Hopf positivity and uniform linear growth on a fixed compact collar

Smooth boundary, the Riemannian metric and the Laplacian convention are those of Weighted exterior elliptic domains and admissible weights. The inward unit normal is the vector orthogonal to $T_{y}S$, of $h$-norm one, pointing into $\mathrm{\Sigma }$. The regularity $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Sigma })$ includes continuous first derivatives up to the boundary in boundary charts.

The tangent-ball derivative conclusion is supplied by A variable-coefficient annular barrier and inward boundary derivative. Smoothness of the metric and potential on a sufficiently small closed chart ball gives its coefficient bounds and uniform ellipticity.

Work in ZFC, including The Axiom of Choice and the choice assumptions inherited through the barrier supplier. In particular the Countable Choice hypothesis of Collar neighborhood theorem holds. Fix its smooth collar $F:S\times [0,ϵ)\to \mathrm{\Sigma }$, a smooth embedding onto a relatively open neighbourhood of $S$ with $F(y,0)=y$. Its coordinate is $\rho (F(y,t))=t$. The vector ${\mathrm{\partial }}_{t}F(y,0)$ is strictly inward and transverse, since $F$ is a collar embedding.

A continuous real function on a nonempty compact space attains its minimum and maximum, and a continuous image of a compact space is compact, by A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism. The interval $[0,\delta ]$ is compact for $\delta >0$ by Closed real intervals are compact by completeness, and its product with compact $S$ is compact by A product of finitely many compact spaces is compact in the product topology. Compactness also means that every open cover has a finite subcover; the common collar width will be obtained by that property directly.

The closed-interval mean value theorem Mean value theorem on a closed real interval applies to a real function continuous on $[a,b]$ and differentiable on $(a,b)$. The supplier Continuous functions have an oriented Riemann integral defines the Riemann integral of continuous real functions on compact intervals, proves its interval additivity, and gives $m(b-a)\le {\int }_a^{b}g(s)ds\le M(b-a)$ whenever $m\le g\le M$ and $a<b$. These are used over the real numbers with the complete ordered-field structure of Dedekind-complete ordered real field.

For the distance formulation, curves are continuous maps $\gamma :[a,b]\to \mathrm{\Sigma }$ that are $C^1$ on each piece of a finite subdivision $a=t_0<\cdots <t_N=b$, up to the piece endpoints in interior or boundary charts; endpoint derivatives are one-sided. They are allowed to lie on or meet $S$. Define $L_h(\gamma )={\sum }_{j=1}^N{\int }_{t_{j-1}}^{t_j}\sqrt{h_{\gamma (s)}(\dot{\gamma }(s),\dot{\gamma }(s))}ds$. Each speed is continuous on its closed piece, so [F5] makes the length finite and nonnegative. At a subdivision point each adjacent piece uses its own one-sided derivative. Interval additivity gives invariance under refinement, and a common refinement gives independence of subdivision. For $a=b$, set $L_h(\gamma )=0$. Define $d_h(x,S)$ to be the infimum of $L_h(\gamma )$ over all these curves starting at any point of $S$ and ending at $x$, with $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. A nonempty family of lengths has an infimum in $[0,\mathrm{\infty })$ by the completeness axiom in math:def-complete-ordered-real-field, applied to the negatives of the nonnegative lengths. Thus this distance-to-subset construction applies to manifolds with boundary and disconnected exteriors, including components not joined to $S$; finiteness on the collar will follow from its explicit curves.