I wanted to know if there was some equivalent of Weyl law for the spectrum of the Jacobi operator of a minimal surface in the non-compact case. If the minimal surface is not closed, for example in $\mathbb{R}^3$, and $\varphi:\Sigma\rightarrow\mathbb{R}^3$ is a smooth minimal branched immersion from some surface $\Sigma$, and $M=\varphi(\Sigma)$ is a finite index minimal surface, the surface $\Sigma$ is not compact (and is actually conformally equivalent to a punctured RIemann surface), and classical Weyl's law seems to break out. Here if $g$ is the pull-back of the Euclidean metric on $\mathbb{R}^3$, then the Jacobi operator considered here is simply $\Delta_g-2K_g$, where $\Delta_g$ is the negative Lalace operator and $K_g$ is the Gauss curvature.
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asked May 9, 2016 at 15:12
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$\begingroup$ I would guess Weyl's law should always break down for complete minimal surfaces, because they have negative Gauss curvature, so behave maybe more like complete surfaces of constant negative curvature, like the hyperbolic plane, for which the law does not apply, since there is essential spectrum: lots of harmonic functions on the upper half plane. $\endgroup$– Ben McKayCommented May 9, 2016 at 16:13
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2$\begingroup$ For a non-compact surface, the spectrum is not expected to be discrete, so already the question about eigenvalue asymptotics no longer makes sense. $\endgroup$– Christian RemlingCommented May 9, 2016 at 16:35
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