  The c(2$\times$2) reconstruction of (001) PbTiO$_3$ surfaces is studied by
means of first principles calculations for paraelectric (non-polar) and
ferroelectric ([001] polarized) films. Analysis of the atomic displacements in
the near-surface region shows how the surface modifies the antiferrodistortive
(AFD) instability and its interaction with ferroelectric (FE) distortions. The
effect of the surface is found to be termination dependent. The AFD instability
is suppressed at the TiO$_2$ termination while it is strongly enhanced,
relative to the bulk, at the PbO termination resulting in a c(2x2) surface
reconstruction which is in excellent agreement with experiments. We find that,
in contrast to bulk PbTiO$_3$, in-plane ferroelectricity at the PbO termination
does not suppress the AFD instability. The AFD and the in-plane FE distortions
are instead concurrently enhanced at the PbO termination. This leads to a novel
surface phase with coexisting FE and AFD distortions which is not found in
PbTiO$_3$ bulk.
