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Let $\Delta$ be the scalar Laplace-Beltrami operator on a compact, connected, orientable 2-manifold without boundary smoothly embedded in $\mathbb{R}^3$ and let $\phi$ be one of its eigenfunctions, i.e.,

$$ \Delta \phi = \lambda \phi $$

for some constant $\lambda \in \mathbb{R}$. Is there any reason to believe that the maximum and minimum pointwise values of $\phi$ have the same magnitude? This statement certainly holds for simple domains like $S^1$ (where the eigenfunctions are just $\cos(nx)$ and $\sin(nx)$, $n \in \mathbb{Z}$) and also appears to be true empirically (e.g., if you numerically compute eigenfunctions of triangulated surfaces). But I have trouble seeing why it would be true in general.

One thing to note is that the eigenfunctions of $\Delta$ are orthogonal with respect to the $L^2$ inner product. In particular, the constant function is always an eigenfunction of $\Delta$, which means all other eigenfunctions will have zero mean. Unfortunately zero mean does not guarantee that the max and min have equal magnitude.

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I may be misunderstanding the question, since this counterexample seems a bit too easy if you've already gone to the trouble of computing eigenfunctions on triangulated surfaces, but isn't $2P_2(\cos\theta)=3\cos^2\theta-1$ on the unit sphere an eigenfunction for $\lambda=-6$ with maximal value $2$ and minimal value $-1$?

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No, you're absolutely right. Good example! –  PolyKnowMeAll Sep 16 '11 at 13:36

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