APPENDIX BOther, Aspects of Ricci Flow and Related Flows
- Convergence to Ricci solitons
Given that convergence to a soliton plays a role in proving the conver-
gence of the Ricci flow on compact surfaces (see Chapter 5 in Volume One),
it is reasonable to ask if noncompact steady solitons play a role as limiting
geometries for the Ricci flow on complete surfaces (or higher-dimensional
manifolds).
Since steady soliton solutions are, by definition, evolving by diffeomor-
phism, even if we start near a soliton metric, we cannot expect pointwise
convergence of the Ricci flow unleSF> we take the diffeomorphisms into ac-
count. We use the following notion of convergence [373]:
DEFINITION B.l. Let g(t) satisfy the Ricci flow on a nonc~m;act man-ifold Mn for 0 ~ t < oo and let g be a metric on M. We say that g(t)
has modified subsequence convergence to g if there exist a sequence of
times ti ----+ oo and a sequence of diffeomorphisms ¢i of M such that ¢ig(ti)
converges uniformly tog on any compact set.
Since the Ricci flow is conformal on surfaces and preserves completeness,
it makes sense to begin with metrics in the conformal class of the Euclidean
metric, i.e., metrics of the form
(B.1) g = eu(x,y) ( dx2 + dy2). '
We can describe the overall 'shape' of such metrics in terms of the aperture
and circumference at infinity. First, the aperture is defined by
A( )= 1. L(8B (0, r))
g. r-->oo lm (^2) 7fr ,
where the B ( 0, r) are geodesic balls about some chosen point 0 E IR^2. (The
choice does not affect the value of the limit.) For example, the flat metric
has aperture one, .the cigar metric has aperture zero, and surfaces in IR^3 that
are asymptotic to cones have aperture a/ (27r), where a is the cone angle.
The circumference at infinity is defined by
C 00 (g) =i= sup inf {L(8D) I K is compact, Dis open and KC D}.
. K D.
For flat and conical metrics, C 00 is infinite, while C 00 is finite for the cigar.
Let R_ = max{-R, O}.
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