Regular energy shell and weighted area measure

Let $O\subset {\mathbb{R}}^m$, $m\ge 2$, be open, let $H\in C^2(O,\mathbb{R})$, and let $J\subset \mathbb{R}$ be open. An energy $E\in J$ is regular if $\mathrm{\nabla }H(x)\mathrm{\ne }0$ for every $x\in {\mathrm{\Sigma }}_E:=H^{-1}(\{E\})$. By the implicit-function theorem, a neighborhood of each point of a nonempty regular level is a graph $x_i=h(z)$.

In a graph chart, use the local area density $\sqrt{1+\mathrm{\mid }\mathrm{\nabla }h(z){\mathrm{\mid }}^2}dz$, where $dz$ is $(m-1)$-dimensional Lebesgue measure. A graph partition on ${\mathrm{\Sigma }}_E$ is a locally finite graph-chart cover with nonnegative smooth functions ${\chi }_a$ supported in their respective charts and summing to one on the level. Whenever such a partition is supplied, define the candidate weighted shell integral by ${\mu }_E(g):={\sum }_a\int g(z,h_a(z)){\chi }_a(z,h_a(z))\mathrm{\mid }{\mathrm{\partial }}_{i}H(z,h_a(z)){\mathrm{\mid }}^{-1}dz$ for nonnegative Borel $g$. Its independence from the chosen chart partition, and its equality with ${\int }_{{\mathrm{\Sigma }}_E}g\mathrm{\mid }\mathrm{\nabla }H{\mathrm{\mid }}^{-1}d{A}_E$, are proved in the theorem on this page. If $0<{\mu }_E(1)<\mathrm{\infty }$, the microcanonical shell probability is ${\widehat{\mu }}_E:={\mu }_E\mathrm{/}{\mu }_E(1)$.

This is an explicit chart-based definition, not a formal delta distribution $\delta (H-E)$. Critical levels are excluded.