# Maximizing metrics for the first eigenvalue

Can someone explain me what does the following abbreviation denote:

$$\lambda_1(\Sigma,g)\mathrm{area}(\Sigma,g)?$$

The context I am learning about it are closed Riemannian surfaces $(\Sigma,g)$ and

where the spectrum of the Laplace operator acting on smooth functions, is purely discrete and can be written as

$$0=\lambda_0<\lambda_1(\Sigma,g)\leq\lambda_2(\Sigma,g)\leq\lambda_3(\Sigma,g)... \to \infty$$

My guess is that if we let $\Delta_g = \nabla_g\cdot\nabla_g$ and consider $\lambda_i,\phi$ to solve the Helmholtz equation $\Delta_g\phi + \lambda_i\phi=0$ for all $i$, then $\lambda_1(\Sigma,g)\in\mathbb{R}$ is simply the $i$th eigenvalue of the spectrum of the Laplace-Beltrami operator, for the Riemannian manifold $(\Sigma,g)$; i.e. it is a single fixed number associated to the manifold.
The "area" of a Riemannian manifold is defined analogously to that of a surface in Euclidean space (which is a type of Riemannian manifold). However, because these "surfaces" generalize to arbitrary dimensions, one generally calls it "volume". Recall that the metric tensor $g$ lets you define the length of vectors, as well as inner products between such vectors, on $\Sigma$ (or more accurately its tangent spaces at each point). Then we can define lengths of curves using $g$. We can also define a volume form (more here, here, or here), which in coordinates $x^j$ is given by: $$\omega_g = \sqrt{\det g\,}\; dx^1\wedge\ldots\wedge dx^n$$ for an $n$ dimensional manifold. Then the volume is: $$\text{Vol}(\Sigma) = \int_\Sigma \omega_g$$ but note that this only makes sense on a single chart where $x^j$ are defined!
As an example, consider $(\Sigma,g)=(S,I)$, where $\{(0,0)\leq (x,y) \leq (a,b)\}=S\subset\mathbb{R}^2$. Then, $\det g = 1$ and we get: $$\text{Vol}(S) = \int_S \omega_g =\int_0^a\int_0^b dx^1\wedge dx^2 = \int_0^a b\, dx^2 = ab = \text{area}(S)$$ as we would expect.
Note there are some interesting connections between $\lambda_1$ and $\text{area}(\Sigma)$. For instance, Kac (1966) showed: $$\sum_{i=1}^\infty \exp(-\lambda_i t) \approx \frac{1}{4\pi t}\left[ \text{area}(\Sigma) - \sqrt{4\pi t\,}\ell(\partial\Sigma) + O(t) \right]$$ where $\ell$ denotes Riemannian length. (See here for a bit more). So for small $t$, $\lambda_1$ is the dominant factor in determining $\text{area}(\Sigma)$ in some sense!