Finite short-geodesic polygon cellulation from curvilinear triangles

By compactness and bounded ambient curvature near $M$, refine the supplied finite strongly convex cover to finitely many small strongly convex normal coordinate disks $U_j$ covering $M$, each with closure in a supplied cover member and ${\int }_{U_j}\mathrm{\mid }K\mathrm{\mid }{\mu }_g<\pi$. Apply [F1] subordinate to this refined cover. For every closed face $P$, choose its assigned small disk $U(P)$ and its containing supplied cover member $V(P)$. Since there are finitely many faces and each $P$ is compact, each has a positive chart margin inside $U(P)$ and $V(P)$. The original finite graph has regular $C^2$ edges, finitely many vertices, and pairwise distinct incident tangent rays; every sector occupied by a face has a positive opening. At a smooth boundary vertex the two boundary germs delimit the inward half-plane, and each incident interior edge germ has a strict inward angle. At an ordinary corner the corresponding sector is the supplied wedge. These assertions follow from the face sector and link clauses of [F1]–[F3], including the strict sectors in [F1]'s construction.

For each regular interior edge choose a finite subdivision by arclength with mesh at most $h$. For $h$ small enough, every subarc and its two endpoints lie in a common strongly convex normal coordinate disk of the ambient metric, so [F2] gives its unique minimizing geodesic chord $c$. Subdivide further to make every chord length less than $r$; uniform continuity of the finitely many edges makes this a finite operation. Leave boundary edges fixed, except that their existing marked endpoints may be retained as subdivision points.

The approximation has explicit uniform bounds. Parametrize an original subarc $gamma:[0,h]\to N$ by arclength and its chord $c:[0,h]\to N$ proportionally to arclength. In one of finitely many normal-coordinate disks covering the original graph, the coordinate acceleration of $gamma$ is bounded by a fixed $A_0$, and $c$ satisfies $c^{\mathrm{\prime }\mathrm{\prime }k}=-{\mathrm{\Gamma }}_{ij}^k(c)c^{\mathrm{\prime }i}{c}^{\mathrm{\prime }j}$ with uniformly bounded Christoffel symbols and speed, hence $\mathrm{\mid }{c}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\mid }\le A_1$. The two curves have the same endpoints. Applying the one-dimensional endpoint Green-function estimate to $c-\gamma$ gives $\sup \mathrm{\mid }c-\gamma \mathrm{\mid }\le A{h}^2$ and $\sup \mathrm{\mid }{c}^{\mathrm{\prime }}-{\gamma }^{\mathrm{\prime }}\mathrm{\mid }\le Ah$, with a common $A$ over the finite edge family after reducing $h$. These coordinate bounds are independent of the subdivisions and include the one-sided tangents at subarc endpoints.

Choose disjoint small ambient disks about the finitely many original vertices. By step 1.1, their incident edge germs occupy pairwise disjoint narrow cones and, at a boundary vertex, every interior germ cone lies strictly inside the inward half-plane or supplied corner wedge. Choose the disks small enough that each germ is a graph over its tangent ray there. Outside these vertex disks, disjoint original edges and the boundary have positive separation after deleting their common endpoint neighbourhoods; each original edge has a narrow embedded tubular strip in which its arclength projection is a coordinate. Reduce $h$ so the errors of step 3.1 are smaller than one quarter of every such separation, chart margin, and cone-angle margin. Then consecutive chords of one edge are graphs with strictly increasing arclength projection, chords on different edges cannot cross outside the vertex disks, and chords of distinct incident edges stay in their disjoint cones inside those disks. At a boundary vertex each interior chord initially points into the strict inward sector, and its $C^1$-close continuation cannot meet the fixed boundary arc in the vertex disk; away from boundary vertices the separation margin prevents boundary crossings. This directly covers a smooth boundary that oscillates under an unrelated straight line: each chosen chord is controlled relative to its own old interior edge.

The new finite graph is embedded in $M$ with the same incidence and cyclic order as the old one. In each edge strip, the old and new arcs are graphs over the same longitudinal coordinate; interpolation of their transverse graph functions, multiplied by a cutoff that vanishes at the strip boundary, gives an ambient isotopy of that strip. In each vertex disk the disjoint ordered germs admit the analogous radial interpolation inside their strict cones, fixed near the disk boundary; on a boundary disk keep the boundary germs fixed. These finitely many local isotopies agree as the identity off their disjoint supports and extend to a homeomorphism of $M$ carrying the old graph to the new graph. Therefore the complements still have finitely many closed polygonal Jordan-disk faces, the same circle/interval links and full-cell intersections, and ordinary positive face sectors. A former face $P$ and its new image remain in $U(P)\subset V(P)$ by the chart margins chosen in step 1.1.

Split each original triangular face boundary at the inserted edge vertices. Its image under the homeomorphism of step 5.1 is a polygonal disk: each interior side is a finite chain of short ambient minimizing geodesic chords and each boundary side is a prescribed regular $C^2$ arc or subarc. The family remains face-to-face, with each shared old edge given the same subdivision and chords from both incident faces. Its finitely many vertices, edges and faces supply the claimed cellulation. In the empty case use the empty cellulation. Full AC is used only through [F1]; the subdivisions, uniform estimates and isotopies are finite.