= Solution
\b[There is no compact boundaryless minimal surface in $\mathbb R^3$.] For an immersion $X$ of a minimal surface, the coordinate functions are harmonic: $\Delta_SX=0$. Consequently $\Delta_S|X|^2=2|\nabla_SX|^2=4$, because the immersion has a two-dimensional orthonormal tangent frame. Integrating over a compact surface without boundary gives zero on the left by the <divergence theorem> and $4\operatorname{Area}(S)>0$ on the right, a contradiction. If surfaces with boundary are allowed, a closed planar disk is a compact minimal example; the boundaryless convention matters.
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