  We investigate the statistics of the maximal fluctuation of two-dimensional
Gaussian interfaces. Its relation to the entropic repulsion between rigid walls
and a confined interface is used to derive the average maximal fluctuation $<m>
\sim \sqrt{2/(\pi K)} \ln N$ and the asymptotic behavior of the whole
distribution $P(m) \sim N^2 e^{-{\rm (const)} N^2 e^{-\sqrt{2\pi K} m} -
\sqrt{2\pi K} m}$ for $m$ finite with $N^2$ and $K$ the interface size and
tension, respectively. The standardized form of $P(m)$ does not depend on $N$
or $K$, but shows a good agreement with Gumbel's first asymptote distribution
with a particular non-integer parameter. The effects of the correlations among
individual fluctuations on the extreme value statistics are discussed in our
findings.
