- BASIC PROPERTIES OF GRADIENT RICCI SOLITONS 3
Define the !-Ricci tensor to be Re t ~ Re + \7^2 f; this is also known as the Bakry-
Emery Ricci tensor. Define the !-scalar curvature to be Rt = R+26.f-l\7 fl^2.
Given a measurable set X CM, define its !-volume to be
(27.8) Vol 1 (X) ~ l e-f dμ.
The operator 6. f is self-adjoint with respect to the L^2 -inner product of functionsusing the measure e-f dμ.
EXERCISE 27.l. Show that for any rp E C^00 (M) we have that( 6. 1 - ~Rt) rp = efl^2 ( 6. _ ~R) (e-ff2rp).Equations for normalized GRS may be conveniently expressed using 6. 1 , Rc 1 ,and Rt· First of all,
by (27.1) and we also haveR1=df-n), forE=±l,
Rt= -1, for E = 0.
We may rewrite the difference of (27.2) and (27.6) as(27.9a)
(27.9b)6.tf=E(f-~) forE=±l,
- ff = -1 for E = 0
and we may rewrite (27.4) as
(27.10) 6.1R = -21 Re I^2 - ER~ --R^2 2 - ER.
n1.2. Scalar curvature lower bound for a GRS structure.
The following result of B.-L. Chen does not require any curvature bound in its
hypothesis. In the case where M is compact, this follows directly from applying
the maximum principle to equation (27.4).THEOREM 27.2 (Lower bound for the scalar curvature ofGRS). Let (Mn, g, f, E),
where E = -1, 0, or l, be a complete GRS structure.
(1) If the GRS is shrinking or steady, then R 2: 0.
(2) If the GRS is expanding, then R 2: -l
REMARK 27.3. Note that the Gaussian soliton shows that part (1) is sharp,whereas the Einstein solutions with Re= -~g show that part (2) is sharp.
PROOF OF THEOREM 27.2. We may assume that Mis noncompact. The proof
consists of locali zing equation (27.4).
STEP l. The Laplacian of R times a cutoff function 71. Fix a point 6 E M
and let r(x) ~ d(x,6). Let b E [2,oo) and define 71: [O,oo) ~ [0,1] to be a C^00
nonincreasing cutoff function with
(27.11) 7l(u) = {^1 for u E [O, 1],
0 for u E [l + b, oo)and 7111 - 2 (711
)2
2: -const
71