Second Eigenvalue of Schrödinger Operators¶and Mean Curvature
- Cite this article as:
- El Soufi, A. & Ilias, S. Comm Math Phys (2000) 208: 761. doi:10.1007/s002200050009
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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 −Δ+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 Alikakos–Fusco conjecture concerning the stability of the interface in the Allen–Cahn reaction diffusion model.