On the Second Eigenvalue of the Laplace Operator Penalized by Curvature

Consider the operator ?r 2 ?q(), where ?r 2 is the (positive) Laplace-Beltrami operator on a closed manifold of the topological type of the two-sphere S 2 and q is a symmetric non-negative quadratic form in the principal curvatures. Generalizing a well-known theorem of J. Hersch for the Laplace-Beltrami operator alone, it is shown in this note that the second eigenvalue 1 is uniquely maximized, among manifolds of xed area, by the true sphere. Dimensionally, the Laplace operator ?r 2 is comparable to the square of curvature , both having dimensions (length) ?2. Thus one might expect to encounter partial diierential operators of the form ?r 2 ? q in applications, where q is a quadratic expression in the principal curvatures. This was recently the case when N. Alikakos and G. Fusco performed a stability analysis of the interfacial surface separating two phases in one of the simpler reaction-diiusion models, the Allen-Cahn system. It was already realized in the rst article about this model 5] that it exhibits interfaces moving according mean-curvature, as a consequence of which the model has attractive geometric features; for current state of mathematical knowledge of this see 9]. While simpliied in comparison to most realistic reaction-diiusion systems, Allen-Cahn is a reasonable model at least for bistable alloys of iron and aluminum. Picture a bubble of material of phase I in a background of phase II. It undergoes slow motions and deformations, and if it is not at an external boundary, the surface smooths out and eventually becomes round. According to Alikakos and Fusco, instabilities of the surface are associated with negative eigenvalues of an operator emerging from linearization at of the form (1) ?r 2 ? 2 X j=1 2 j ; where j are the principal curvatures at any given point of , and r 2 is the Laplace-Beltrami operator on. This can be thought of as a geometric Schrr odinger operator with a negative potential determined by curvature. It is evident that (1) is a highly symmetric, reasonable object, and that the second eigenvalue is special because it equals 0 when attains its target shape of a sphere. (While the analysis by Alikakos and Fusco is not accessible in print at the time of this writing, the lower dimensional analogy is worked out in the recent thesis of V. Stephanopoulos 12, see Proposition 5.4 and Theorem 7.1; see also 2]. Related work and …