Convexity and absolute continuity of the Laplace-Beltrami operator (Q1112409)

Let \(\Delta\) be the Laplace-Beltrami operator; adopt the sign convention that makes \(\Delta\) a negative operator on the smooth functions of compact support. \(\Delta\) has a unique self-adjoint extension. The author shows the existence of a convex \(C^ 4\) function on M implies the absolute continuity with respect to Lebesgue measure of the spectral measures of \(\Delta\).