Inverse spectral positivity for surfaces

Pierre Bérard (1) , Philippe Castillon (2)

1 IF - Institut Fourier
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier

Let $(M,g)$ be a complete non-compact Riemannian surface. We consider operators of the form $\Delta + aK - q$, where $\Delta$ is the non-negative Laplacian, $K$ the Gaussian curvature, $q$ a non-negative function, and $a$ a positive real number. We address the question ''What conclusions on $(M,g)$ and $q$ can one draw from the fact that the operator $\Delta + aK - q$ is non-negative'' and we improve earlier results in particular in the cases $a = \4$ and $a \in (0,\4)$. We also show that the non-negativity is preserved under normal Riemannian covering with amenable covering group.

Mots clés: Spectral theory; Positivity; Minimal surface; Constant mean curvature surface

Domaines: Géométrie différentielle [math.DG]

Pierre Bérard, Philippe Castillon. Inverse spectral positivity for surfaces. 2011. ⟨hal-00644783v1⟩