1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. DYNAMIC STABILITY RESULTS OBTAINED BY OTHER METHODS 305

THEOREM 35.37 ([152]). L et (Mn , g) be a closed simply-connected R iemannian

manifold of positive scalar curvature. There exists On depending only on n 2 4 such

that if the irreducible components of the curvature tensor satisfy

IVl

2

+ IWl

2

< on IUl

2

=

20

n R^2

n (n - 1)

pointwise, then the volume normalized R icci flow with initial data g converges to a

unique metric of constant positive curvature.

Also see Margerin [217] and Nishikawa [294].

REMARK 35.38. Note that the hypotheses above only need pointwise pinching,

in sharp contrast to the global pinching assumptions necessary for classical sphere

theo rems in comparison geometry; e.g., Kmax(Mn , g) < 4Kmin(Mn , g). See the

survey [2].

Margerin subsequently improved 64 to its optimal value:

THEOREM 35 .39 ([218]). L et (M4,g) be a closed simply-connected Riemannian

manifold of positive scalar curvature. If the irreducible components of the curvature

tensor satisfy

IVl

2

+ IWl

2

< IUl

2

= ~R^2

6

pointwise, then the volume normalized R icci flow with initial data g converges to

a unique metric of constant positive curvature. I n this case, M^4 is diffeomorphic

to the standard 4-sphere if it is orientable and to IRIP'^4 otherwise. The equality

IVl

2

+ IWl

2

= IUl

2

implies that M^4 is isometric either to CIP'^2 with its Fu bini-

Study metric or else to a quotient of !RxS^3.

R ecall that the Riemannia n curvature operator may be regarded as a self-

adjoint map Rm : A^2 T xMn --+ A^2 TxMn at each x E Mn. One says t hat Rm is

2-positive if the sum of its two sm allest eigenvalues is positive. Bohm and Wilking

prove the following remarkable convergence result in all dimensions n 2 5.

THEOREM 35.40 ([30]). If (Mn, g) is a closed simply-connect ed Riemannian

manifold with 2-positive curvature operator, then the volume normalized Ricci flow

with initial data g converges to a unique metric of constant positive curvature.

The work of Brendle and Schoen [35] on the general n-dimensional ~-pinching
theo rem (along with the earlier work by Chen [65] on t he 4-dimensional case)
provides another noteworthy stability (in fact, co nvergence) result for positively
curved sp ace forms.

4.2. Dynamic stability of Ricci-flat metrics.
Haslhofer and Muller have recently established a strong stability res ult for
Ricci fl.ow at Ricc i fl.at metrics by exploiting P erelman's >.-functional (35.8) and a
Lojas iewicz- Simon inequality:

THEOREM 35.4 1 ([147]). L et (Mn , g) be a compact Ricci-fiat manifold. If g

is a local maximizer of>., then for every Ck,O'._neightborhood U of g (k 2 2), there

exists a Ck,O'._neighborhood V c U such that Ricci flow starting at any metric in V

exists for all times and converges (modulo diffeomorphisms) to a R icci-fiat metric

inU.