# Convergence of spectrum

Let $$M$$ be a compact manifold and $$g_k$$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $$g$$.

Let $$\{\lambda^k_j\}$$ be the spectrum of the Laplacian of the Riemannian metric $$g_k$$, and $$\{\lambda_j\}$$ be the spectrum of the Laplacian of $$g$$ (with values in increasing order, i.e., $$\lambda^k_{j+1} \geq \lambda^k_{j}$$).

I expect that $$\lambda^k_j \to \lambda_j$$ as $$k \to \infty$$. Is this true? If yes, what is a reference? If I replace smooth convergence with $$C^k$$-convergence, what is the lowest $$k$$ such that convergence of spectrum holds?

• The lowest $k$ for which this is true is $0$. Roughly: write down the min-max formula for $\lambda_j^k$ and $\lambda_j$, take bases of eigenfunctions $\{u_j\}$, $\{u_j^k\}$ for the two problems, and use $E = \text{span}\{u_1, \ldots, u_j\}$ as a competitor for the $\lambda_j^k$ problem and vice versa; this gives $\lambda_j^k \leq R_{g_k}(u)$ for some $u \in E$, where $R$ is the Rayleigh quotient. As this only contains the metric coefficients and volume forms if you write it out, you can estimate like $R_{g_k}(u) \leq R_g(u) + C\|g - g_k\|_{C^0}R_g(u)$. But $R_g[u] \leq \lambda_j$ as $u \in E$. Sep 2 at 18:26
• I see, so basically the finite dimensional argument works! Sep 2 at 23:03

$$C^0$$-convergence is sufficinent.
Note that $$\lambda_i$$ can be defined as the least lower bound on numbers $$\lambda$$ such that the following property holds:
• There is an $$i$$-dimensional subspace $$W$$ of smooth functions on $$M$$ such that $$\int\limits_{(M,g)} |d f|_g^2<\lambda\cdot \int\limits_{(M,g)} f^2$$ for any $$f\in W$$.
Fix $$\lambda>\lambda_i$$ and such a subspace $$W$$. Note that $$W$$ meets the same property for $$g_n$$ with large $$n$$. It follows that $$\lambda_i\geqslant \limsup_{n\to\infty} \lambda_i^n$$.
Nearly the same argument shows that $$\lambda_i\leqslant \liminf_{n\to\infty} \lambda_i^n$$.
• Isn't $C^0$ sufficient? The equation you write doesn't depend on derivatives of the metric Sep 15 at 13:15