An application of min-max characterization of eigenvalues

Question

Let $(M,g_0)$ be a $n$-dimensional closed Riemannian manifold with a Riemannian covering $(\widetilde{M},\widetilde{g}_0)$. Let

$$ \mathcal{V}_{ab}=\{g\colon a^2 g_0\leq g\leq b^2 g_0\}, \quad \text{where} \ 0<a^2<1<b^2. $$

My question is that: for all $g\in\mathcal{V}_{ab}$ and for all $k\in\mathbb{N}$, why does the following estimate of $k$-th eigenvalue of Laplacian hold

$$ \frac{a^n}{b^{n+2}} \lambda_k(\widetilde{g}_0) \leq \lambda_k (\widetilde{g}) \leq \frac{b^n}{a^{n+2}} \lambda_k(\widetilde{g}_0) $$

by min-max characteristic of spectrum of Laplacian? Could you please give me more details? Thanks in advance.

Answer 1

There is no need to consider a Riemannian cover here so I’m not sure what role that plays. As such, I will drop the tildes from the metrics.

The key idea is the classical result that it is possible to characterize the $k$-th eigenvalue as the following min-max problem.

$$\lambda_k = \inf_{V \subset C^2(M), \, dim(V) = k} \max_{f \in V \backslash \{0\}} \frac{\int_M |\nabla f|^2 \,dVol} {\int_M f^2\, dVol}$$

Intuitively, the infimum is realized when $V$ is the subspace spanned by the first $k$-eigenfunctions and then the maximization picks out the largest value of the Raleigh quotient. For simplicity denote the integral quotient in the previous equation as $R(f)$.

To prove the inequalities in your question, first consider a single function $f \in C^2(M)$. We then have that the Raleigh quotient satisfies $$ \frac{\int_M \frac{1}{b^2} |\nabla f|_g^2 \, \frac{1}{b^n} dVol_g} {\int_M f^2\, \frac{1}{a^n} dVol_g} \leq \frac{\int_M |\nabla f|_{g_0}^2 \,dVol_{g_0}} {\int_M f^2\, dVol_{g_0}}, $$ which follows from bounds on the metric and the way that the volume form and gradient scale with the metric. Note that the left hand side is simply $\frac{a^n}{b^{n+2}}$ times the Rayleigh quotient with respect to $g$.

To obtain the first inequality, we then consider the space $V$ spanned by the first $k$ eigenfunctions with respect to $g_0$. From the previous inequality, we have that for every $f \in V$, $\frac{a^n}{b^{n+2}} R_g(f) \leq R_{g_0}(f)$. For the function $f_k \in V$ which maximizes the quantity $$\frac{\int_M |\nabla f_k|_g^2 \, dVol_g} {\int_M f_k^2\, dVol_g},$$ we must have that $$\frac{a^n}{b^{n+2}} R_g(f_k) \leq R_{g_0}(f_k) \leq \max_{f \in V} R_{g_0}(f) = \lambda_k(g_0),$$ where the final inequality holds by the definition of $V$. To finish the proof, we note that $V$ might not be the subspace which realizes the infimum of the Rayleigh quotient with respect to $g$, so we need to minimize the left hand side over all subspaces. However, doing so makes the Rayleigh quotient smaller, so we find the desired inequality $$\frac{a^n}{b^{n+2}} \lambda_k(g) \leq \lambda_k(g_0).$$

For the other inequality, we simply switch the roles of $g$ and $g_0$ and argue in the same way.