= Vanishing harmonic function by level-set integration
{title2=$U\to0\text{ at infinity},\quad\Delta U=0\ \Longrightarrow\ U=0$}
For a smooth <harmonic function> on a manifold without boundary, assume its nonzero superlevel sets are compact. On a regular positive superlevel set $\{U>\varepsilon\}$, <integration by parts> gives $\int|\nabla U|^2=-\varepsilon\int_{\partial\{U>\varepsilon\}}|\nabla U|\leq0$. Applying the same argument to $-U$ proves $U=0$. This avoids assuming an unstated decay rate for the boundary flux at infinity.
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