Context

Gromov explicitly recorded the question of smooth isometric immersion of arbitrary smooth Riemannian surfaces into \(\mathbb{R}^4\) (Gromov 2000, sec. I, Remark (d), p. 3 of the author version). He later attributed the four-space surface question to Chern around 1950, while expressing uncertainty about the compactness qualification (Gromov 2015, 4, footnote 6). Theorem 1 concerns the compact boundaryless case of this question.

Nash’s smooth embedding theorem gives a global realization of a compact \(n\)-dimensional Riemannian manifold in dimension \(n(3n+11)/2\), which is \(17\) for surfaces (Nash 1956, Theorem 2, p. 59). A closer surface comparison is Gromov’s construction of smooth isometric immersions of closed surfaces into \(\mathbb{R}^5\): his retrospective explicitly recalls this result and refers to §3.2.4 of Partial Differential Relations (Gromov 2015, 59)(Gromov 1986). Theorem 1 lowers this immersion dimension to four.

The local analytic theory reaches a smaller dimension. The theorem of Janet and Cartan places an analytic Riemannian surface locally and isometrically in \(\mathbb{R}^3\) (Janet 1926; Cartan 1927); see also the formulation in (Gromov 2015, sec. 1.2). Both the analytic hypothesis and the local conclusion matter here. For comparison, the companion (OpenAI 2026, Theorem A) constructs a smooth positive definite metric on \((-1,1)^2\) for which no neighborhood of the origin admits a smooth isometric immersion into \(\mathbb{R}^3\).

There is also a smooth four-dimensional theory on planar domains. Poznyak proved that \(C^{3,\alpha}\) data admit \(C^{2,\alpha}\) isometric immersions into \(\mathbb{R}^4\) on compact portions of a simply connected planar domain with regular boundary (Poznyak 1973, Theorem 4, p. 67). The smooth version of this construction is stated and explained in Lewicka’s modern account (Lewicka 2026, Theorem 9.1, p. 57). Passing from planar domains to a closed surface requires the corrections to remain compatible through a global construction. Here we achieve this by preserving the normal geometry needed for successive local metric corrections.

Regularity gives a second distinction. An immersion is strictly short if the prescribed metric minus its induced metric is positive definite. Nash’s oscillatory construction and Kuiper’s codimension-one refinement show that a strictly short immersion in positive codimension can be uniformly approximated by \(C^1\) isometric immersions (Nash 1954; Kuiper 1955); the contemporaneous account (Choquet 1958, Theorems 1–2) states the immersion form explicitly. Lewicka proves a finer flexibility theorem in \(\mathbb{R}^4\) for metrics of low Hölder regularity. In the August 2026 version, the domain is bounded, planar, and diffeomorphic to a disk, the metric is \(C^{r,\beta}\) on its closure with \(0<r+\beta\le2\), and strictly short \(C^1\) immersions admit uniform approximation by \(C^{1,\alpha}\) isometric immersions for \(0<\alpha<\min\{(r+\beta)/2,1\}\) (Lewicka 2026, Theorem 1.1, p. 2). These domain and regularity hypotheses differ from the smooth closed-surface statement of Theorem 1.

The proof uses the established ideas of positive rank-one metric decomposition, oscillatory metric correction, and smoothing to control derivative loss. We give the corresponding attribution where these ideas enter the construction. The mode equations, finite correctors, smooth convergence argument, and propagation of the normal geometry are proved below. Whitney’s ordinary immersion theorem supplies the starting map (Whitney 1944, Theorem 8, p. 274); Sard’s equal-dimensional critical-value theorem is used to choose the support boundaries and phases (Sard 1942, Theorem 4.1, p. 885).

The construction

The proof has four stages. First prepare a smooth immersion into a small round three-sphere in \(\mathbb{R}^4\) together with a finite decomposition of its metric deficit. Its induced metric \(g_0\) is strictly below \(g\), and the negative radial direction is a globally defined normal, including when \(M\) is nonorientable. The decomposition has the form \[ g-g_0=\sum_{j=1}^{J_0} a_j^2\,dx_j^2, \tag{1}\] where each \(x_j\) is a coordinate function on a disk and \(a_j\) is smooth, positive in that disk, and zero outside it. The decomposition is arranged so that each phase has a positive transverse Hessian for the metric immediately preceding its addition.

Second, let \(F\) be the current immersion and consider one addition \(a^2dx^2\). Write \(y\) for a complementary coordinate. The relevant plane is perpendicular to both \(F_y\) and \(F_{yy}\). A periodic path on a circle in this plane has the prescribed average and adds \(a^2dx^2\) to the leading metric. Orthogonality to \(F_{yy}\) removes the first transverse metric error. A finite expansion then cancels the oscillatory errors to an arbitrarily prescribed order. Its remaining mean terms form a finite differential perturbation of the induced-metric map. Small corrections on a slower scale reduce this perturbed map’s error to the required finite accuracy.

Third, turn the resulting approximate metric into an exact one. We prove a smooth correction theorem under the pointwise hypothesis that the vector-valued second fundamental form is nonzero. A finite construction solves the linearized equation on individual oscillatory modes to arbitrarily high finite accuracy, with suitably chosen phase directions. Smoothing between successive steps controls the lost derivatives; increasing the accuracy in successive blocks gives convergence in every smooth norm. No global inverse of the linearized metric map is required.

Finally, retain the geometric conditions needed for the remaining disks. The large oscillation may rotate the second fundamental form. Conditions on the later boundary curves, and a choice of large transverse derivatives at the turns of the velocity loop, prevent its relevant normal directions from vanishing or becoming opposite to the preferred normal. The small \(C^2\) exact correction preserves the required derivative asymptotics. Together with the exact Gauss equation, these asymptotics establish the strict boundary and crossing conditions. After the finite list of additions, the induced metric is \(g\) by (1).

The two analytical devices used here have separate roles. The small-mode construction supplies high-order metric accuracy while its displacement is small in \(C^2\). The large circle construction changes the metric by a prescribed positive primitive tensor while retaining selected second-derivative information. Their combination is what permits a finite global construction in codimension two. The number of global primitive additions is finite; within each addition, however, the exact correction uses an infinite smoothing iteration converging to one map in every \(C^m\) norm.

Organization

Section 2 sets up normal geometry and admissible phases. Section 3 states the primitive-addition theorem, including the boundary conditions it preserves. Sections 4 and 5 prove the small increment and exact correction results. Sections 6 and 7 construct the velocity loop and realize the primitive metric. Section 8 completes the primitive-addition proof. Section 9 prepares the finite list and proves Theorem 1.

Weighted bounds and the increment statement

For a smooth function, vector field, or tensor on the fixed coordinate atlas, set \[|f|_{m,s}=\sum_{j=0}^m s^j\|f\|_{C^j},\qquad 0<s\leq1.\] For a map \(G\), the notation \(|j^2G|_{m,s}\) means \(\sum_{j=0}^m s^j\|G\|_{C^{j+2}}\). We write \(f=O_s(b)\) if, for every fixed \(m\), \(|f|_{m,s}\leq C_m b\). In statements concerning a family of inputs, the constants may depend on its indicated weighted bounds and on the fixed coordinate data. Product estimates and smooth composition estimates hold in these norms by the product and chain rules. In particular, \[ |\partial^rG|_{m,s}\leq C_m s^{2-r} \quad(r\geq2),\qquad\text{if }j^2G=O_s(1). \tag{8}\]

Fix positive phase data from Lemma 3: phases \(\phi_\nu\), cutoffs \(\psi_\nu\), and linear forms \(q_\nu\). Choose concentric \(C^0\) balls \(\mathcal U_0,\mathcal U_1\) about a metric \(h_0\), with radii \(0<r_0<r_1\), so that for \(A\in\mathcal U_1\), \[ b_\nu(A)=\psi_\nu\sqrt{q_\nu(A)},\qquad A=\sum_\nu b_\nu(A)^2\,d\phi_\nu^2. \tag{9}\] All square-root arguments have a uniform positive lower bound for \(A\in\mathcal U_1\) on their coordinate neighborhoods. The prescribed tensor belongs to \(\mathcal U_0\); the larger ball allows the finite intermediate tensor adjustments below. The differentials of \(\phi_\nu\), \(2\phi_\nu\), and \(\phi_\nu\pm\phi_\mu\) are good wherever the corresponding supports meet. For each such phase choose local coordinates with that phase as the first coordinate. All these finitely many coframes and their inverses have fixed bounds. We require the relevant immersion 2-jets to remain in a compact subset of the open set where these conditions hold.

A differential operator is called polynomial above order two if each of its coordinate components is a finite sum of terms \[ c(x,j^2G)\prod_{\ell=1}^d\partial^{\alpha_\ell}G^{k_\ell}, \qquad |\alpha_\ell|>2, \tag{10}\] where \(c\) is smooth; the empty product is allowed. This definition permits arbitrary smooth dependence on the low jets within the prescribed open set.

The proof has three finite steps: solve the nonzero linear modes to high order, adjust their quadratic mean, and cancel the remaining quadratic modes. There is no infinite series of differential operators in this construction.

The conjugated linearized equation

Fix one permitted phase and call its coordinates \(x,y\). Complexify all vector spaces linearly. Let \(g=\gamma(G)\) and \(D_G=\gamma'(G)\). Decompose a complex amplitude \(Z\) as \[Z=G_k g^{k\ell}u_\ell+v, \qquad u_i=\langle G_i,Z\rangle, \qquad v\perp\operatorname{span}(G_x,G_y).\] Since \(G_{ij}=\Gamma^k_{ij}G_k+B_{ij}\), direct differentiation gives \[(D_GZ)_{ij} =\partial_i u_j+\partial_j u_i-2\Gamma^k_{ij}u_k -2\langle B_{ij},v\rangle.\] Here and in this subsection \(B=B_G\). This identity already accounts for the normal component; no derivative of the normal frame is omitted. Conjugating by \(e^{ix/\tau}\) gives \[ (D_G^xZ)_{ij} =\partial_i u_j+\partial_j u_i-2\Gamma^k_{ij}u_k +\frac{i}{\tau}(\delta_{ix}u_j+\delta_{jx}u_i) -2\langle B_{ij},v\rangle. \tag{12}\]

The good-phase condition says that \(b=B_{yy}\) does not vanish. Given a target amplitude \(h\) and a normal amplitude \(v_0\) perpendicular to \(b\), define \[ c_{ij}=\frac{\langle B_{ij},b\rangle}{|b|^2},\qquad v=v_0+\frac b{|b|^2} \left(\partial_yu_y-\Gamma^k_{yy}u_k-\frac{h_{yy}}2\right). \tag{13}\] The \(yy\) equation in \(D_G^xZ=h\) is then exact. The \(xx\) equation, divided by 2, and the \(xy\) equation become \[\begin{align*} \frac i\tau u_x+\partial_xu_x-\Gamma^k_{xx}u_k -c_{xx}(\partial_yu_y-\Gamma^k_{yy}u_k) &=\frac{h_{xx}}2+\langle B_{xx},v_0\rangle -\frac{c_{xx}h_{yy}}2, \tag{14}\\ \frac i\tau u_y+\partial_xu_y+\partial_yu_x-2\Gamma^k_{xy}u_k -2c_{xy}(\partial_yu_y-\Gamma^k_{yy}u_k) &=h_{xy}+2\langle B_{xy},v_0\rangle-c_{xy}h_{yy}. \tag{15}\end{align*}\] Thus the remaining system is \[\frac i\tau u+Lu=k(h,v_0),\] where \(L\) is first order and \(k\) is algebraic and linear. All coefficients are smooth functions of the 2-jet of \(G\) on the permitted compact set.

Define \(S(h,v_0)\) by taking \(u=(\tau/i)k(h,v_0)\) and reconstructing \(v,Z\) from (13). Weighted differentiation gives \[\begin{align*} |S(h,v_0)|_{m,s} &\leq C_m\bigl(|h|_{m+1,s}+|v_0|_{m+1,s}\bigr), \tag{16}\\ |D_G^xS(h,v_0)-h|_{m,s} &\leq C_m\frac\tau s \bigl(|h|_{m+1,s}+|v_0|_{m+1,s}\bigr). \tag{17}\end{align*}\] Indeed the residual is \(Lu\), and the reconstructed normal component contains at most one derivative of \(u\). In the homogeneous case the leading term is exactly the chosen normal amplitude: \[ |S(0,v_0)-v_0|_{m,s} \leq C_m\frac\tau s|v_0|_{m+1,s}. \tag{18}\] These are differential formulas, so they preserve supports.

A fixed exponent for polynomial perturbations

We next quantify the effect of the operators \(P_i\). This is the point at which independence from the approximation order \(q\) is needed.

For a monomial (10), let \[E_\alpha=\sum_{\ell=1}^d(|\alpha_\ell|-2), \qquad E=\max E_\alpha,\] where the maximum runs over all monomials of all \(P_i\). For an empty collection take \(E=0\). A permissible choice will be \[ p=E+6. \tag{19}\]

To verify this, consider a variation of order \(k\leq3\). Suppose \(h\) arguments replace higher-jet factors of orders \(r_1,\ldots,r_h\) and the other \(k-h\) arguments differentiate the smooth low-jet coefficient. The unvaried factors have excess weight \(E_\alpha-\sum_{j=1}^h(r_j-2)\). The variation arguments carry at most \(\sum_jr_j+2(k-h)\) derivatives. The total short-scale power is therefore at most \[E_\alpha-\sum_{j=1}^h(r_j-2)+\sum_{j=1}^hr_j+2(k-h) =E_\alpha+2k\leq E+6.\] Weighted differentiation does not increase this exponent: (8) controls each differentiated higher-jet factor, and the chain rule controls the smooth low-jet coefficient. For conjugated amplitudes, each derivative of an exponential costs a power of \(\tau^{-1}\), while a slow derivative costs \(s^{-1}\leq\tau^{-1}\).

Put \(\mathcal R=\mathcal P-\gamma\). For \(1\leq k\leq3\), this proves, with a fixed finite derivative loss \(\ell\), the bounds \[\begin{align*} &\left|e^{-i(\phi_1+\cdots+\phi_k)/\tau} \mathcal R^{(k)}(G) \bigl(e^{i\phi_1/\tau}Z_1,\ldots,e^{i\phi_k/\tau}Z_k\bigr) \right|_{m,s} \\[-1mm] &\hspace{35mm}\leq C_m\epsilon\tau^{-p}\prod_{j=1}^k|Z_j|_{m+\ell,s}. \tag{20}\end{align*}\] The phases in this estimate belong to the fixed finite collection. At scale \(\tau\), the same estimate without exponentials holds for arbitrary variations at every map \(K\) with \(j^2K=O_\tau(1)\) in the permitted low-jet domain. The constants use finitely many corresponding weighted bounds. The inequality \(\epsilon^i\leq\epsilon\) handles all the perturbation coefficients.

Let \(\mathcal L=(\mathcal P'(G))^x\) and \(S_0(h)=S(h,0)\). Combining (16)–(17) with (20) gives \[|\mathcal L S(h,v_0)-h|_{m,s} \leq C_m\eta\bigl(|h|_{m+\ell,s}+|v_0|_{m+\ell,s}\bigr),\] after increasing \(\ell\). Starting with \(Z_0=S(h,v_0)\), define \[r_j=\mathcal LZ_j-h,\qquad Z_{j+1}=Z_j-S_0(r_j).\] The exact identity \[r_{j+1}=-(\mathcal L S_0-\operatorname{Id})r_j\] shows by finite induction that, for any integer \(q\), the resulting amplitude \(Z\) satisfies \[\begin{align*} |Z|_{m,s}&\leq C_{m,q} \bigl(|h|_{m+\ell_q,s}+|v_0|_{m+\ell_q,s}\bigr), \tag{21}\\ |\mathcal LZ-h|_{m,s}&\leq C_{m,q}\eta^q \bigl(|h|_{m+\ell_q,s}+|v_0|_{m+\ell_q,s}\bigr). \tag{22}\end{align*}\] For \(h=0\) it also satisfies \[ |Z-v_0|_{m,s}\leq C_{m,q}\eta|v_0|_{m+\ell_q,s}. \tag{23}\] The integer \(\ell_q\) and the constants may increase with \(q\), but the exponent \(p\) in (19) has already been fixed. Every step is differential and support preserving.

We have now solved every allowed nonzero mode to any fixed power of \(\eta\). It remains to select the free amplitudes so that their quadratic mean is the desired tensor and then remove the oscillatory quadratic error.

The quadratic mean and its finite adjustment

Finite inner corrections to oscillatory amplitudes also occur in Källén’s low-regularity embedding construction (Källén 1978, Lemma 3.3 and its proof). We derive the mean equation and its finite substitution estimates for the differential operators considered here.

In a phase chart choose a smooth unit normal \(n_\nu\) perpendicular to \((B_G)_{yy}\). One explicit choice is the normalized ambient perpendicular product of \(G_x,G_y,(B_G)_{yy}\). The vectors in this product are independent. The choice depends smoothly on the 2-jet and uses only the orientation of this chart and of \(\mathbb{R}^4\).

For a trial tensor \(A\in\mathcal U_1\), use \[v_{0,\nu}=\sqrt2\,\delta\tau b_\nu(A)n_\nu,\qquad h=0,\] in the finite mode construction, and denote its amplitude by \(Z_\nu(A)\). Define the real displacement \[U_1(A)=\sum_\nu\operatorname{Re} \bigl(e^{i\phi_\nu/\tau}Z_\nu(A)\bigr).\] Then \(U_1=O_\tau(\delta\tau)\). Before multiplication by the exponential, the leading complex amplitude of \(dU_1/\delta\) in the \(\nu\)th mode is \[i\sqrt2\,b_\nu(A)n_\nu\,d\phi_\nu,\] with error \(O_s(\eta)\). This follows from (23); a slow derivative of the leading amplitude contributes \(O_s(\tau/s)\). Furthermore, \[ \mathcal P'(G)U_1=O_\tau(\delta\tau\eta^q). \tag{24}\]

Expand the quadratic tensor \[Q(A)=\tfrac12\mathcal P''(G)(U_1(A),U_1(A))\] using the positive and negative mode labels. Its zero-phase part, denoted \(Q_0(A)\), consists of a mode paired with its own complex conjugate. All other terms have phases \(\pm2\phi_\nu\) or \(\pm\phi_\nu\pm\phi_\mu\) with \(\nu\ne\mu\). The latter phases have nonzero differentials on their supports, by hypothesis; in particular they do not contribute another zero-phase term. Repeated nonzero phases may simply be grouped.

For one leading real mode, the metric quadratic term is \[2\delta^2b_\nu(A)^2\sin^2(\phi_\nu/\tau)\,d\phi_\nu^2 =\delta^2b_\nu(A)^2 \bigl(1-\cos(2\phi_\nu/\tau)\bigr)\,d\phi_\nu^2.\] Its zero-phase term is \(\delta^2b_\nu(A)^2d\phi_\nu^2\). Thus (9) and the amplitude error give \[ \delta^{-2}Q_0(A)=A+T(A),\qquad T(A)=O_s(\eta). \tag{25}\] The contribution of \(\mathcal R''\) has the same bound, since (20) bounds it by \(C\epsilon\tau^{-p}(\delta\tau)^2\leq C\delta^2\eta\). Every nonzero-mode coefficient of \(Q\) is \(O_s(\delta^2)\).

The map \(T\) has a finite derivative-loss difference estimate: \[ |T(A)-T(\bar A)|_{m,s} \leq C_m\eta|A-\bar A|_{m+L_q,s} \tag{26}\] on bounded sets of weighted norms in \(\mathcal U_1\). To see this, the square roots in \(b_\nu\) have positive lower bounds, so their chain-rule difference estimates apply. The finite mode construction is linear in \(b_\nu(A)n_\nu\). In the difference of the two sides of (25), the leading contribution \(A-\bar A\) cancels exactly. Every remaining bilinear term contains either one \(O_s(\eta)\) mode error or the factor \(\epsilon\tau^{-p}\) from the perturbation. The product rule then gives (26) with a finite \(L_q\).

Set \(A_0=H\) and successively define \[A_{j+1}=H-T(A_j).\] For any fixed number of substitutions, all these tensors remain in \(\mathcal U_1\) and have bounded weighted norms when \(\eta\) is sufficiently small. The positive gap \(r_1-r_0\) and the estimate \(T(A_j)=O_s(\eta)\) give this domain condition using only finitely many input derivatives: for a fixed number of substitutions, work backward through their finite derivative losses to obtain the required \(C^0\) bounds. Once all iterates lie in \(\mathcal U_1\), the finite formulas give bounds in each higher weighted norm by the chain rule, without a further domain restriction. Their successive differences gain one power of \(\eta\) by (26). More explicitly, for \(j\geq1\), \[A_j+T(A_j)-H=T(A_j)-T(A_{j-1})=O_s(\eta^{j+1}).\] After finitely many substitutions, choose the resulting \(A\) so that \[ Q_0(A)=\delta^2H+O_s(\delta^2\eta^q). \tag{27}\] This is a finite algebraic procedure on smooth tensors, not a contraction argument in a function space with derivative loss.

Cancellation and the cubic remainder

Use the amplitude \(A\) just obtained. On each nonzero quadratic phase, apply (21)–(22) with target the negative corresponding coefficient of \(Q-Q_0\) and with \(v_0=0\). Taking real parts gives \(U_2\) such that \[ U_2=O_\tau(\delta^2),\qquad \mathcal P'(G)U_2+(Q-Q_0)=O_\tau(\delta^2\eta^q). \tag{28}\] The supports are intersections of the original supports, or original supports for doubled phases. Since \(\delta/\tau\) is small, \[U=U_1+U_2=O_\tau(\delta\tau),\qquad dU_1=O_\tau(\delta),\qquad dU_2=O_\tau(\delta^2/\tau).\] The metric quadratic terms involving \(U_2\) are bounded by \[C_m\left(\frac{\delta^3}{\tau}+\frac{\delta^4}{\tau^2}\right) \leq C_m\frac{\delta^3}{\tau}.\] The corresponding terms from \(\mathcal R''\) are, by (20) at scale \(\tau\), bounded by \[C_m\epsilon\tau^{-p}(\delta^3\tau+\delta^4) \leq C_m\frac{\delta^3}{\tau}.\] Here \(\epsilon\tau^{-p}\leq\eta\leq1\) and \(\delta/\tau\leq1\) suffice.

For completeness, the third-order Taylor estimate is uniform along the whole segment \(G+tU\), \(0\leq t\leq1\). Indeed \[|j^2U|_{m,\tau}\leq C_m\left(\frac \delta\tau+\frac{\delta^2}{\tau^2}\right),\] so smallness of \(\delta/\tau\) keeps the low jets inside the permitted domain. Also \(j^2(G+tU)=O_\tau(1)\), since \(\tau\leq s\). The arbitrary-variation version of (20) therefore bounds the integral Taylor remainder of \(\mathcal R\) by \[C_m\epsilon\tau^{-p}(\delta\tau)^3 \leq C_m\frac{\delta^3}{\tau}.\] There is no third-order remainder for \(\gamma\) itself.

Finally combine (24), (27), and (28). Their errors are bounded by \(C_m \delta\eta^q\), because \(\delta,\tau\leq1\). The remaining terms have just been bounded by \(C_m\delta^3/\tau\). This proves (11) and Proposition 6. Every local formula is smooth and differential in cutoff amplitudes, so extension by zero gives the asserted global smooth displacement and support properties. None of the local normal choices requires orientability of \(M\).

Smoothing with a finite input bound

We use the weighted norms introduced in the preceding section. The following elementary smoothing facts also apply to tensor fields.

We record how these facts will be used. Let \(G:M\to\mathbb{R}^4\) be smooth, let \(H\) be a smooth symmetric tensor, and define \[ \mathcal S_m(t;G,H)= \sum_{j=0}^m t^j\bigl(\|G\|_{C^{j+2}}+\|H\|_{C^j}\bigr). \tag{35}\] Suppose \(\mathcal S_r(t;G,H)\le B\), and let \(0<\tau\le s\le t\le1\). Set \(G_s=S_sG\) and \(H_s=S_sH\) using Lemma 9. Then, for every fixed \(m\), \[\begin{align*} |j^2G_s|_{m,s}+|H_s|_{m,s}&\le C_{m,r}B, \tag{36}\\ |j^2(G-G_s)|_{m,\tau}+|H-H_s|_{m,\tau} &\le C_{m,r}\left[ B(s/t)^r+\mathcal S_m(t;G,H)(\tau/t)^r\right]. \tag{37}\end{align*}\] For the first inequality use (32) with \(q=2\) and \(q=0\), respectively. To prove the second, split the derivatives at \(r\). A derivative of order \(j\le r\) of \(j^2G\) or \(H\) contributes at most \[C B t^{-r}s^{r-j}\tau^j =C B(s/t)^r(\tau/s)^j \le C B(s/t)^r.\] For \(j>r\), boundedness of smoothing in the original derivative norm gives a contribution at most \[C_m(\tau/t)^j\mathcal S_m(t;G,H) \le C_m(\tau/t)^r\mathcal S_m(t;G,H).\] In particular, the constants in (36) do not use any derivative of the true input above order \(r+2\); the possibly large higher derivatives occur only linearly in (37).

A recurrence at every derivative order

Fix a small \(z\) for the moment. Lemma 4 provides phase data for \(f_z\) with reference metric \(h_*\). They remain admissible in the \(C^2\) ball of radius \[\rho_z=z^4\] about \(f_z\). Their bounds in every fixed order are polynomial in \(z^{-1}\). Put \(B=z^{-4}\). The current integer \(k\ge10\) controls the amplitude and the number of derivatives to be retained. We use \[ \beta=\frac65,\qquad \lambda=\frac{11}{10},\qquad \tau=t^\beta,\qquad s=t^\lambda,\qquad \delta=t^k,\qquad r(k)=q(k)=40(k+1). \tag{38}\] Here \(q(k)\) is the accuracy order in Proposition 6. The integer \(r(k)\) is the order controlled at the current step; the estimates below will also hold at every fixed observation order \(m>r(k)\), without assuming a bound there.

For a current map \(G\), define the normalized defect \[ H=\delta^{-2}(h_*-\gamma(G)),\qquad \mathcal S_m=\mathcal S_m(t;G,H). \tag{39}\] The step will be applied under the finite-order hypotheses \[ \|G-f_z\|_{C^2}\le\rho_z/2,\qquad \|H-h_*\|_{C^0}<\varepsilon_*,\qquad \mathcal S_{r(k)}\le B, \tag{40}\] where \(\varepsilon_*>0\) is a fixed sufficiently small radius for the positive tensor decompositions.

The same construction therefore has two different uses. At the controlled order \(r(k)\), a small scale preserves its bound. At any higher fixed order, the current norm may be large, but it is multiplied by a coefficient that tends to zero. The latter fact will let us increase \(k\) after a finite number of steps.

It remains to initialize this step from the finite assumptions of Proposition 8, keep the maps in the allowed \(C^2\) ball, and schedule increases of \(k\).

Initialization and passage through the derivative orders

Because \(\|f_z\|_{C^2}=O(z^{-2})\), for small \(z\) the distance condition in (40) implies \[ \|G\|_{C^2}+\|h_*\|_{C^0}<B/4. \tag{47}\] For each fixed \(z,k\), choose a positive scale threshold that makes Lemma 10 applicable, preserves the defect neighborhood, and, by (42) with \(m=r(k)\), preserves \(\mathcal S_{r(k)}\le B\). Impose also \[ \|U\|_{C^{\lfloor k/2\rfloor}}\le\rho_z t. \tag{48}\] This last threshold is available because (41) gives the upper bound \[C(z,k)t^{k+(6/5)(1-\lfloor k/2\rfloor)},\] and its exponent is greater than one for every \(k\ge10\). The thresholds can cover both choices of \(\delta'\) at once.

For the initial block \(k=10\), only finitely many orders and inequalities have been used. Their constants are polynomial in \(z^{-1}\), and every smallness condition has a strict positive power of \(t\). Therefore there is a fixed \(\mu>1\) such that all these conditions hold for \(t\le z^\mu\) and sufficiently small \(z\). Enlarge \(\mu\) if necessary, and set \[R=r(10)=440,\qquad N>20\mu+1,\] with \(N\) an integer. These choices use polynomial exponents, not the implicit constants in the assumptions of the proposition.

Start at \(t=z^\mu\), \(k=10\), \(\delta=t^{10}\), and \[G=\sqrt{1-\delta^2}\,f_z.\] Its normalized defect has the exact form \[H=h_*+(1-\delta^{-2})(\gamma(f_z)-h_*).\] It therefore tends to \(h_*\) in \(C^R\). The weighted derivatives of \(G\) through the required order are bounded by \[C z^{-2}\sum_{j=0}^R z^{(\mu-1)j},\] so \(\mathcal S_R<B/2\) for small \(z\). The initial displacement is \(O(z^{20\mu-2})\), which is at most \(\rho_z/4\) for sufficiently small \(z\), since \(\mu>1\) and \(20\mu-2>4\). This verifies all the initial conditions.

At every subsequent step replace \(t\) by \(t^{6/5}\). Thus, if \(t_n\) is the scale at step \(n\), \[t_n=t_0^{(6/5)^n},\qquad \sum_{n=0}^\infty t_n<1/4\] for small enough \(t_0\). Equations (48) and the initial displacement keep all maps within \(\rho_z/2\) of \(f_z\) in \(C^2\). Hence the phase data and (47) remain valid.

We explain why no derivative order prevents the construction from advancing. Suppose that \(k\) is held fixed and let \(m\) be any fixed integer. Apply Lemma 10 repeatedly with \(\delta'=\tau^k\). Its all-order recurrence says that the numbers \(x_n=\mathcal S_m(t_n;G_n,H_n)\) satisfy \[x_{n+1}\le B/4+c_n(1+x_n),\qquad c_n=C_m(z,k)t_n^{1/5}\longrightarrow0.\] Every \(x_n\) is finite because all finite-stage maps are smooth. Once \(c_n\le1/4\), this recurrence first bounds the sequence; subsequently \(c_n\to0\) gives \(\limsup x_n\le B/4\). In particular, after finitely many further steps \(x_n\le B\).

Apply this observation with \(m=r(k+1)\). Wait in the block with integer \(k\) until that higher bound holds and the scale is below the threshold for the block with integer \(k+1\). Wait also until \[C_{r(k+1)}(z,k)t^{1/5}(1+B)\le B/2,\] where the constant is the one for the current block in (42). All three conditions hold after finitely many steps. Now choose \(\delta'=\tau^{k+1}\). That recurrence, already proved for this smaller target amplitude, bounds the new higher norm by \(B/4+B/2<B\). The new scale \(\tau<t\) also satisfies the next block’s threshold. Thus each block has finite length and \(k\) tends to infinity.

For every fixed \(m\), all sufficiently late blocks satisfy \(\lfloor k/2\rfloor\ge m\). Equation (48) then makes the tail of the increments absolutely summable in \(C^m\). The maps converge in every \(C^m\) to a smooth map \(\widetilde f_z\). Their defects tend to zero in \(C^0\), since \(\delta=t^k\to0\) and \(H\) stays in a fixed \(C^0\) neighborhood of \(h_*\). Continuity of the induced metric in \(C^1\) yields \(\gamma(\widetilde f_z)=h_*\). Finally, \[\|\widetilde f_z-f_z\|_{C^2} \le\rho_z/4+\rho_z\sum_{n=0}^\infty t_n \le\rho_z/2\longrightarrow0.\] This proves Proposition 8.

The plane and the leading metric

For a 2-jet near that of \(F\), define \[ Y=G_y,\qquad C=G_{yy},\qquad P=\operatorname{span}(Y,C)^\perp,\qquad v=PG_x,\qquad R_0=(|v|^2+a^2)^{1/2}. \tag{49}\] We use \(P\) also for orthogonal projection to this plane. The following facts justify these definitions and the frame we will use.

For an angular function \(\alpha\) to be chosen, set \[ p=\cos\alpha\,e_1+\sin\alpha\,e_2,\qquad V=R_0p,\qquad w=V-v,\qquad X=G_x+w. \tag{51}\] Since \(G_x-v\in\operatorname{span}(Y,C)\) and \(V,v\in P\), \[ |X|^2=|G_x|^2+a^2,\qquad X\cdot Y=G_x\cdot G_y. \tag{52}\] Thus the pair \((X,Y)\) has exactly the required leading metric. What remains is to make the periodic mean of \(w\) zero, while controlling the places where the velocity stops along its circle.

A prescribed mean and smooth vanishing

Initially use an angular parameter \(t\in\mathbb{R}/(2\pi\mathbb Z)\) and angles \[ \alpha=h_{\mathrm{turn}}+A\cos t. \tag{53}\] Here \(h_{\mathrm{turn}}\) will be a smooth function supported in a fixed compact subset of \(D\) and small in value. We first choose \(A\), independently of that function.

On a sufficiently small collar, \(e_1=\widehat v\) and \(a/|v|\) is small. There is a nonnegative smooth amplitude \(A_{\mathrm{sm}}\) satisfying \[ \frac1{2\pi}\int_0^{2\pi}\cos(A_{\mathrm{sm}}\cos t)\,dt =\frac{|v|}{R_0}. \tag{54}\] Indeed the left side is \(1-A_{\mathrm{sm}}^2/4+O(A_{\mathrm{sm}}^4)\). The implicit function theorem applied to the squared amplitude gives \[ A_{\mathrm{sm}}=\frac a{|v|}\, c\left(\frac{a^2}{|v|^2}\right),\qquad c(0)=\sqrt2, \tag{55}\] where \(c\) is positive and smooth. Thus \(A_{\mathrm{sm}}\) is smooth and flat on \(C_0\), and it is zero on the exterior. When \(h_{\mathrm{turn}}=0\), the sine average vanishes as well, so the unweighted mean of \(R_0p\) is exactly \(v\) on this collar.

Farther inward, while still in a region where \(e_1=\widehat v\) and the small formula is defined, interpolate \(A\) from \(A_{\mathrm{sm}}\) to a fixed number larger than \(\pi\), keeping \(A\ge A_{\mathrm{sm}}\). Use that constant on the remaining interior. For \(a>0\) the point \(v/R_0\) lies strictly inside the convex hull of the path \(p\) when \(h_{\mathrm{turn}}=0\). On the small arc this follows because its unweighted mean lies strictly between the endpoint chord and the point \((1,0)\) in the \((e_1,e_2)\) coordinates. Increasing \(A\) retains this arc; when \(A>\pi\) the path covers the circle and \(|v|/R_0<1\). On any fixed compact subset of the interior, this strict membership persists for sufficiently small \(\|h_{\mathrm{turn}}\|_{C^0}\), uniformly on a compact neighborhood of the permitted jet data.

We use the following elementary fact to adjust the mean.

Apply the lemma in the frame \((e_1,e_2)\), with \(c=v/R_0\), and use \(\rho=1\) on a smaller collar where (54) applies and \(h_{\mathrm{turn}}=0\). Here the parameters are position and the \(2\)-jet of \(G\). The frame identifies all the velocity planes with \(\mathbb{R}^2\), and the remaining parameter set is compact and lies in \(a>0\). The interior hypothesis therefore has the required uniform margin, and the lemma patches a smooth density with the prescribed collar solution. Put \[\theta=\int_0^t\rho(s)\,ds\] and replace \(t\) by the smooth inverse of this orientation-preserving circle diffeomorphism. We henceforth use \(\theta\) as periodic parameter. Its uniform mean satisfies \[ \operatorname{avg}_\theta V=v,\qquad \operatorname{avg}_\theta w=0. \tag{56}\] All data are smooth functions of position and the 2-jet of \(G\). On the exterior \(A=0\), \(h_{\mathrm{turn}}=0\), and \(\rho=1\), so \(V=v\). Equations (55) and (54) show that \(w\) is smooth across \(C_0\) and vanishes there with every spatial derivative.

In the interior \(A>0\). There are exactly two zeros of \(\alpha_\theta\), corresponding to \(t=0,\pi\). At these turns, \[ \alpha=h_{\mathrm{turn}}\pm A, \qquad \alpha_y=(h_{\mathrm{turn}})_y\pm A_y. \tag{57}\] In this formula spatial differentiation holds the new parameter \(\theta\) fixed. The additional chain-rule term from the inverse reparametrization is zero because it is multiplied by \(\sin t\). Figure 1 separates the velocity circle from its angular parametrization. The two stationary angles persist under the positive reparametrization; their transverse derivatives are the quantities we will prescribe along the later curves.

A small function with a prescribed transverse derivative

We will need a large value of \((h_{\mathrm{turn}})_y\) along the later curves away from the boundary, while keeping \(h_{\mathrm{turn}}\) small in value. The finite-tangency hypothesis of Proposition 5 gives exactly this freedom.

Uniform normal bounds and the choice at turns

For the rest of this section evaluate the slow data at \(G=F\). Let \(S_0\) be the length of the projection of \(C\) to \(\operatorname{span}(X,Y)^\perp\), and let \(n_0^*\) be its unit direction. This projection never vanishes: if \(C\) belonged to \(\operatorname{span}(X,Y)\), applying \(P\) would force its \(X\) coefficient to be zero since \(PX=V\ne0\), contradicting independence of \(C,Y\). Compactness gives positive lower and upper bounds for \(S_0\), uniformly over all angular values. A complementary unit normal is \[m_0^*=-\sin\alpha\,e_1+\cos\alpha\,e_2.\] It is perpendicular to \(X,Y,C\), and hence also to \(n_0^*\). Set \[ D_0^*=\langle X_y,n_0^*\rangle, \qquad N_0^*=\langle X_y,m_0^*\rangle. \tag{58}\] The derivative here holds the periodic parameter fixed. Directly, \[X_y=(F_x-v)_y+(R_0)_yp+ R_0\bigl(\cos\alpha(e_1)_y+\sin\alpha(e_2)_y +\alpha_y m_0^*\bigr).\] The term containing \(\alpha_y\) drops out of \(D_0^*\), so \[ |D_0^*|\le C_0, \qquad N_0^*=R_0\alpha_y+E_0,\quad |E_0|\le C_0, \tag{59}\] where \(C_0\) is independent of \(h_{\mathrm{turn}}\), its derivatives, and the reparametrization. These estimates hold over all angular values. The same independence holds for the bounds on \(S_0\) and \(1/S_0\).

Write \[k_+=K_{\gamma_+}\det(\gamma_+)\] with determinant in \((x,y)\) coordinates, and write \(Y_i=b_i\partial_x+c_i\partial_y\) on the disk. At a crossing \(C_i\cap C_0\), the loop is constant and \(X=F_x\). Since \(a\) is flat, the prescribed metric \(\gamma_+\) has the old curvature there. Consequently the strict condition \(Q_{0i}(F)>0\) is exactly \[ (D_0^*b_i+S_0c_i)^2+k_+b_i^2>0. \tag{60}\] It remains valid on the later curves in a sufficiently thin collar for all path angles. Indeed \(A_{\mathrm{sm}}\) tends to zero at the boundary, and \(D_0^*\) contains no angular derivative. Compactness confines the later curve portions of a thin collar to neighborhoods of their finitely many crossings with \(C_0\). The collar can also avoid every later mutual crossing, because there are no triple intersections. The same inequality holds if the path angles are translated by any sufficiently small value of \(h_{\mathrm{turn}}\). Thus this collar estimate remains available even if the eventual interior support neighborhood of \(h_{\mathrm{turn}}\) overlaps its inner part.

Let \(K\) be the compact union of the later curve portions in the disk outside this collar. By Lemma 14, we may prescribe \((h_{\mathrm{turn}})_y=B_*\) on \(K\) with arbitrarily small values of \(h_{\mathrm{turn}}\). The support can lie in a fixed compact neighborhood of \(K\) inside \(D\), while a smaller boundary collar remains unchanged. On any overlap with the collar estimate, smallness in value preserves that estimate. For any fixed threshold \(T\), choose \[ B_* > \|A_y\|_{C^0} +\frac{T+C_0+1}{\min R_0}. \tag{61}\] Equations (57) and (59) then give \(N_0^*>T+1\) at both turns on \(K\). After making the values of \(h_{\mathrm{turn}}\) small enough to retain convex-hull membership, choose the positive density as above. That reparametrization does not change the turn derivative identity.

We summarize the construction in the form needed by the next section.

All choices in this section are now fixed before the fast oscillation scale is sent to zero. In particular their higher derivatives may be large but are finite. The next section will use this fixed smooth loop to produce exact immersions. We will choose \(T\) from explicit crossing bounds in Section 8; those bounds use only the quantities just shown to be independent of \(T\).

Removing the oscillating coefficients

Finite-order asymptotic isometric constructions also occur in Poznyak’s planar immersion problem in \(\mathbb{R}^3\) (Poznyak 1973, sec. 2.3, Theorem 5). For the present velocity loop, we give the periodic correctors and their support properties by an explicit recursion.

For a periodic scalar or vector \(b\), define \[\operatorname{avg}b=\frac1{2\pi}\int_0^{2\pi}b(\theta)\,d\theta, \qquad \Pi b=b-\operatorname{avg}b.\] For \(\operatorname{avg}b=0\), let \(\mathcal I b\) be its unique periodic primitive with mean zero. These operations commute with differentiation in the slow variables \(x,y\).

Correcting the mean and selecting the scales

The preceding lemma leaves two errors: a finite mean operator \(\mathcal P_z-\gamma\), and a remainder beginning at \(z^L\). We first record how the latter behaves when the slow map depends on \(z\). Suppose \[ G\longrightarrow F\text{ in }C^3, \qquad j^2G=O_s(1),\qquad s=z^\sigma, \qquad 0<\sigma<1. \tag{74}\] A jet of order \(k>2\) is \(O_s(s^{2-k})\). Since the expansion has finitely many coefficients, Lemma 17 supplies an integer \(p_1\ge0\) such that they and their required periodic derivatives are all \(O_s(s^{-p_1})\). This exponent is independent of the weighted differentiation order; the constants may depend on that order. Indeed each higher-jet monomial loses the sum of the quantities \(k-2\) for its factors, while weighted differentiation of its coefficients loses no additional power of \(s\).

Substitution of \(\theta=x/z\) changes these estimates to \(O_z(s^{-p_1})\): a derivative in \(\theta\) costs \(z^{-1}\), and a slow derivative costs \(s^{-1}\le z^{-1}\). In particular, \[ \gamma(f_z[G])-a^2dx^2-\mathcal P_z(G) =O_z(z^Ls^{-p_1}). \tag{75}\]

For later use, differentiating the finite ansatz gives \[\begin{align*} f_x&=X+\sum_{j=1}^L z^jU_{j,x} +\sum_{j=2}^L z^{j-1}U_{j,\theta},\\ f_y&=Y+\sum_{j=1}^L z^jU_{j,y}, &f_{yy}&=C+\sum_{j=1}^L z^jU_{j,yy},\\ f_{xy}&=X_y+\sum_{j=1}^L z^jU_{j,xy} +\sum_{j=2}^L z^{j-1}U_{j,\theta y},\\ zf_{xx}&=V_\theta+zG_{xx} +\sum_{j=1}^L z^{j+1}U_{j,xx} +2\sum_{j=1}^L z^jU_{j,x\theta} +\sum_{j=2}^L z^{j-1}U_{j,\theta\theta}. \end{align*}\] All coefficients on these right sides are evaluated at \(G\) and \(\theta=x/z\). Each error contains a positive power of \(z\) and at most two further slow derivatives. Thus \[ \sigma(p_1+2)<1 \tag{76}\] makes these errors uniformly small. The leading terms converge uniformly to their values at \(F\), including \(X_y\), because of the \(C^3\) convergence in (74). The same estimates show \(\|f_z[G]\|_{C^k}=O(z^{-k})\) for each fixed \(k\).

We now construct a slow map satisfying (74) for which the mean error is small enough for exact correction.

The primitive metric is now realized exactly. The next section uses the unscaled \(yy\) and \(xy\) limits together with the exact Gauss equation to determine the component of \(B_{xx}\) along \(B_{yy}\). The scaled \(xx\) limit controls the complementary normal component. These are precisely the second-form data needed to preserve the normal and crossing conditions for the remaining disks.

The exact second form

Choose the loop of Lemma 15, with the turn threshold to be specified below, and let \(\widetilde f_z\) be the resulting exact immersions from Proposition 16. On the disk their \(yy\) second form has the representation \[B_{\widetilde f_z}(\partial_y,\partial_y)=S n^*,\qquad |n^*|=1,\qquad S>0,\] for all sufficiently small \(z\). Indeed its limiting value is \(S_0n_0^*\) and \(S_0\) has a positive minimum. Project \(m_0^*\) to the line in the new normal plane perpendicular to \(n^*\) and normalize; the resulting \(m^*\) is well defined and close to \(m_0^*\). Write \[B_{\widetilde f_z}(\partial_x,\partial_y)=D^* n^*+N^* m^*,\qquad k=K_{\gamma_+}\det(\gamma_+).\] The determinant is in the \((x,y)\) coordinates. The Gauss equation gives \[ B_{yy}=(S,0),\qquad B_{xy}=(D^*,N^*),\qquad B_{xx}=\left(\frac{k+(D^*)^2+(N^*)^2}{S},\,L^*\right) \tag{80}\] in the orthonormal frame \((n^*,m^*)\). We have used \(\langle B_{xx},B_{yy}\rangle-|B_{xy}|^2=k\). Thus the exact metric determines the first component of \(B_{xx}\) from the unscaled \(yy\) and \(xy\) data. Only its second component, \(L^*\), requires the scaled \(xx\) asymptotic from the realization theorem. The derivative asymptotics of the exact realization imply, uniformly on \(\overline D\), \[ \begin{gathered} S=S_0+o(1),\qquad n^*=n_0^*+o(1),\qquad D^*=D_0^*+o(1),\qquad N^*=N_0^*+o(1),\\ zL^*=R_0\alpha_\theta+o(1). \end{gathered} \tag{81}\] The final identity follows because \(V_\theta=R_0\alpha_\theta m_0^*\) is already normal to the limiting pair \((X,Y)\).

For later use, write \(Y_i=b_i\partial_x+c_i\partial_y\). Formula (80) gives the exact identities \[ \begin{aligned} S\langle B(Y_i,Y_i),n^*\rangle &=(D^*b_i+Sc_i)^2+(k+(N^*)^2)b_i^2,\\ \langle B(Y_i,Y_i),m^*\rangle &=L^*b_i^2+2N^*b_ic_i. \end{aligned} \tag{82}\] These two expressions separate the two mechanisms of the construction: a large angular velocity gives a large \(L^*\), whereas at a turn a large transverse derivative gives a large \(N^*\).

Crossing thresholds without circular choices

At a mutual later crossing inside \(D\), both \(b_i,b_l\) are nonzero. Put \[u_i=\partial_x+t_i\partial_y,\qquad u_l=\partial_x+t_l\partial_y, \qquad t_i=c_i/b_i,\quad t_l=c_l/b_l.\] The finitely many slopes are bounded. From (80), the pure and mixed vectors are \[ \begin{aligned} P_i:=B(u_i,u_i) &=\left(\frac{(N^*)^2+k+(D^*+St_i)^2}{S},\ L^*+2N^*t_i\right),\\ P_l:=B(u_l,u_l) &=\left(\frac{(N^*)^2+k+(D^*+St_l)^2}{S},\ L^*+2N^*t_l\right),\\ Z_{il}:=B(u_i,u_l) &=\left(\frac{(N^*)^2+k+(D^*+St_i)(D^*+St_l)}{S},\ L^*+N^*(t_i+t_l)\right). \end{aligned} \tag{83}\] In particular \[ Z_{il}-P_i=(t_l-t_i)(D^*+St_i,N^*). \tag{84}\]

Choose bounds \(0<s_-\le S\le s_+\), \(|D^*|\le d_+\), and \(|k|\le K_0\) on the entire closed disk and over all angular values, together with a fixed bound on the finitely many crossing slopes. They can be chosen before the turn derivative: the limiting bounds for \(S_0,1/S_0,D_0^*\) are independent of that derivative, and after the loop is fixed, (81) transfers slightly enlarged bounds to the exact immersions for sufficiently small \(z\). There are constants \(C_1,C_2\) depending only on these bounds such that \[ \begin{aligned} \langle P_j,n^*\rangle&\ge (N^*)^2/s_+-C_1,\\ \left\langle Z_{il},\frac{P_j}{|P_j|}\right\rangle &\ge (N^*)^2/s_+-C_2(1+|N^*|),\qquad j=i,l, \end{aligned} \tag{85}\] whenever the right side of the first line is positive. To see the second line, use \(\langle Z_{il},\widehat P_j\rangle \ge |P_j|-|Z_{il}-P_j|\) and (84). In particular these estimates do not depend on \(L^*\).

Also \[K_{\gamma_+}|u_i\wedge u_l|_{\gamma_+}^2 =k(t_i-t_l)^2.\] We can therefore fix a finite threshold \(T\) so large that \(T^2>K_0+1\) and, when \(|N^*|>T\), both pure vectors have positive dot with \(n^*\) and both mixed projections in (85) exceed \[\sqrt{\max\{0,-k(t_i-t_l)^2\}}+1\] at every crossing. Use a margin, for example \(T+2\), as the turn threshold in Lemma 15, and then fix the loop. No constant used to select this threshold depends on the resulting large value of \((h_{\mathrm{turn}})_y\). The sufficiently small choice of \(z\) may depend on this now fixed loop.

We next show positivity of both ordered crossing invariants for all small \(z\). If a sequence of crossing phases tends to a turn, (81) gives \(|N^*|>T\), so the preceding estimate applies. If its limiting phase is not a turn, then \(|L^*|\to\infty\), while \(N^*\) stays bounded for the fixed loop. All three vectors in (83) have the same leading second component \(L^*\); hence \[\left\langle Z_{il},\widehat P_i\right\rangle\to+\infty, \qquad \left\langle Z_{il},\widehat P_l\right\rangle\to+\infty.\] These two cases and compactness of the phase circle give positivity at the finitely many crossings uniformly for small \(z\). Returning to the original directions changes each ordered invariant by the positive factor \(b_i^2b_l^2\). Indeed pure second forms scale by squares, so their unit directions do not change even if either \(b\) is negative. We have proved \[ Q_{il}(\widetilde f_z)>0,\qquad Q_{li}(\widetilde f_z)>0 \quad\text{at all later mutual crossings in }D. \tag{86}\]

Avoidance along the full curves

We now prove that on every later curve inside the disk, \(B(Y_i,Y_i)\) is nonzero and its unit direction is not \(-n^*\). Suppose otherwise along a sequence \(z\downarrow0\). At a point of failure the second component in (82) is zero and the first is nonpositive, so \[ (D^*b_i+Sc_i)^2+(k+(N^*)^2)b_i^2\le0, \qquad L^*b_i^2+2N^*b_ic_i=0. \tag{87}\] Take a subsequence with limiting position and periodic phase. The limiting \(b_i\) cannot be zero: the vector field \(Y_i\) is nonzero, so then \(c_i\ne0\) and the first inequality would give \(S_0^2c_i^2\le0\). Multiplying the second equality by \(z\) and using (81) now forces \(\alpha_\theta=0\) at the limiting phase.

In the boundary collar, (60) rules out the first inequality of (87), even without its nonnegative \((N^*)^2b_i^2\) term. On the remaining compact curve portions, the limiting phase must be one of the two turns and \(k+(N_0^*)^2>0\) by the threshold just chosen. This again contradicts the first inequality. Therefore, for all sufficiently small \(z\), \[ B(Y_i,Y_i)\ne0, \qquad \frac{B(Y_i,Y_i)}{|B(Y_i,Y_i)|}\ne-n^* \quad\text{on }C_i\cap\overline D. \tag{88}\] This argument needs boundedness of \(N^*\) only for the fixed loop while \(z\) tends to zero. Such a bound may depend on the chosen turn size; it is distinct from the derivative-independent bounds used to choose that size.

Adjusting the normal at crossings

It remains to give both pure vectors positive dot product with a preferred normal at each crossing. If their dots with \(n^*\) are already positive, no change is needed. At each crossing, choose small neighborhoods of the two turning phases on which \(|N_0^*|>T+1\). The uniform limits and (85) make both dot products positive on these neighborhoods for small \(z\). Consequently any crossing requiring adjustment has phase in their compact complement, where \(|\alpha_\theta|\) has a positive lower bound. There \(|L^*|\to\infty\) uniformly, and the two pure unit vectors are arbitrarily close to the same sign of \(m^*\). For small \(z\), denote this sign by \(m\) and use the path \[ n_t=\frac{(1-t)n^*+t\widehat P_i} {|(1-t)n^*+t\widehat P_i|},\qquad 0\le t\le1. \tag{89}\] This path is defined and avoids the opposite of either pure unit vector. Its numerator has nonnegative \(m\) component, strictly positive for \(t>0\), whereas the opposite pure vectors have negative \(m\) component. At \(t=0\), use \(n^*\perp m\). At the endpoint both required dot products are positive because \(\widehat P_i\) and \(\widehat P_l\) are close.

The vectors in this construction extend smoothly to a neighborhood of the crossing. Strictness, together with compactness of \([0,1]\), gives a neighborhood on which the whole path remains defined and avoids the opposite pure directions. Replacing \(t\) by a smooth spatial cutoff, equal to one at the crossing, makes a smooth local normal adjustment. Since there are no triple intersections, its support can avoid every other curve. The finitely many crossing neighborhoods can be chosen pairwise disjoint and disjoint from the boundary collar. These changes preserve (88); they do not affect (86), whose quantities are independent of the preferred normal.

Extension through the exterior collar

We finish with a global preferred normal. The exact realization converges to \(F\) in \(C^2\) on the exterior of \(D\). Thus the normalized projection of the old normal \(n\) to the new normal bundle defines a smooth unit normal \(n_{\mathrm{out}}\) there for small \(z\). On a thin exterior collar, the new \(yy\) second form is nonzero and its unit direction \(n^*\) is close to the old direction \(\widehat A_0\). By (6), \(\widehat A_0\) and \(n\) are not antipodal on \(C_0\). After narrowing the collar, the normalized convex interpolation between \(n^*\) and \(n_{\mathrm{out}}\) is consequently defined. Use it to match the unchanged \(n^*\) near the disk boundary to \(n_{\mathrm{out}}\) farther outside.

We check the later curves during this interpolation. At an old crossing \(C_i\cap C_0\), the Gauss interpretation (5) gives \[ \langle A_0(F),A_i(F)\rangle =|B_F(Y_0,Y_i)|^2+ K_\gamma|Y_0\wedge Y_i|_\gamma^2 \ge Q_{0i}(F)>0. \tag{90}\] Thus both old endpoint normals have positive dot product with the later pure vector: one by this inequality, and the other by (6). Continuity and exterior \(C^2\) convergence retain positivity near these crossings for the new immersion, and convex interpolation preserves it. A sufficiently thin collar meets later curves only in these crossing neighborhoods. It can avoid all later mutual crossings. Away from the disk and collar, every old strict condition persists by exterior \(C^2\) convergence and compactness. The resulting normal \(n_+\) is global and satisfies all of (6) for the later indices.

The second fundamental form is nonzero on the disk because \(S>0\), and it is nonzero on the exterior by \(C^2\) convergence to \(F\). Choose \(z\) small enough for these properties, all the finite crossing conditions, and the prescribed exterior error \(\varepsilon\). The exact metric is \(\gamma_+\) by Proposition 16. This completes the proof of Proposition 5.

All frame orientations in this argument were chosen inside a coordinate disk. The global object carried from one step to the next is only the unit section \(n_+\) of the normal bundle. The construction therefore applies equally to orientable and nonorientable surfaces.

A spherical starting immersion

Every closed smooth surface admits an ordinary smooth immersion into \(S^3\). Indeed, the two-dimensional case of Whitney’s immersion theorem gives a \(C^1\) immersion into \(\mathbb{R}^3\) (Whitney 1944, Theorem 8, p. 274). Approximation in \(C^1\) by a smooth map preserves injectivity of the differential on a compact domain. Such an approximation can be obtained by mollifying in finitely many charts and combining the approximations with a smooth partition of unity. Finally, inverse stereographic projection is a diffeomorphism from \(\mathbb{R}^3\) onto a punctured \(S^3\). Neither step requires an orientation of the surface.

For an immersion \(I:M\to S^3\), denote its second fundamental form in \(S^3\) by \(B_I^{S^3}\), regarded as an \(\mathbb{R}^4\)-valued tensor. This notation avoids any choice of unit normal to the surface in \(S^3\).

Only the first two derivatives are controlled uniformly in this lemma. Higher derivatives may grow when the points are close together. The preparation below uses only the stated bounds; constants requiring higher derivatives will be chosen after the finite set has been fixed.

For later use, consider \(F^0=\lambda I_P\), where \(\lambda>0\), and write \(\gamma^0=\gamma(F^0)\). The vector field \(n^0=-I_P\) is a global unit normal in \(\mathbb{R}^4\). The second fundamental forms satisfy \[ B_{F^0}(v,w) =\lambda B_{I_P}^{S^3}(v,w) +\lambda^{-1}\gamma^0(v,w)n^0. \tag{91}\] In particular, \(\langle B_{F^0}(v,v),n^0\rangle>0\) for \(v\ne0\). At each point of \(P\), the first term vanishes and the Gauss equation gives \(K_{\gamma^0}=\lambda^{-2}\). For any nonzero vectors \(Y_i,Y_l\) at such a point, the quantity in (6) is consequently \[ Q_{il} =\lambda^{-2}\bigl( \gamma^0(Y_i,Y_l)^2+|Y_i\wedge Y_l|_{\gamma^0}^2\bigr) =\lambda^{-2}\gamma^0(Y_i,Y_i)\gamma^0(Y_l,Y_l)>0. \tag{92}\] This identity also holds when the two vectors are proportional.

A finite list with prescribed intermediate metrics

Completion of the global construction