Second Eigenvalue of Schrödinger Operatorsand Mean Curvature
Abstract
Let M be a compact immersed submanifold of the Euclidean space, the hyperbolic space or the standard sphere. For any continuous potential q on M, we give a sharp upper bound for the second eigenvalue of the operator ∆+q in terms of the total mean curvature of M and the mean value of q. Moreover, we analyze the case where this bound is achieved. As a consequence of this result we obtain an alternative proof for the AlikakosFusco conjecture concerning the stability of the interface in the AllenCahn reaction diffusion model.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2000
 DOI:
 10.1007/s002200050009
 Bibcode:
 2000CMaPh.208..761E