Restriction of formal-immersion data has the parametric lifting property

First take $m=k\ge 1$. Write the boundary homotopy as $(b_v,a_v)$, where $a_v$ is the outward radial column and $d{b}_v$ supplies the other $k-1$ directions. There is a continuous, spatially smooth unit normal field $u_v$ to their $k$-plane. To construct it, the initial plane field extends over $P\times D^k$ by the initial immersion; contracting this contractible space and transporting a vector in the positive-rank orthogonal complement by successive orthogonal projections gives an initial field. Along the boundary homotopy, use finitely many time intervals on which orthogonal projection from the previous complement into the next is an isomorphism, and normalize after each transport. Such intervals exist uniformly because the boundary parameter space is compact and the plane fields are continuous in their spatial jets. This constructs $u_v$ for the entire time interval before choosing any disk lifts. Its spatial derivatives are bounded on every compact parameter piece.

Fix an interval starting at $v_0$ and its current disk lift $g$. For $x\in S^{k-1}$ put $y=b_{v_0}(x)$, ${\mathrm{\Delta }}_v={\log }_y{b}_v(x)$, and $T_v=d_w\mathrm{\ell }(y,{\mathrm{\Delta }}_v)$. Treat $d{b}_v$ as an operator on $T_x{S}^{k-1}$; all singular-value bounds use unit vectors in $\mathbb{R}\oplus T_x{S}^{k-1}$ and do not choose a global sphere frame. Pull the boundary columns and normal into $T_{y}N$, writing $A_v=T_v^{-1}d{b}_v$, $a_v^{\ast }=T_v^{-1}{a}_v$, and $u_v^{\ast }=T_v^{-1}{u}_v$, then renormalize the last vector and, if needed, project it orthogonally off the first $k$ columns. At $v_0$ the radial column is $a_0={\mathrm{\partial }}_{r}g(1,x)$. Choose a smooth linear map $Q_v$ close to the identity with $Q_v{a}_0=a_v^{\ast }$: the rank-one formula $Q_v=I+(a_v^{\ast }-a_0)\otimes a_0^{\ast }\mathrm{/}\mathrm{\parallel }{a}_0{\mathrm{\parallel }}^2$ works on a sufficiently short time interval. On a thin inner collar define $R(r,x)={\log }_{y}g(r,x)$. Thus $R(1,x)=0$, ${\mathrm{\partial }}_{r}R(1,x)=a_0$, and the logarithm is defined uniformly there.

Here is the rank estimate used below. For every boundary point and time, $[a_v^{\ast },A_v]$ has rank $k$ and $u_v^{\ast }$ is unit and normal to its image. Consequently the matrices $[\frac{a_v^{\ast }+\lambda u_v^{\ast }}{\sqrt{1+{\lambda }^2}},A_v],\lambda \in \mathbb{R},$ have a common positive least singular value $c$. Indeed replace $\lambda$ by the compact pair $(t,z)=(1,\lambda )\mathrm{/}\sqrt{1+{\lambda }^2}$, including its limits $(0,\pm 1)$. The radial column $t{a}_v^{\ast }+z{u}_v^{\ast }$ has nonzero projection off $\mathrm{im}{A}_v$: its squared normal length is $t^2\mathrm{\parallel }{a}_{v,\perp }^{\ast }{\mathrm{\parallel }}^2+z^2$, bounded away from zero. Compactness, including unit input vectors, gives $c>0$. Therefore changing each normalized column by operator norm less than $c\mathrm{/}3$ preserves rank. Choose $L>6\mathrm{/}c$. This estimate includes arbitrarily skew initial frames; no universal numerical angle lemma is needed.

Let $r_0<r_1<1$ be sufficiently close to $1$. Choose a nondecreasing smooth $\alpha$ which is zero for $r\le r_1$, one near $1$, and has $\mathrm{\mid }{\alpha }^{\mathrm{\prime }}\mathrm{\mid }\le C\mathrm{/}(1-r_0)$, with $C$ independent of collar width. Choose a smooth bump $\beta$ which rises from zero between $r_0$ and $r_1$ to $L$, and equals $L(1-\alpha )$ for $r\ge r_1$; thus $\beta ={\beta }^{\mathrm{\prime }}=0$ near $1$, $\mathrm{\mid }\beta \mathrm{\mid }\le L$, and $\mathrm{\mid }{\beta }^{\mathrm{\prime }}\mathrm{\mid }=L\mathrm{\mid }{\alpha }^{\mathrm{\prime }}\mathrm{\mid }$ wherever ${\alpha }^{\mathrm{\prime }}\mathrm{\ne }0$. Put $M(v,p)={\max }_x\mathrm{\parallel }{\mathrm{\Delta }}_v(p,x)\mathrm{\parallel }$. Define on the collar $W_v=(I+\alpha (Q_v-I))R+\alpha {\mathrm{\Delta }}_v+\beta M(v,p)u_v^{\ast },G_v(r,x)=\mathrm{\ell }(y,W_v(r,x)),$ and set $G_v=g$ for $r\le r_0$. The formulas paste smoothly because both cutoffs vanish near $r_0$. They give $G_{v_0}=g$, $G_v(1,x)=b_v(x)$, and ${\mathrm{\partial }}_r{G}_v(1,x)=T_v{Q}_v{a}_0=a_v$. Every coefficient depends continuously on parameters in all spatial derivatives.

Choose the time step uniformly small in the prescribed boundary homotopy, then the collar uniformly thin for the current compact family of disk lifts. The formula has rank $k$. After applying the inverse vertical differential $d_w\mathrm{\ell }(y,W_v{)}^{-1}$, its tangential columns equal $A_v+E_v$ and its radial column equals $a_v^{\ast }+{\beta }^{\mathrm{\prime }}M{u}_v^{\ast }+e_v+{\alpha }^{\mathrm{\prime }}{\mathrm{\Delta }}_v$, where $\mathrm{\parallel }{E}_v\mathrm{\parallel }$ and $\mathrm{\parallel }{e}_v\mathrm{\parallel }$ can be made arbitrarily small uniformly. To verify these bounds, differentiate step 4.1. Tangential errors are sums of the differences of the initial collar jets from their boundary jets, $R{d}_x{Q}_v$, $\beta M{d}_x{u}_v^{\ast }$, and the smooth local-addition coefficient differences at $W_v$ and ${\mathrm{\Delta }}_v$. The first two tend to zero with collar width; the last two tend to zero with the size of the prescribed boundary change. Radial errors are the initial radial-jet difference, $Q_v-I$ times bounded radial jets, and ${\alpha }^{\mathrm{\prime }}(Q_v-I)R$. Since $\mathrm{\parallel }R\mathrm{\parallel }\le C_0(1-r_0)$, the last term is bounded by $C{C}_0\mathrm{\parallel }{Q}_v-I\mathrm{\parallel }$ independently of collar width. The local-addition vertical differential is invertible near the compact diagonal. Writing $\lambda ={\beta }^{\mathrm{\prime }}M$, the remaining radial error has norm $\mathrm{\parallel }{\alpha }^{\mathrm{\prime }}{\mathrm{\Delta }}_v\mathrm{\parallel }\le \mathrm{\mid }\lambda \mathrm{\mid }\mathrm{/}L$; it vanishes wherever ${\alpha }^{\mathrm{\prime }}=0$. Divide the radial column by $\sqrt{1+{\lambda }^2}$ and use step 3.1: its last error is at most $1\mathrm{/}L<c\mathrm{/}6$, and all other normalized errors can be made less than $c\mathrm{/}6$. Thus the differential is injective. Outside the collar it is $dg$.

The allowed time step in step 5.1 depends only on the prescribed boundary homotopy, the preconstructed normal field and the compact boundary-frame conditioning; it does not depend on the current lift's interior injectivity margin. Its collar width may depend on that lift. Choose a finite uniform time subdivision and repeat steps 2.1–5.1, using the preceding endpoint lift as initial disk data. This produces a lift on all of $I$, continuous in the weak smooth topology. If the boundary homotopy is constant for a parameter, then ${\mathrm{\Delta }}_v=0$, $Q_v=I$ and $M(v,p)=0$ there on every interval, so the lift is constant there. When $k=1$ there are no tangential columns, and the same normalized rank estimate uses the nonzero radial column and its independent normal vector.

For $m>k$, first lift the first $k$ columns by steps 1.1–6.1. The remaining columns are a transversal $(m-k)$-frame, together with tangential components. Transport the changing tangent and orthogonal normal bundles of the lifted core immersion to fixed bundles by finitely many sufficiently close orthogonal projections; this is possible on the compact already constructed core family. In these fixed bundles the boundary normal-frame homotopy lifts as follows. For nearby frames $V_0,V_1$, the operator $S=I+(V_1-V_0)(V_0^{\ast }{V}_0{)}^{-1}{V}_0^{\ast }$ sends $V_0$ to $V_1$ and is close to the identity. Identify the current normal bundles on a thin radial collar with their boundary bundles by orthogonal projection, uniformly invertible by compactness. In these radial identifications extend $S$ over a collar of $S^{k-1}\subset D^k$ by $I+\chi (r)(S-I)$ and act on the current interior normal frame; the operator remains invertible, so the frame stays independent throughout. The boundary normal-frame family is compact, hence finitely many time steps suffice with a uniform bound $\mathrm{\parallel }S-I\mathrm{\parallel }<1\mathrm{/}2$. Tangential components extend by the constant-along-radii linear collar operator $\chi (r)B(x)$. No normal boundary derivatives of these fields are prescribed, so no boundary-jet extension operator is required. The resulting full $m$ columns agree with all prescribed boundary columns.

The formal restriction is proved by the same operator transport. Identify nearby target tangent fibres by projection in the ambient Euclidean embedding. Correct the base map by $\mathrm{\ell }$ and a collar cutoff, retaining its prescribed independent radial derivative $C$. In logarithm coordinates for the current base map on a collar put $R_0={\log }_{y}f$, $\mathrm{\Delta }={\log }_{y}b$, $T=d_w\mathrm{\ell }(y,\mathrm{\Delta })$ and $K=T^{-1}C-{\mathrm{\partial }}_r{R}_0(1,x)$. Replace $R_0$ by $R_0+\chi (r)\mathrm{\Delta }+(r-1)\chi (r)K$, with $\chi =1$ near the boundary. This has boundary value $\mathrm{\Delta }$ and radial derivative $T^{-1}C$, so after $\mathrm{\ell }$ its actual boundary jet is exactly $(b,C)$. A thin collar keeps its values in the local-addition domain; no derivative rank is required of a formal base map. Extend the boundary monomorphism change by the invertible operators $S$ of step 7.1 acting on the current full interior frame; its rank is preserved without an evolving injectivity-margin assumption. Compact boundary data allow a fixed finite time subdivision; the collar width for base-map correction may vary at the successive stages. These constructions depend continuously in all spatial jets, and give the formal lifting property. Finally a holonomic core datum is realized near the core by $\mathrm{\ell }(g(x),{\sum }_{j>k}{B}_j(x)v_j)$; its derivative at $v=0$ is the full given monomorphism, hence it is an immersion on a compact-family neighbourhood. This proves both cubical restriction assertions and the germ realization.

Compact smooth parameter lifting follows without assuming a global normal field over the parameter manifold. Choose a finite parameter cover on which the initial normal bundle over the centre of the disk has a frame, and a subordinate finite partition ${\rho }_i$ with closed supports inside those sets. Extend each frame over the disk by radial projection transport. For a boundary homotopy $b(p,t)$ perform finitely many segments $b_i(p,t,s)=b(p,t({\sum }_{j<i}{\rho }_j(p)+s{\rho }_i(p)))$, $0\le s\le 1$. At segment $i$ the initial lift is the preceding segment's endpoint, depending on $(p,t)$. Over the closed support of ${\rho }_i$, its normal bundle is trivialized by transporting the initial local frame along the auxiliary $t$ coordinate and the radial disk contraction; those transports start at the original lift when $t=0$. The preceding proof then lifts this segment on that parameter support. Where ${\rho }_i=0$, the boundary segment is constant and the formula is exactly the initial lift, so the formulas paste to the unchanged family outside the parameter set. Closed support containment and compactness give uniform constants, including along its boundary. Concatenating the finite segments gives a compact-parameter lift of the original homotopy. It is constant on every parameter region where the input homotopy is constant. Formal and transverse-frame transports have the same localization. Thus the compact smooth parameter lifting used in the relative comparisons is supplied as well.