On the L_r-operators penalized by (r+1)-mean curvature

2016/11/09 by Souza, Leo Ivo S.
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1611.02998

Abstract

In this paper, we establish the non-positivity of the second eigenvalue of the Schrödinger operator -\textrmdiv( P_r ∇⋅) - W_r² on a closed hypersurface Σⁿ of ℝⁿ⁺¹, where W_r is a power of the (r+1)-th mean curvature of Σⁿ. In the case that this eigenvalue is null we have a characterization of the sphere. This generalizes a result of Evans and Loss proved for the Laplace-Beltrame operator penalized by the square of the mean curvature.

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