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Let $D^2$ be a smooth disk and for any Riemannian metric in $D$, let $\mu_1(g)$ be the first positive Neumann eigenvalue of the Laplacian on $(D, g)$. …

Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$ …

For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $k=0,1,2, \dots$ be the eigenvalues of the Laplacian $\Delta …

The spectrum of the Laplacian operator $\Delta_g = -\operatorname{div} \nabla$ consists of an increasing and diverging sequence of positive eigenvalues: $$0 = \lambda_0(M,g) < \lambda_1(M,g) \leq \lambda …

Let $(M,g)$ be a compact Riemannian manifold with boundary. It is well known that the first positive Steklov eigenvalue $\sigma_1$ of $M$ has the following variational characterization: $$\sigma_1(M,g …

It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$, and the associated eigenfunctions are the coordinate functions $x,y,z$ restricted to the sphere. … I wonder if the functions $x$ and $y$ generate the first eigenspace of the Laplacian the upper hemisphere with Neumann boundary condition. …

\Lambda(\Sigma) := \sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ is a smooth Riemannian metric on $\Sigma$} \}$$ where $\lambda_1(\Sigma,g)$ denotes the first eigenvalue of the Laplacian …

Here, $\lambda_1(\Sigma,g)$ denotes the first eigenvalue of the Laplacian associated to the metric $g$ with Dirichlet ($=0$) boundary condition. …