6. GRADIENT SHRINKERS WITH NONNEGATIVE RICCI CURVATURE 29
By Perelman's proof of his no loca l collapsing theorem, one can demonstrate
the following.COROLLARY 27.49. If g =(Mn, g , f, -1) is a complete noncompact shrinking
GRS, then there exists a constant n, > 0 depending only on C 1 (9) satisfying the
following property. If xo E M is a point and r 0 > 0 are such that R ::; r 02 in
Bx 0 (ro), then VolBx 0 (ro) 2 n,r 0.The logarithmic Sobolev inequality plays a crucial role in the following result
of Munteanu and J. Wang.THEOREM 27.50 (Shrinkers must have at least linear volume growth). If g =
(Mn,g,f,-1) is a complete noncompact shrinking GRS and p EM, then there
exists a constant c = c(n, f(p), J e-f dμ) > 0 such thatAreaS(p, r);::: 2cHence
VolBp(r);::: er for r 2 l.
The constant c is nondecreasing in f (p) and non increasing in J e-f dμ. Hence, ifg is also a singularity model that is n,-noncollapsed below all scales, then by taking
p to be a minimum point off, we have that c depends only on n and n,.
Finally, we state a pair of important results on the geometry at infinity of
shrinkers. First, we have the following uniqueness result of Kotschwar and L. Wang.THEOREM 27.51. Any two complete noncompact shrinking GRS which have a
pair of ends which are asymptotic to the same Euclidean cone over a smooth (n - 1)-
dimensional closed Riemannian manifold must have isometric universal covers.Second, we have the following result of Munteanu and J. Wang.THEOREM 27.52. Let (Mn, g , f , -1) be a complete noncompact shrinking GRS.
If !Rel (x)---+ 0 as x---+ oo, then (M,g) is asymptotic to a Euclidean cone over a
smooth (n - 1)-dimensional closed Riemannian manifold.6. Gradient shrinkers with nonnegative Ricci curvature
If we consider shrinkers which are non-Ricci fiat with nonnegative Ricci cur-
vature, then we have the following improvement for the lower bound of the scalar
curvature due to one of the authors.
THEOREM 27.53 (Re 2 0 and Re =f:. 0 shrinker::::} R 2 o > 0). Let (Mn,g, f , -1)
be a complete noncompact non-Ricci fiat shrinking GRS with nonnegative Ricci cur-vature. Then there exists 0 > 0 such that
(27.132) R;::: o on M.
PROOF OF THE THEOREM (MODULO A CLAIM). Let 6 EM and let Xo EM-
B0(8r1), where the constant r 1 < oo is chosen to satisfy (27.147) below. Let u be
an integral curve of \l f with u (0) = x 0. Since \l f is a complete vector field (by
Corollary 27.7), u (u) is defined for all u ER Using \l R = 2 Re (\l !), we h ave
d
(27.133) du R (u (u)) = (\l R , \l f) = 2 Re (\l f , \l J) 2 0