  The topological surface states of mercury telluride (HgTe) are studied by ab
initio calculations assuming different strains and surface terminations. For
the Te-terminated surface, a single Dirac cone exists at the $\Gamma$ point.
The Dirac point shifts up from the bulk valence bands into the energy gap when
the substrate-induced strain increases. At the experimental strain value
(0.3%), the Dirac point lies slightly below the bulk valence band maximum. A
left-handed spin texture was observed in the upper Dirac cone, similar to that
of the Bi$_2$Se$_3$-type topological insulator. For the Hg-terminated surface,
three Dirac cones appear at three time-reversal-invariant momenta, excluding
the $\Gamma$ point, with nontrivial spin textures.
