A metric-gradient critical crossing preserves the pointed disk pair

Straighten the actual stable and unstable disks as in [F1] and use [F3] on the unstable axis so that $f(0,b)=f(q)-\mathrm{\mid }b{\mathrm{\mid }}^2$. Choose the derivative of this axis coordinate change as the positive diagonal Hessian scaling in a spectral basis; it commutes with the diagonal hyperbolic linear part, so the hypotheses of [F1] remain valid in these coordinates. On the entry level the incoming normal neighbourhood is a graph $s=h(u,\xi )$, $\xi \in Q$, with $h$ smooth in the corner charts and $\mathrm{\parallel }Dh\mathrm{\parallel }$ uniformly bounded on a smaller neighbourhood. Choose the entry sufficiently small that its stable coordinates are inside the boundary-data ball of [F1]. Choose $\delta$ small enough that each entry point with $u\mathrm{\ne }0$ crosses a fixed lower height before leaving the critical box: the unstable norm grows and the stable norm decays, and at the outer unstable sphere the height is below that lower level. The $u=0$ orbit converges to $q$.

For an open terminal unstable ball $\mathrm{\mid }b\mathrm{\mid }<3R$, solve $u={\alpha }_T(h(u,\xi ),b)$ for every $T\ge 0$. Axis invariance gives $D_a{\alpha }_T(a,0)=0$. The uniform mixed second-derivative bound in [F1], integrated in $b$, gives $\mathrm{\parallel }{D}_a{\alpha }_T(a,b)\mathrm{\parallel }\le C\mathrm{\mid }b\mathrm{\mid }{e}^{-\beta T}$, and the first-derivative bound gives $\mathrm{\mid }{\alpha }_T(a,b)\mathrm{\mid }\le C^{\mathrm{\prime }}\mathrm{\mid }b\mathrm{\mid }{e}^{-\beta T}$. Choose $R$ so small that $3CR\mathrm{\parallel }Dh\mathrm{\parallel }<1\mathrm{/}2$ and $3C^{\mathrm{\prime }}R<\delta \mathrm{/}4$. The equation is then a contraction on the $u$ ball of radius $\delta \mathrm{/}2$, uniformly in $T,\xi$. Denote its smooth solution by $u=F_T(b,\xi )$ and its stable endpoint by $s_T={\zeta }_T(h(F_T(b,\xi ),\xi ),b)$. It is the exact field passage. At $T=0$, $F_0(b,\xi )=b$. The same equations in $\rho =1\mathrm{/}T$ have identity unknown derivative at zero, and [F1] makes $F_{1\mathrm{/}\rho }$ and $s_{1\mathrm{/}\rho }$ smooth and flat there.

For each finite $T$, $b\mapsto F_T(b,\xi )$ is injective: its initial point is $(h(u,\xi ),u)$, and finite-time uniqueness recovers its endpoint $b$ from $u$. Differentiating the equation of step 2.1 gives $D_b{F}_T=(I-D_a{\alpha }_{T}Dh{)}^{-1}{D}_b{\alpha }_T$. The first factor has positive determinant by its norm distance from the identity, and the second does by [F2]. Thus this is a positive local diffeomorphism and therefore an embedding of the prescribed ball, depending smoothly on $T,\xi$. All images lie in the inner part of the normal disk. This supplies a genuine finite-time isotopy of these disks from the identity, rather than an unproved disk-unknotting assertion.

Fix a sufficiently large finite $T_0$. On the moving image $F_T(B_{3R},\xi )$ define the velocity $V_T(F_T(b,\xi ),\xi )={\mathrm{\partial }}_T{F}_T(b,\xi )$, multiplied by a cutoff in $b$ equal to one on $D_{2R}$ and supported in $B_{3R}$. Extend it by zero outside this image. Its space-time inverse is smooth by step 3.1; the cutoff has compact support away from the image boundary, so the zero extension is smooth. For $0\le T\le T_0$ its support is a compact subset of the interior normal disk, uniformly over compact $Q$. Its finite-time flow therefore gives ambient diffeomorphisms of that normal disk, fixed near its outer boundary, carrying $b$ to $F_T(b,\xi )$ on $D_{2R}$. These diffeomorphisms keep $\xi$ fixed and preserve all its old corner faces. This explicit compact velocity extension avoids any smooth Schoenflies or isotopy-extension hypothesis.

Take the lower cutoff $B=f(q)-R^2$. At late times the endpoint height is $f(s_T(b,\xi ),b)$, converging flatly to $f(q)-\mathrm{\mid }b{\mathrm{\mid }}^2$. On $\mathrm{\mid }b\mathrm{\mid }\le R\mathrm{/}2$ it is above $B$ for all sufficiently large $T$; on the annulus about $\mathrm{\mid }b\mathrm{\mid }=R$ its radial derivative is negative and nonzero. Thus [F3] gives a unique smooth cutoff radius $r_B(\rho ,\theta ,\xi )$, tending flatly to $R$ as $\rho \to 0$. The endpoint domain is the disk bounded by that radius. A smooth radial collar adjustment, equal to the identity for $\mathrm{\mid }b\mathrm{\mid }\le R\mathrm{/}2$ and outside $\mathrm{\mid }b\mathrm{\mid }\ge 2R$, identifies it with $D_R$; take $(r,\theta )\mapsto (r+(r_B-R)\chi (r),\theta )$ with $\chi (R)=1$ and $\mathrm{\mid }{r}_B-R\mathrm{\mid }\mathrm{\parallel }{\chi }^{\mathrm{\prime }}\mathrm{\parallel }<1\mathrm{/}2$. Hence the portion $T\ge T_0$, completed at $T=\mathrm{\infty }$, is $Q\times D_R\times [0,1\mathrm{/}{T}_0]$. At zero its endpoint is the labelled lower unstable disk, including $b=0$.

The section at $T=T_0$ attaches this late collar along a disk $K_{T_0}$ in the original incoming normal disk. The radial adjustment of step 4.2 is itself isotopic to the identity by scaling its correction. Composing it with the ambient diffeomorphism of step 4.1 shows that $K_{T_0}$ is an ambient image of the standard inner disk, by an isotopy fixed near the outer boundary. Indeed $K_{T_0}$ consists of the entries whose height at $T_0$ is at least $B$; any such entry has terminal $\mathrm{\mid }b\mathrm{\mid }<2R$ for large $T_0$, by the height inequality and the small stable endpoint, so it is in the exact graph of step 2.1. This proves that no extra attachment component is omitted.

For $u\mathrm{\ne }0$ let ${\tau }_B(u,\xi )$ be its first passage to height $B$; it is finite and smooth by strict descent and step 1.1. It tends to infinity as $u\to 0$, uniformly over compact $Q$, since bounded passage times would contradict finite-time convergence to a stable orbit ending at $q$. Choose $T_0$ above all outer-boundary passage times. The finite portion, from a fixed short backward-time top cut to $\min (T_0,{\tau }_B)$, is a cylinder over the entire normal disk: normalize its finite, positive interval length. Its bottom disk consists of $K_{T_0}$ at time $T_0$ and an exit annulus outside it. The late collar of step 4.2 is attached exactly along that inner bottom disk. By step 5.1 straighten the disk by a fibrewise ambient diffeomorphism. A cylinder with a collar attached to a standard inner part of its bottom is again a cylinder topologically, fixed on its top and outer side: in meridian coordinates its shape is the union of two rectangles with nested radial widths, a star-shaped region. Prescribe the boundary homeomorphism taking the old bottom first to the new bottom disk and then to the exit annulus, keeping the top and outer side, and extend by rays from an interior centre. Retain angular directions; the axis collapses continuously. This gives the required disk-pair homeomorphism and matches the regular outside flow-height transport. For $k=0$ it is just extension of an interval by an endpoint collar.

The finite-cylinder coordinates and the late fixed-anchor coordinates describe the same marked trajectories on their overlap by uniqueness of the exact passage. Their inverse data are their entry point, marked endpoint and passage time; at infinite time they are the incoming history and lower unstable endpoint. The flat estimates give geometric continuity there. All maps above retain the history coordinate, so they agree on the old broken faces and extend through an exit cutoff. Away from the tube the ordinary regular transport applies. Thus the global crossing comparison is continuous and bijective on compact pointed pieces, has continuous inverse in these charts, fixes the chosen high cap and carries the old interior to the new unbroken interior. It is the claimed homeomorphism of disk pairs for the actual metric field, without replacing its eigenvalues by normalized rates.