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In this paper, we obtain a generalization, in dimension $3$, of a theorem ofDavid Rees about joint reductions of the bigraded filtration $\{\overline{I^rJ^s}\}$ of complete ${\mathfrak m}$-primary ideals and vanishingof the second normal Hilbert coefficient $\overline{e}_2(IJ)$ where $R$ is atwo-dimensional Cohen-Macaulay analytically unramified local ring with maximalideal $\mathfrak m.$ This generalization is obtained as a consequence of aformula for the third local cohomology module of the extended Rees algebras ofthe $\mathbb Z^3$-graded filtration $\{\overline{I^rJ^sK^t}\}$ with support inthe ideal $(x_1t_1,x_2t_2,x_3t_3)$ where $(x_1,x_2,x_3)$ is a good jointreduction of $\{\overline{I^rJ^sK^t}\}.$

Local Cohomology of Multi-Rees Algebras with Applications to Joint Reductions and Complete Ideals