Spherical Hessian identity (${\int }_{S^2}|{\nabla }_S^{2}w{|}^2={\int }_{S^2}({\Delta }_{S}w{)}^2-{\int }_{S^2}|{\nabla }_{S}w{|}^2$)

For a smooth real function on the unit two-sphere, integration of the covariant-derivative commutation formula gives the displayed identity. Explicitly, ${\nabla }^a{\nabla }_a{\nabla }_{b}w={\nabla }_b{\Delta }_{S}w+{\mathrm{Ric}}_b{}^c{\nabla }_{c}w$, and the unit sphere has $\mathrm{Ric}=g$. 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.