  We prove that $\mathcal{C}^r$ maps with $r>1$ on a compact surface have
symbolic extensions, i.e. topological extensions which are subshifts over a
finite alphabet. More precisely we give a sharp upper bound on the so-called
symbolic extension entropy, which is the infimum of the topological entropies
of all the symbolic extensions. This answers positively a conjecture of
S.Newhouse and T.Downarowicz in dimension two and improves a previous result of
the author \cite{burinv}.
