, [Z, X] =2Y. The canonical metric has A =sintcos t and B = C =cost, ut we are going to consider general A, B, C. The curvature tensor in this presentation is listed on page 165 of this volume. Assuming now that B C, we deduce from those formulas that RiX,X)= A /A 2A B /AB +2A 2 /B 4 ), RiY, Y ) = RicZ, Z) = B /B A B /AB B 2 /B 2 +4B 2 2A 2 )/B 4 ), RiT,T)= A /A 2B /B), and all off-diagonal entries vanish. Now let A = sint cos t, B = 1 cosht/100). Clearly this choice gives a 100 smooth metric. We have, for 0 <t< 1 10, A /A =4, B /B = 10 4, 0 <A B /AB < 1 10, 0 <B /B < 1 100, and 4B 2 2A 2 )/B 4 > 100 provided that A B.

3 BIG POSITIVELY CURVED MANIFOLDS WITH LARGE BETTI NUMBERS 159 Thus if 0 <t 0 < 1 10 is such that At 0)=Bt 0 ), At) <Bt) fort<t 0,then Ricci curvatures are positive for 0 t t 0. Such t 0 exists ecause A0) = 0 < B0), whereas A 1 10 ) > 1 20 >B 1 10 ). Therefore the core is constructed. Construction of the neck. We first prove the following result. Assertion. Let S n,g) e a rotationally symmetric metric of sectional curvature > 1, distance etween the poles πr and waist 2πr; that is, g can e expressed as ds 2 = dt 2 +B 2 t) dσ 2, where dσ 2 is the standard metric of S n 1, t [0,πR], and max t Bt) =r. Let ρ>0 e such that ρ<rand r n 1 <ρ n. Then there exists a metric of positive Ricci curvature on S n [0, 1] such that a) the oundary component S n {1} has intrinsic metric g and is strictly convex; moreover, all its normal curvatures are > 1; ) the oundary component S n {0} is concave, with normal curvatures equal to λ, and is isometric to a round sphere of radius ρλ 1, for some λ>0. Note that, if ρ is small enough, the core constructed earlier in this section can e glued, after rescaling, to the neck along S n {0}, so that the resulting space can e smoothed with positive Ricci curvature; see Section 4.) To prove the assertion, we start y rewriting our metric g in the form ds 2 = r 2 cos 2 xdσ 2 + A 2 x) dx 2, where π/2 x π/2anda is a smooth positive function satisfying A±π/2) = r, A ±π/2) = 0. Clearly Ax) R>rfor some x, so we can write Ax) =r1 ηx)+ηx) a ), where max x ηx) =1anda R/r, η±π/2) = 0, and η x ±π/2) = 0. Consider a metric on S n [t 0,t ]oftheform ds 2 = dt 2 + A 2 t, x) dx 2 + B 2 t, x) dσ 2, where B = tt)cosx, A = tt) 1 ηx) +ηx)at) ), at 0 )=1,a t 0 )=0, t 0 )=ρ, t 0 )=0,at )=a > 1, t ) >r. That metric will satisfy the conditions of our assertion after rescaling y a multiple r/t t )), provided that the functions at) and t) are chosen appropriately. The curvatures of our metric can e computed as follows: KT X) = A tt /A KT Σ) = B tt /B KX Σ) = A t B t /AB + A x B x /A 3 B B xx /BA 2 KΣ 1 Σ 2 )=1/B 2 B 2 x/a 2 B 2 B 2 t /B 2, and the Ricci tensor has only one nonzero off-diagonal term, namely RiT,X)=n 1)A t B x /A 2 B B xt /AB).

4 160 G. PERELMAN Further computations give ) RiT,T)= n + 2 t RiT,X)= n 1) tg x for the Ricci tensor, and ηa + 2ηa t) +2 ηa t) 2 ) ηa, KX Σ) = 1 1 t 2 2 ) η ) x tg x a 1) 2 ) 3 ) ) 1 t + 1 t + + ηa, KΣ 1 Σ 2 )= 1 ) ) 1 t 2 2 cos 2 x tg 2 2 x 1 ) 2 t + for the sectional curvatures. Note that the first terms in the last two formulas are the intrinsic curvatures of the hypersurfaces t =const,k i X Σ), K i Σ 1 Σ 2 ). We construct the functions t) and at) as follows: / = βt t 0 )/2t 2 0 log 2t 0 for t 0 t 2t 0, / = β log 2t 0 /t log 2 t for t 2t 0 must e smoothed near t =2t 0 ), a /a = α / for t t 0. Here β =1 ε)log ρ log r)/1 + 1/4 log 2t 0 )) and α =1+δ)β 1 log a /1 + 1/4 log 2t 0 )), so that t 0 / =1 ε)log r log ρ) and t 0 a /a =1+δ)loga ; the small positive numers ε, δ are to e determined later. At this point we still have freedom in the choice of t 0,ε,δ,andt.Note,however, that t is determined y δ from the relation at )=a.wechooseε>0 first in such a way that the metric ρ/r) ε g still has sectional curvatures > 1. In this situation we prove elow that the curvatures K i X Σ) and K i Σ 1 Σ 2 ) can e estimated from elow y c/t 2 for some c>1 independent of t 0, t,and δ. Then we show that the Ricci curvatures of our metric are strictly positive if t 0 is chosen large enough, independently of δ, ifδ is small. Finally, we choose δ in such a way that the normal curvatures of S n {t } are sufficiently close to t 1. First of all we need to check that 1 <α<n. Indeed, at the maximum point of ηx) wehavetgxη x =0,soK g X Σ) = 1/r 2 a 2. Thus a < 1/r and log a <nlog ρ log r) sincer n 1 <ρ n. On the other hand a R/r > ρ/r, so log a > log ρ log r. Now we are in a position to check that K i X Σ) c/t 2 and K i Σ 1 Σ 2 ) c/t 2 for some c>1. Let ψ =log t 2 K i Σ 1 Σ 2 ) ). We know that ψ t=t0 > 0and

5 BIG POSITIVELY CURVED MANIFOLDS WITH LARGE BETTI NUMBERS 161 ψ t=t > 0. A computation gives 1+ ψ t = 2 αηa sin 2 x )) 2 sin 2 x) This expression is positive if η 0, and it is decreasing in a if η<0. Therefore ψ cannot have a local minimum on t 0,t ) for any fixed x, soψ>0asrequired. Now let φ =log t 2 K i X Σ) ). Again consider the ehavior of this function for t [t 0,t ] and fixed x. Suppose that tg xη x 0. A computation gives φ t = = [ ] 1 αa tg xη x 21 η)+2ηa1 α) 1 η +tgxη x + aη tg xη x ) [ ]) 2 2η +. αa tg xη x 1 η +tgxη x + aη tg xη x ) If η 0 the first expression in large rackets is clearly decreasing in a, whereas if η<0 the second expression in rackets, and also αa/), are clearly increasing in a, soφ cannot have a local minimum in t 0,t ), and therefore φ>0sinceφ t=t0 > 0andφ t=t > 0. Now suppose that tg xη x < 0. Then φ> 2loga > 0, ecause at )t ) < 1 and loga) isincreasingint note that loga) = /)α 1) > 0). The estimate for K i X Σ) and K i Σ 1 Σ 2 ) is proved. Now we can estimate the Ricci curvatures of our metric. Note that the normal curvatures 1/t + / and 1/t + / + ηa /) can e estimated y 1/t + O1/t log t). It follows that KX Σ) c/t 2 and KΣ 1 Σ 2 ) c/t 2, for c>0. It is also clear that RiT,X), KX T ), and KΣ T ) are all ounded aove y c log t 0 /t 2 log 2 t); hence, in particular, RiX, X) c/t 2 and RiΣ, Σ) c/t 2,withc>0. To estimate RiT,T), write it as ) αηa RiT,T)= n /) + 2 t ) n /) 2 ). ηa 2 a a ) 2 ) a +. a Note that, since η 1andα<n, the first multiple is negative and ounded away from 0. Therefore, the first term is c log t 0 /t 2 log 2 t), for c>0. The remaining terms are of order c log 2 t 0 /t 2 log 4 t), so we get RiT,T) c log t 0 t 2 log 2 t. Since RiT,X) crit,t) RiX, X), we conclude that Ricci curvature is positive, if t 0 was chosen large enough. To complete the construction it remains to choose δ>0sosmall,andcorre- spondingly t so large, that normal curvatures of S n {t } are > r/ρ) ε t 1 ; this is possile since they are estimated y 1/t + O1/t log t).

6 162 G. PERELMAN 3. Construction of the Amient Space The metric of the doule suspension of a sphere of radius R 0 near singular circle can e written as ds 2 = dt 2 +cos 2 tdx 2 + R 2 t) dσ 2, where t 0 is the distance from the singular circle, Rt) =R 0 sin t. Wesmooth this metric y modifying Rt) in such a way that R0) = 0, R 0) = 1, and R 0) = 0. The curvatures of this metric are easily computed to e KT X) = 1, KT Σ) = R /R, KX Σ) = R /R)tgt, KΣ 1 Σ 2 )=1 R 2 )/R 2, and all mixed curvatures in this asis vanish. We will choose Rt) insuchaway that 1 >R > 0andR < 0whent>0, thus making all sectional curvatures positive. Consider a metric all of small radius r 0 centered on the axis t =0. We assume that R R for 0 t r 0,sothatKT Σ) 1. In this case the normal curvatures of the oundary of this all do not exceed ctg r 0. We ll show that the function Rt) can e chosen so that the intrinsic curvatures of the oundary are strictly igger than ctg 2 r 0. It would follow that for sufficiently small R 0 > 0, we can construct a uilding lock that, after rescaling y tg r 0, can e glued using the results of Section 4) in our amient space instead of the metric all. At a point of the oundary of our metric all, which is at a distance t from the axis, we can compute the intrinsic curvatures as follows: K i Σ 1 Σ 2 )=1/R 2 )1 R 2 sin 2 φ), K i Y Σ) = R R sin2 φ + R R tg t R cos2 φ +ctgr 0 cos φ, R where Y X T is a tangent vector to the oundary, φ is the angle etween T and the normal vector, cos φ =ctgr 0 tg t. Since KT Σ) 1, it follows from the comparison theorem that K i Σ 1 Σ 2 ) 1/ sin 2 t ctg 2 t1 cos 2 φ)=1+ctg 2 r 0 > ctg 2 r 0. To make sure that K i Y Σ) > ctg 2 r 0,wehavetochooseRt) more carefully. Namely, let Rt) =R 0 sin t + δγt/r 0 1)) for t ε, whereγ is a standard smooth function interpolating etween γx) = 1forx 0andγx) = 0for x 1. Extend Rt) tothesegment[0,ε] in such a way that R0) = 0, R 0) = 1, and R /R 2r0 2 on [0,ε]. This is possile with an appropriate choice of ε and δ for example, if ε = 1 2 r2 0 and δ = 1 4 r4 0.) Now it is clear that already the first term in the expression for K i Y Σ) is igger than ctg 2 r 0 if 0 t ε. For ε t r 0 we have K i Y Σ) = sin 2 φ +ctgt + δ)tgtcos 2 φ +ctg 2 r 0 ctgt + δ)tgt ctgt + δ)tgt ctg 2 r 0 +1).

7 BIG POSITIVELY CURVED MANIFOLDS WITH LARGE BETTI NUMBERS 163 So we need to check that tgt + δ)/ tg t<1+tg 2 r 0.Thisistruefort ε = 1 2 r2 0 and δ = 1 4 r4 0 and for small r 0.Thus,othK i Σ 1 Σ 2 )andk i X Σ) are igger than ctg 2 r 0, and the construction of the amient space is complete. 4. Gluing and Smoothing To justify gluing and smoothing in our construction we need the following fact. Let M 1,M 2 e compact smooth manifolds of positive Ricci curvature, with isometric oundaries M 1 M 2 = X. Suppose that the normal curvatures of M 1 are igger than the negatives of the corresponding normal curvatures of M 2. Then the result M 1 X M 2 of gluing M 1 and M 2 can e smoothed near X to produce a manifold of positive Ricci curvature. To prove this, express the metric of M 1 X M 2 in normal coordinates with respect to X; let t e the normal coordinate. Introduce a small parameter τ>0, and for aritrary coordinate vectors X 1,X 2 tangent to X replace our given function X 1,X 2 x, t) y its interpolation on the segment [ τ, τ]. At first we can take a C 1 interpolation given y a cuic polynomial in t, whose coefficients are linear functions of X 1,X 2 x, ±τ) and X 1,X 2 tx, ±τ). Clearly this procedure is independent of the choice of coordinates in X, anditgivesac 1 metric, which is C 2 outside two hypersurfaces X τ,x τ, corresponding to t = ±τ. It is easy to see that in the segment [ τ, τ] wegetallkt X) positive of order cτ 1 and all KX 1 X 2 )andrit,x) ounded, so the Ricci curvature is positive. Now we can use a similar procedure to smooth our manifold near hypersurfaces t = ±τ. This time we choose another τ τ and construct a C 2 -interpolation. It is clear that only the components RT,,,T) of the curvature tensor were discontinuous on, say, X τ, and, up to an error of order τ, these components now interpolate linearly etween their original values on the different sides of X τ. Since positivity of Ricci curvature is open and convex condition, the smoothed manifold has positive Ricci curvatures. G. Perelman St. Petersurg Branch V. A. Steklov Institute of Mathematics POMI) Russian Academy of Sciences Fontanka 27 St. Petersurg, Russia perelman@pdmi.ras.ru