Given two measured laminations
and
in a hyperbolic surface which fill up the surface, Kerckhoff [Lines of Minima in
Teichmueller space, Duke Math J. 65 (1992) 187–213] defines an associated
line of minima along which convex combinations of the length functions of
and
are
minimised. This is a line in Teichmüller space which can be thought as analogous to
the geodesic in hyperbolic space determined by two points at infinity. We show that
when
is uniquely ergodic, this line converges to the projective lamination
, but that when
is rational, the line
converges not to ,
but rather to the barycentre of the support of
.
Similar results on the behaviour of Teichmüller geodesics have been proved by
Masur [Two boundaries of Teichmueller space, Duke Math. J. 49 (1982)
183–190].
Keywords
Teichmüller space, Thurston boundary, measured geodesic
lamination, Kerckhoff line of minima
