  We prove that the eigencurve associated to a definite quaternion algebra over
$\QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and
Buzzard--Kilford: (a) over the boundary annuli of weight space, the eigencurve
is a disjoint union of (countably) infinitely many connected components each
finite and flat over the weight annuli, (b) the $U_p$-slopes of points on each
fixed connected component are proportional to the $p$-adic valuations of the
parameter on weight space, and (c) the sequence of the slope ratios form a
union of finitely many arithmetic progressions with the same common difference.
In particular, as a point moves towards the boundary on an irreducible
connected component of the eigencurve, the slope converges to zero.
