Isoperimetric inequalities for the eigenvalues of natural Schrödinger operators on surfaces

Ahmad El Soufi (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

This paper deals with eigenvalue optimization problems for a family of natural Schrödinger operators arising in some geometrical or physical contexts. These operators, whose potentials are quadratic in curvature, are considered on closed surfaces immersed in space forms and we look for geometries that maximize the eigenvalues. We show that under suitable assumptions on the potential, the first and the second eigenvalues are maximized by (round) spheres.

Mots clés: Laplacian; Schrödinger operator; eigenvalues; isoperimetric inequalities for eigenvalues

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]

Ahmad El Soufi. Isoperimetric inequalities for the eigenvalues of natural Schrödinger operators on surfaces. Indiana University Mathematics Journal, 2009, 58 (1), pp.335--350. ⟨hal-00360863⟩