O que é Volatility surface, log-moneyness, total implied variance?

The volatility surface of an underlying is the function $(K,T)\mapsto\sigma_{\mathrm{imp}}(K,T)$ that gives, for every strike and expiry, the implied volatility of the European option, computed with Black’s formula on the forward $F_{0,T}$ of that expiry. Its natural coordinates are the log-moneyness $k=\ln(K/F_{0,T})$, which measures a strike against the forward of its own expiry, and the total implied variance $w(k,T)=\sigma_{\mathrm{imp}}^2T$, the variance of the log-return to expiry that the price implies.