On the second eigenvalue of the Laplace operator penalized by curvature (Q1356796)

Let \(M\) be a compact simple surface immersed in \(\mathbb{R}^3\); \(M\) is topologically a sphere. Let \(q(k_1,k_2)\) be a non-negative symmetric quadratic function of the principal curvatures of \(M\) and let \(L:=d\delta-q(k_1,k_2)\) on \(C^\infty(M)\). Let \(\lambda_0<\lambda_1\leq\lambda_2\dots\) be the eigenvalues of \(L\). The author characterizes the standard sphere using the second eigenvalue \(\lambda_1\) by proving the Theorem: Adopt the notation above. Then \(\lambda_1\leq 4\pi(2-q(1,1))\cdot \text{Area}(M)^{-1}\). Equality holds if and only if \(M\) is a sphere of radius \(r\) for some \(r\).