We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We prove that on any compact manifold 
$M^{n}$ with boundary, there exists a conformal class 
$C$ such that for any Riemannian metric 
$g\in C$ of unit volume, the first positive eigenvalue of the Neumann Laplacian satisfies 
${\it\lambda}_{1}(M^{n},g)<n\,\text{Vol}(S^{n},g_{\text{can}})^{2/n}$. We also prove a similar inequality for the first positive Steklov eigenvalue. The proof relies on a handle decomposition of the manifold. We also prove that the conformal volume of 
$(M,C)$ is 
$\text{Vol}(S^{n},g_{\text{can}})$, and that the Friedlander–Nadirashvili invariant and the Möbius volume of 
$M$ are equal to those of the sphere. If 
$M$ is a domain in a space form, 
$C$ is the conformal class of the canonical metric.
