Choosing a smooth separating regular level from an average bound

Choose $\alpha =(c_{-}+a)\mathrm{/}2$ and $\beta =(b+c_{+})\mathrm{/}2$, and define $P(s)=\min \{\beta ,\max \{\alpha ,s\}\}$ and $v=P\circ f$. For $r\le s$, the three intervals cut out by $\alpha ,\beta$ give $0\le P(s)-P(r)\le s-r$; hence $P$ is $1$-Lipschitz. By compactness the smooth gradient of $f$ is bounded, and integrating $df$ along piecewise smooth curves proves the length-Lipschitz bound used in [F3]. Thus $v$ is length-Lipschitz on each component. Continuity and the exact boundary values give neighborhoods of $B_{-}$ and $B_{+}$ on which $f<\alpha$ and $f>\beta$, respectively. On these neighborhoods $v=\alpha$ and $v=\beta$. The full boundary is exhausted by these parts, so [F3] applies to $v$ on $(a,b)$. For every $s\in (a,b)$, $\{v=s\}=\{f=s\}$, and this set lies in $R^{\circ }$ because the only boundary values of $f$ are $c_{-}$ and $c_{+}$. Consequently the full level-area function in the hypothesis is measurable, with the normalization fixed in [F10] and the interior interpretation supplied by coarea.

By [A1] and [F7], [F1] applies to $f{\mathrm{\mid }}_{R^{\circ }}:R^{\circ }\to \mathbb{R}$. Its regular values in $(a,b)$ therefore have full Lebesgue measure $b-a$. Put $M=A+ϵ$; the strict hypothesis and nonnegativity of area imply $M>0$. If every regular value in $(a,b)$ had area greater than $M$, monotonicity on that full-measure set and the null-set rule in [F8] would give ${\int }_a^b{\mathcal{H}}_k^2(\{f=s\})ds\ge M(b-a)$, contrary to the hypothesis. Hence choose a regular $t\in (a,b)$ with level area at most $M$.

Since $\mathrm{\partial }R=B_{-}\bigsqcup B_{+}$ and $c_{-}<t<c_{+}$, $D=f^{-1}(t)$ misses all of $\mathrm{\partial }R$. It is closed in compact $R$ and hence compact. If nonempty, it is an embedded codimension-one submanifold of $R^{\circ }$ by [F2], so it is a smooth surface without boundary. If empty, the empty surface has all these properties. No connectedness is required.

For any continuous path $\gamma :[0,1]\to R$ with $\gamma (0)\in B_{-}$ and $\gamma (1)\in B_{+}$, the function $f\circ \gamma$ is continuous with endpoint values $c_{-}$ and $c_{+}$. By [F4], it takes the value $t$, so the path meets $D$. In particular the existence of such a path implies $D\mathrm{\ne }\mathrm{\varnothing }$.

The sublevel $\mathrm{\Omega }=f^{-1}((-\mathrm{\infty },t])$ is closed in compact $R$, so it is compact. At a point with $f<t$ it agrees locally with $R$, while points with $f>t$ have a neighborhood disjoint from it. At every point of $D\subset R^{\circ }$, regularity and [F5] give a smooth transverse coordinate $y_1=f-t$ (compose the projection coordinate with the inverse target chart); thus $\mathrm{\Omega }$ has the half-space chart $y_1\le 0$ and boundary $y_1=0$. Near $B_{-}$ it agrees with $R$ by continuity and $c_{-}<t$, so its boundary there is exactly the original boundary part $B_{-}$. Near $B_{+}$ it is empty by $c_{+}>t$. These exhaust the boundary cases. The charts are restrictions or smooth changes of the given charts, so they give the induced smooth structure and a smooth embedding into $R$. Therefore $\mathrm{\partial }\mathrm{\Omega }=B_{-}\bigsqcup D$, as asserted.