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Among bounded domains in ${\mathbb{R}}^n$ of a prescribed volume, the ball minimizes the first Dirichlet eigenvalue of the negative Laplace operator. For sufficiently regular $\Omega$ and a ball $B$ with $|B|=|\Omega |$,This is the Faber-Krahn inequality. The variational formula shows why rearrangement helps: replacing a function by its symmetric decreasing rearrangement preserves its squared integral while reducing its gradient energy.^[ref]

For a stretched domain $r\Omega$, the eigenvalue scales as $r^{-2}{\lambda }_1(\Omega )$, so the volume normalization is essential. For a membrane with fixed wave speed and area, the disk has the lowest fundamental frequency because frequency is proportional to $\sqrt{{\lambda }_1}$. For diffusion with an absorbing boundary, it instead gives the slowest asymptotic decay among shapes of that volume.

The hyperbolic Laplacian is the Laplace-Beltrami operator for the metric of hyperbolic space. In upper-half-space coordinates $(x_1,\dots ,x_{n-1},y)$ with $y>0$,Using the nonnegative convention $\Delta =-\mathrm{div}\mathrm{grad}$ givesIndeed, insert $g^{ij}=y^2{\delta }^{ij}$ and $\sqrt{\det g}=y^{-n}$ into the coordinate formula $-{\left(\det g\right)}^{-1/2}{\partial }_i(\sqrt{\det g}{g}^{ij}{\partial }_j)$. The choice of overall sign must be specified when comparing spectral formulas.^[ref]

In dimension two the first-derivative term disappears. For $f(y)=y^s$, direct differentiation gives $\Delta f=s(n-1-s)f$, and in particular $s(1-s)f$ in the hyperbolic plane. This is the eigenvalue expression underlying many Maass form conventions.

Despite its name, this is an elliptic differential operator on $y>0$: its leading symbol in the chosen sign convention is $y^2|\xi {|}^2$. The adjective hyperbolic refers to the geometry, not to the classification of a hyperbolic partial differential equation.

Yaiza Canzani works in spectral geometry, studying how the geometry of a space controls eigenfunctions and eigenvalues of its Laplace operator. Her research account includes bounds for eigenfunction concentration, averages over submanifolds and spectral counting remainders.^[ref]

Her Analysis on Manifolds via the Laplacian by Yaiza Canzani develops the geometry behind this spectral language. On a circle, for example, the modes $1$, $\cos (k\theta )$ and $\sin (k\theta )$ have nonnegative Laplace eigenvalues $0$ and $k^2$. Their oscillation scale grows with $k$, making the link between shape, frequency and eigenvalue tangible. On a general Riemannian manifold, the corresponding questions concern how waves concentrate and how geometry changes their allowed modes.^[ref]