Smooth isometric immersions of closed surfaces into Euclidean four-space

1 Introduction

Let M be a closed smooth surface: a compact smooth two-dimensional manifold without boundary. We do not assume that M is orientable or connected. For a smooth positive definite metric g on M, an isometric immersion into Euclidean four-space is a smooth map F:M\to\R^4 satisfying

Positive definiteness then makes dF_p injective at every point. The question is whether every such metric can be realized in this dimension.

The result concerns immersion; self-intersections are allowed. There is no restriction on Gaussian curvature or on the topology of the closed surface. In particular, the construction applies directly to nonorientable surfaces.

1.1 Context

Gromov explicitly recorded the question of smooth isometric immersion of arbitrary smooth Riemannian surfaces into \R^4 [4, §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 [5, p. 4, footnote 6]. Theorem 1.1 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 [13, Theorem 2, p. 59]. A closer surface comparison is Gromov’s construction of smooth isometric immersions of closed surfaces into \R^5: his retrospective explicitly recalls this result and refers to §3.2.4 of Partial Differential Relations [5, p. 59][3]. Theorem 1.1 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 \R^3 [7, 1]; see also the formulation in [5, §1.2]. Both the analytic hypothesis and the local conclusion matter here. For comparison, the companion [14, 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 \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 \R^4 on compact portions of a simply connected planar domain with regular boundary [15, Theorem 4, p. 67]. The smooth version of this construction is stated and explained in Lewicka’s modern account [10, 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 [12, 9]; the contemporaneous account [2, Theorems 1–2] states the immersion form explicitly. Lewicka proves a finer flexibility theorem in \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\} [10, Theorem 1.1, p. 2]. These domain and regularity hypotheses differ from the smooth closed-surface statement of Theorem 1.1.

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 [17, Theorem 8, p. 274]; Sard’s equal-dimensional critical-value theorem is used to choose the support boundaries and phases [16, Theorem 4.1, p. 885].

1.2 The construction

The proof has four stages. First prepare a smooth immersion into a small round three-sphere in \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

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 (1.1).

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.

1.3 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 1.1.

2 Normal geometry and admissible phases

We begin with the geometric information used by both corrections. A nonzero second fundamental form supplies enough directions for the small oscillations; it need not satisfy any rank condition beyond being nonzero as a tensor.

Fix a smooth background metric and a finite collection of precompact coordinate patches on M. All C^m norms and unmarked tangent norms use these fixed data. For a smooth immersion F:M\to\R^4, write

The last expression is tensorial; in coordinates B_{F,ij} is the normal projection of F_{ij}. Let K_\gamma denote the Gaussian curvature of \gamma=\gamma(F). Our sign convention for the Gauss equation is

In coordinates x,y, the right side is K_\gamma\det\gamma for V=\partial_x,W=\partial_y. The Hessian convention gives \Hess_\gamma x(\partial_y,\partial_y)=-\Gamma^x_{yy}. To check (2.1), choose Riemannian normal coordinates for \gamma, orthonormal at the point under consideration. There \Gamma=0, so F_{xx},F_{xy},F_{yy} are normal vectors, and the coordinate curvature formula is

Substituting \gamma_{ij}=\langle F_i,F_j\rangle cancels the third derivatives and gives K_\gamma=\langle F_{xx},F_{yy}\rangle-|F_{xy}|^2 at that point. Tensoriality gives (2.1) for arbitrary V,W.

A preferred normal is a smooth unit section of N_F. This is less than a normal frame: we will use complementary normal vectors only within coordinate disks. None of the following local constructions requires an orientation of M.

If B_F(p)\ne0, there are only finitely many bad projective covectors at p. Indeed, take a normal component of B_F(p) that is a nonzero scalar quadratic form. Its zeros in the projective tangent line are finite, and a zero of the vector-valued quadratic form must be one of these zeros.

We choose these good directions for the small oscillatory corrections. They need not be the phases x_j of the prescribed primitive additions in (1.1): those phases will satisfy a transverse Hessian condition, and their transverse second fundamental form may vanish in the interior of the support disk.

Writing a positive metric as a sum of positive rank-one forms is a basic part of Nash’s construction [12]; see [5, §3.5.4, p. 32 and footnote 30] and [8, Lemmas 2.2–2.3] for related decompositions. Here the individual phases and their relevant pairwise combinations must also be good for the current second fundamental form. The following construction supplies those additional conditions.

The small-oscillation argument is local on these coframes. If a combination is not globally injective as a coordinate function, the dual coordinate vector fields still make sense where its differential and the chosen complement are independent. Subdivision gives actual coordinate neighborhoods when necessary. Amplitudes are compactly supported there, so extension by zero is smooth.

For the exact correction, the phase data must also be controlled when the input immersion varies with a small parameter. We record this quantitative version separately. In the statement, a quantity is polynomially bounded if its absolute value is at most Cz^{-A} for some finite A\ge0; the exponent may depend on a specified finite derivative order, but not on z.

3 The primitive metric addition

The global construction will add finitely many positive tensors of rank one. We first state the local step, including the boundary conditions that allow one step to be followed by the next. Only a preferred unit normal is propagated; a full normal frame is chosen locally when needed.

Let F:M\to\R^4 be a smooth immersion, and write \gamma=\gamma(F). Suppose C_0,\ldots,C_m are smooth embedded closed curves with nonzero smooth vector fields Y_i defined on neighborhoods of C_i. The fields Y_i are not required to be tangent to the curves; in the application they are the transverse coordinate directions of later phases. Set

For i\ne l, wherever the curves intersect and A_i(F)\ne0, define

Here K_\gamma is Gauss curvature, and |u\wedge v|_\gamma^2=\gamma(u,u)\gamma(v,v)-\gamma(u,v)^2. The Gauss equation gives a useful interpretation. With \widehat A_i=A_i(F)/|A_i(F)| and Z_{il}=B_F(Y_i,Y_l),

Thus Q_{il}>0 forces the two pure second-form vectors to have positive inner product. It also shows that Q_{il} depends continuously on the 2-jet of F wherever A_i\ne0, despite its curvature expression in (3.1). Both observations will be used when the preferred normal is joined across the support boundary. For a preferred normal n, the conditions to be preserved are

The crossing conditions are imposed in both orders. In particular they require positivity of the dot product with n for each of the two pure second-form vectors. The numbers Q_{il} do not depend on the chosen preferred normal.

Because a is smooth and vanishes outside D, it and all its derivatives vanish on C_0. This makes the extended tensor smooth. The proposition allows the map itself to change on the exterior; the stated C^2 control is what will preserve the boundary data of the later steps.

The hypotheses have different roles. Table 1 records how they are established initially and passed to the next primitive step. In particular, the phase-Hessian condition depends on the current metric, whereas the boundary conditions also depend on the immersion and normal.

We prove the proposition in three parts. After the analytic correction results, Section 6 constructs a periodic velocity whose averaged displacement is zero and whose leading metric adds a^2dx^2. Section 7 realizes this velocity by exact smooth immersions with controlled derivatives. Section 8 proves the boundary conditions and constructs the new global preferred normal.

4 Small oscillatory metric increments

We first construct a correction whose amplitude and wavelength are both small. A good phase, in the sense of Section 2, permits an oscillation in the normal direction perpendicular to its transverse second form. Finite differential corrections make this oscillation solve the homogeneous linearized equation to high order. Its leading quadratic mean is a positive rank-one tensor. Combining finitely many such oscillations will add a prescribed positive tensor to finite accuracy while changing the map by a quantity small in C^2.

We carry out the construction for the metric map and for a finite polynomial differential perturbation of it. In Section 7, removing the large velocity loop’s oscillating errors leaves mean terms that are differential operators of the slow map, multiplied by powers of the fast scale. Here we allow those terms as part of the operator being corrected. The perturbation changes the required relation between scales. We will bound it using a power of the short scale that is independent of the requested finite accuracy.

4.1 Weighted bounds and the increment statement

For a smooth function, vector field, or tensor on the fixed coordinate atlas, set

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,

Fix positive phase data from Lemma 2.2: 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,

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

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.

4.2 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

Since G_{ij}=\Gamma^k_{ij}G_k+B_{ij}, direct differentiation gives

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

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

The yy equation in D_G^xZ=h is then exact. The xx equation, divided by 2, and the xy equation become

Thus the remaining system is

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 (4.6). Weighted differentiation gives

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:

These are differential formulas, so they preserve supports.

4.3 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 (4.3), let

where the maximum runs over all monomials of all P_i. For an empty collection take E=0. A permissible choice will be

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

Weighted differentiation does not increase this exponent: (4.1) 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

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 (4.9)–(4.10) with (4.13) gives

after increasing \ell. Starting with Z_0=S(h,v_0), define

The exact identity

shows by finite induction that, for any integer q, the resulting amplitude Z satisfies

For h=0 it also satisfies

The integer \ell_q and the constants may increase with q, but the exponent p in (4.12) 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.

4.4 The quadratic mean and its finite adjustment

Finite inner corrections to oscillatory amplitudes also occur in Källén’s low-regularity embedding construction [8, 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 \R^4.

For a trial tensor A\in\mathcal U_1, use

in the finite mode construction, and denote its amplitude by Z_\nu(A). Define the real displacement

Then U_1=O_\tau(\delta\tau). Before multiplication by the exponential, the leading complex amplitude of dU_1/\delta in the \nuth mode is

with error O_s(\eta). This follows from (4.16); a slow derivative of the leading amplitude contributes O_s(\tau/s). Furthermore,

Expand the quadratic tensor

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

Its zero-phase term is \delta^2b_\nu(A)^2d\phi_\nu^2. Thus (4.2) and the amplitude error give

The contribution of \mathcal R'' has the same bound, since (4.13) 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:

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 (4.18), 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 (4.19) with a finite L_q.

Set A_0=H and successively define

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 (4.19). More explicitly, for j\geq1,

After finitely many substitutions, choose the resulting A so that

This is a finite algebraic procedure on smooth tensors, not a contraction argument in a function space with derivative loss.

4.5 Cancellation and the cubic remainder

Use the amplitude A just obtained. On each nonzero quadratic phase, apply (4.14)–(4.15) with target the negative corresponding coefficient of Q-Q_0 and with v_0=0. Taking real parts gives U_2 such that

The supports are intersections of the original supports, or original supports for doubled phases. Since \delta/\tau is small,

The metric quadratic terms involving U_2 are bounded by

The corresponding terms from \mathcal R'' are, by (4.13) at scale \tau, bounded by

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

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 (4.13) therefore bounds the integral Taylor remainder of \mathcal R by

There is no third-order remainder for \gamma itself.

Finally combine (4.17), (4.20), and (4.21). 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 (4.4) and Proposition 4.1. 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.

5 An exact smooth correction

The small increment of Proposition 4.1 leaves a metric error. We now remove that error while keeping the immersion close in C^2. Smoothing is used only to construct each increment; the current map itself is never replaced by its smoothing. This preserves its higher derivatives until a sufficiently late stage can control them.

Smoothing in an iteration with derivative loss originates in Nash’s smooth embedding argument [13, Part A] and was developed by Moser [11, Chapter I, §§1 and 5]; Hamilton gives a systematic inverse-function formulation [6, Part III]. Here we establish the correction estimates and the passage through the derivative orders directly, without assuming a smooth family of inverses of the full linearized metric map.

The proof has two parts. First we obtain a recurrence valid in every fixed derivative order, from a bound in only one finite order. We then let that controlled order increase, waiting a finite number of steps whenever necessary. The distinction between these two orders is what gives a smooth limit from the finite assumptions above.

5.1 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\R^4 be smooth, let H be a smooth symmetric tensor, and define

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 5.2. Then, for every fixed m,

For the first inequality use (5.4) 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

For j>r, boundedness of smoothing in the original derivative norm gives a contribution at most

In particular, the constants in (5.8) do not use any derivative of the true input above order r+2; the possibly large higher derivatives occur only linearly in (5.9).

5.2 A recurrence at every derivative order

Fix a small z for the moment. Lemma 2.3 provides phase data for f_z with reference metric h_*. They remain admissible in the C^2 ball of radius

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

Here q(k) is the accuracy order in Proposition 4.1. 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

The step will be applied under the finite-order hypotheses

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 5.1, keep the maps in the allowed C^2 ball, and schedule increases of k.

5.3 Initialization and passage through the derivative orders

Because \|f_z\|_{C^2}=O(z^{-2}), for small z the distance condition in (5.12) implies

For each fixed z,k, choose a positive scale threshold that makes Lemma 5.3 applicable, preserves the defect neighborhood, and, by (5.14) with m=r(k), preserves \mathcal S_{r(k)}\le B. Impose also

This last threshold is available because (5.13) gives the upper bound

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

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

Its normalized defect has the exact form

It therefore tends to h_* in C^R. The weighted derivatives of G through the required order are bounded by

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,

for small enough t_0. Equations (5.20) and the initial displacement keep all maps within \rho_z/2 of f_z in C^2. Hence the phase data and (5.19) 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 5.3 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

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

where the constant is the one for the current block in (5.14). 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 (5.20) 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,

This proves Proposition 5.1.

6 A velocity circle with controlled turns

We now construct the leading oscillation for Proposition 3.1. Throughout this section its data F,D,a,x,y,n and the later curves are fixed. The basic choice is to oscillate in the plane perpendicular to both G_y and G_{yy}, for a slow map G near F. Orthogonality to the second derivative will remove the first error in the yy metric coefficient. A separate choice at the two turning points of the velocity path will preserve the later boundary conditions. For a 2\pi-periodic function b we write \avg b=(2\pi)^{-1}\int_0^{2\pi}b(\theta)\,d\theta. For a nonzero vector b, write \widehat b=b/|b|.

The use of a periodic velocity to add a rank-one metric tensor belongs to the oscillatory constructions of Nash and Kuiper [12, 9]; see also [2, §4] and [5, §§3.2–3.3]. The second-derivative constraint on the velocity plane and the behavior prescribed at its turns are part of the construction proved here.

6.1 The plane and the leading metric

For a 2-jet near that of F, define

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

Since G_x-v\in\operatorname{span}(Y,C) and V,v\in P,

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.

6.2 A prescribed mean and smooth vanishing

Initially use an angular parameter t\in\R/(2\pi\mathbb Z) and angles

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

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

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 (6.6) 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 \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

and replace t by the smooth inverse of this orientation-preserving circle diffeomorphism. We henceforth use \theta as periodic parameter. Its uniform mean satisfies

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 (6.7) and (6.6) 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,

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.

6.3 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 3.1 gives exactly this freedom.

6.4 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

It is perpendicular to X,Y,C, and hence also to n_0^*. Set

The derivative here holds the periodic parameter fixed. Directly,

The term containing \alpha_y drops out of D_0^*, so

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

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

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 6.3, 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

Equations (6.9) and (6.11) 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.

7 Realizing the primitive metric addition

The velocity loop gives the desired metric at leading order. We now remove its oscillating errors to arbitrarily high finite order, correct the remaining mean error by small increments, and apply Proposition 5.1. The resulting exact immersions retain the first and second derivatives needed to control the later boundary curves.

All compact neighborhoods of jets used below are chosen inside the open sets in this statement. Thus the stated nondegeneracies have positive uniform margins. Derivatives of periodic coefficients in x,y will always hold \theta fixed. We write b_z=o_s(c_z) when |b_z|_{k,s}/c_z\to0 for every fixed weighted differentiation order k.

7.1 Removing the oscillating coefficients

Finite-order asymptotic isometric constructions also occur in Poznyak’s planar immersion problem in \R^3 [15, §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

For \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.

7.2 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

A jet of order k>2 is O_s(s^{2-k}). Since the expansion has finitely many coefficients, Lemma 7.2 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,

For later use, differentiating the finite ansatz gives

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

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 (7.13). The same estimates show \|f_z[G]\|_{C^k}=O(z^{-k}) for each fixed k.

We now construct a slow map satisfying (7.13) 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.

8 Preserving the later boundary conditions

We complete Proposition 3.1. The analytic construction in Proposition 7.1 supplies exact immersions for \gamma_+ with the leading derivatives of the chosen loop. We must show that their second forms preserve the later curve conditions and that their preferred normals extend globally. The crossing calculation below also specifies a turn threshold using only the bounds that were fixed before the loop’s large derivative.

8.1 The exact second form

Choose the loop of Lemma 6.4, with the turn threshold to be specified below, and let \widetilde f_z be the resulting exact immersions from Proposition 7.1. On the disk their yy second form has the representation

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

The determinant is in the (x,y) coordinates. The Gauss equation gives

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,

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 (8.1) gives the exact identities

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^*.

8.2 Crossing thresholds without circular choices

At a mutual later crossing inside D, both b_i,b_l are nonzero. Put

The finitely many slopes are bounded. From (8.1), the pure and mixed vectors are

In particular

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, (8.2) 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

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 (8.5). In particular these estimates do not depend on L^*.

Also

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 (8.6) exceed

at every crossing. Use a margin, for example T+2, as the turn threshold in Lemma 6.4, 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, (8.2) 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 (8.4) have the same leading second component L^*; hence

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

8.3 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 (8.3) is zero and the first is nonpositive, so

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 (8.2) now forces \alpha_\theta=0 at the limiting phase.

In the boundary collar, (6.12) rules out the first inequality of (8.8), 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,

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.

8.4 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 (8.6) 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

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 (8.9); they do not affect (8.7), whose quantities are independent of the preferred normal.

8.5 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 (3.3), \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 (3.2) gives

Thus both old endpoint normals have positive dot product with the later pure vector: one by this inequality, and the other by (3.3). 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 (3.3) 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 7.1. This completes the proof of Proposition 3.1.

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.

9 Global preparation and finite assembly

We now arrange a finite sequence of primitive additions to which Proposition 3.1 applies. The main point is the order of the choices. The initial immersion will be modified at the crossings of the support boundaries, although those boundaries depend on the number of primitive additions. Uniform second-derivative bounds for the initial modification allow us to choose the number of additions first.

The assertion is immediate if M is empty. The connected components of a smooth manifold are open. Compactness therefore implies that M has only finitely many components. It suffices to carry out the construction on each component: the resulting maps together define one smooth isometric immersion on M. We may thus assume in this section that M is connected.

9.1 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 \R^3 [17, 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 \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 \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 \R^4. The second fundamental forms satisfy

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 (3.3) is consequently

This identity also holds when the two vectors are proportional.

9.2 A finite list with prescribed intermediate metrics

9.3 Completion of the global construction