Lemma 5.27.21.C3F45D193092 After shrinking $U$, there is a constant $C>0$ such that
\[ |\partial f|\leq C|f|^{1-1/m_0}\quad \text{on }U\setminus Z. \]
For each compact set $K\subset U$, there are $C_ K,r_ K>0$ with
\[ \int _{K\cap \{ |f|=r\} }|\partial f|\, \mathrm dS\leq C_ Kr \qquad (0<r<r_ K), \]
where $\mathrm dS$ is the Euclidean surface measure on the real hypersurface $\{ |f|=r\} $. In particular, for every nonnegative measurable function $\Psi $,
\[ \int _{K\cap \{ 0<|f|<r_ K\} }\Psi (|f|)|\partial f|^2\, \mathrm dV \leq C_ K\int _0^{r_ K}\Psi (r)r\, \mathrm dr. \]
Proof.
Shrink $U$ so that $|h_ j|\leq 1$ and the unit $g$, its inverse and the first derivatives of $g$ and of all $h_ j$ are bounded. For each $j$, $|f|^{1/m_0}\leq C|h_ j|^{m_ j/m_0}\leq C|h_ j|$. Logarithmic differentiation of the factorization therefore gives
\[ \frac{|\partial f|}{|f|} \leq C\left(1+\sum _ j|h_ j|^{-1}\right) \leq C|f|^{-1/m_0}, \]
which proves the first bound.
For small $\zeta \in \mathbb C$, let $\mu _\zeta $ be the Euclidean volume measure of the fiber divisor $\{ f=\zeta \} $, counted with multiplicity. As $\zeta \to 0$, we have $\log |f-\zeta |\to \log |f|$ in $L^1_{\mathrm{loc}}$: by the local compactness theorem recalled at the beginning of section 5.25, it suffices to identify subsequential limits, and the convergence is locally uniform off $Z$, which identifies every such limit and excludes convergence to $-\infty $. The Poincaré–Lelong formula therefore gives weak convergence of the fiber currents, and hence of their Euclidean trace measures. Pairing with a fixed nonnegative cutoff function equal to one near $K$ shows that $\sup _{|\zeta |<r_ K}\mu _\zeta (K)<\infty $ for some $r_ K>0$.
The critical locus of $f$ has finitely many local components, and $f$ is constant on each of them. After shrinking $U$, every remaining critical component passes through $0$, where $f$ vanishes. Thus the small nonzero fibers are smooth. On $\{ |f|=r\} $, the coarea formula for the map $f$ to the circle of radius $r$ gives
\[ \int _{K\cap \{ |f|=r\} }|\partial f|\, \mathrm dS =\int _{|\zeta |=r}\mu _\zeta (K)\, \mathrm dS(\zeta ) \leq C_ Kr. \]
The coarea formula for the function $|f|$, together with $|\nabla |f||=|\partial f|$, proves the last inequality. The same formulas hold in dimension one, where the fiber volume is the counting measure.
$\square$
Pome
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