= Spherical Hessian identity
{title2=$\int_{S^2}|\nabla_S^2w|^2=\int_{S^2}(\Delta_Sw)^2-\int_{S^2}|\nabla_Sw|^2$}
For a smooth real function on the unit two-sphere, integration of the covariant-derivative commutation formula gives the displayed identity. Explicitly, $\nabla^a\nabla_a\nabla_bw=\nabla_b\Delta_Sw+\operatorname{Ric}_b{}^c\nabla_cw$, and the unit sphere has $\operatorname{Ric}=g$. Integrating by parts once and then again proves the formula. It controls all angular second <derivatives> by the <Laplace-Beltrami operator> without singular estimates at coordinate poles.
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