- MATCHED ASYMPTOTIC STUDIES OF DEGENERATE NECKPINCHES 319
to the rotation axis (through the poles).) The initial geometries satisfy the fol-
lowing curvature conditions: (i) the Ricci curvature is positive in a neighborhood
of each pole; (ii) the sectional curvature of each of the planes tangent to each ofthe sn sections is positive; (iii) the scalar curvature is everywhere positive. In a
neighborhood of one of the poles (which we label the right pole), there is at least
one bump-neck-bump sequence, consisting of a pair of local maxima of the function
V(z) (the bumps) sandwiching a local minimum (the neck) for V(z). (Here V(z)specifies the diameter of the sn section a parameter distance z along the axis.) A
singularity, marked by unbounded curvature, occurs in finite time T at the right
pole, with no singularity occurring anywhere prior to T.We note that these conditions are chosen so that the initial geometries are close
to those which, according to the numerical simulations, form degenerate neckpinch
singularities.
2.2. Coordinates and Ricci flow equations.
Following [11], we discuss the formal matched asymptotic studies using the
(rotationally symmetric) metric representation
(36.11)
(36.12)g(t) = ¢^2 (x, t)dx^2 + 'lj?(x, t)ground
= ds^2 + '1/J^2 (s(x, t), t)ground·Here ground is the round metric on the sphere sn, x E ( -1, 1) is a coordinate
running between the poles PL and PR and labeling the sn cross sections, and
s(x, t) := fox¢(~, t)d~ is the corresponding arclength coordinate. One readily shows
that the Ricci flow equations for these metrics takes the form
(36.13) 8t¢ = n o~'lj; ¢,
(36.14) Ut<p n.J,_n - Uss<p · '·-( n -1)(1-(8s'l/J)2) 'ljJ '
where we note that 05 '1/J := iBx'l/J, so that [8t, 05 ] = -~""Bs.
Based on the Primary Ansatz and arguments from [13] (see also [11]) which
indicate that the bump-neck-bump configuration is preserved until immediately
before the singularity occurs at time T , we divide the formal analysis into four
overlapping regions: the tip (a neighborhood of the right pole PL), the interme-
diate region (a neighborhood of the bump closest to PL), the parabolic region (a
neighborhood of the neck), and the outer region (which extends from the far side
of the neck outwards). We now discuss each of these regions in turn, describing
the coordinate dilations, the auxiliary ansatze, the series expansions and PDE lin-
earization, and the approximate solutions characteristic of each region. We do this
in an order- parabolic, intermediate, tip, outer-which best clarifies the analysis,
though it is not the sequential geometric order of the regions.
2.3. The parabolic region.
In the region surrounding the center of the neck, the geometry (suitably dilated)
is approximately cylindrical. The quadratic deviation from cylindricality motivates
the labeling of this portion of the geometry as the "parabolic region".