Math Help - Minimal functions

1. Minimal functions

There are many examples of minimal functions/surfaces throught internet (Minimal Surfaces -- from Wolfram MathWorld). And i even found smth like parametrization of ALL minimal surfaces (Minimal Surface -- from Wolfram MathWorld). But i cannot get properties of such space of minimal functions. As, let say, subspace of integrable functions in $R^2$. I'm interest in next questions. Will be this space infinite-dimensional (it's trivial, yes - there are many pairs of analytic and meromorphic functions).
Will it be connected in topological space?

2. To find the maximum value on an interval, you need to take the derivative and find all relative maximums/minimums and also check the endpoints.
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