A Geometrical View of the Nehari Manifold

We study the Nehari manifold $N$ associated to the boundary value problem \[-\Delta u=f(u)\,,\quad u\in H^1_0(\Omega)\] where $\Omega$ is a bounded regular domain in $\mathbb{R}^n$. Using elementary tools from Differential Geometry, we provide a local description of $N$ as an hypersurface of the Sobolev space $H^1_0(\Omega)$. We prove that, at any point $u\in N$, there exists an exterior tangent sphere whose curvature is the limit of the increasing sequence of principal curvatures of $N$. Also, the $H^1$-norm of $u\in N$ depends on the number of principal negative curvatures. Finally, we study basic properties of an angle decreasing flow on the Nehari manifold associated to homogeneous non-linearities.