Flexibility of isometric immersions of two-dimensional metrics into $\mathbb{R}^4$: direct convex integration on the Poznyak curvature equation

Lewicka's full-flexibility theorem (arXiv:2511.16305, 2025/2026) shows that every strictly short $C^1$ immersion of a two-dimensional $C^{r,\beta}$ metric ($0<r+\beta\le2$) on a simply connected planar domain into $\mathbb{R}^4$ - the minimal target dimension of Poznyak's classical theorem - can be uniformly approximated by exact isometric immersions of class $C^{1,\alpha}$ for every $\alpha<\min\{(r+\beta)/2,1\}$, so that rigidity disappears completely in codimension two. This record concerns the open question that theorem leaves, stated explicitly at the end of its Appendix (Section 9, 'Appendix: Poznyak's theorem'): whether the main result can be recovered by applying convex integration directly to the scalar Gaussian-curvature equation $\kappa(\tilde g)=0$ for the conformal factor in Poznyak's proof scheme, rather than by the corrugation scheme applied to the full first-order system $(\nabla u)^T\nabla u=g$.

Background

An isometric immersion of a $d$-dimensional Riemannian metric $g$ is a map $u$ into a Euclidean space $\mathbb{R}^n$ that pulls the Euclidean metric back to $g$; in coordinates this is the first-order, fully nonlinear system $(\nabla u)^T\nabla u=g$, and $n-d$ is called the codimension. For analytic metrics, classical local solvability holds in the Janet dimension $d_*=d(d+1)/2$, the number of independent entries of a symmetric $d\times d$ matrix ($d_*=3$ when $d=2$). At low regularity, Nash and Kuiper proved that every strictly short immersion - a $C^1$ map with $(\nabla u)^T\nabla u<g$ in the sense of positive definite matrices, obtained for instance by scaling an arbitrary immersion down - can be uniformly approximated by exact $C^1$ isometric immersions into $\mathbb{R}^{d+1}$, through an iteration of highly oscillatory corrections known as convex integration; each correction is a Kuiper corrugation, a one-dimensional oscillatory perturbation that reduces the defect $g-(\nabla u)^T\nabla u$. The boundary between such flexibility and rigidity is understood only partially. In codimension one, isometric immersions of two-dimensional $C^2$ metrics into $\mathbb{R}^3$ are rigid for Hölder exponents $\alpha>2/3$ (Borisov, with a complete proof by Conti, De Lellis and Székelyhidi), while flexibility is known for $\alpha<1/3$ (Cao, Hirsch and Inauen), after the earlier milestones $\alpha<1/7$ (Borisov's exponent, confirmed by Conti, De Lellis and Székelyhidi) and $\alpha<1/5$ (De Lellis, Inauen and Székelyhidi); the exact threshold remains open and is often compared to Onsager's conjecture in fluid dynamics (the analogous conjectured flexibility/rigidity split for weak solutions of the Euler equations). Here $C^{1,\alpha}$ denotes continuously differentiable maps whose gradient is $\alpha$-Hölder continuous, and $C^{1,1}$ those with Lipschitz (bounded) gradient. In higher codimension, Källén constructed $C^{1,\alpha}$ isometric immersions, $\alpha<(r+\beta)/2$, of $C^{r,\beta}$ metrics into sufficiently high-dimensional targets, and Martínez reached $C^{1,1-\epsilon}$ locally in codimension $d(d+1)/2$; Cao and Székelyhidi globalized the Nash-Kuiper theorem to compact manifolds in the low-exponent codimension-one regime.

In dimension $d=2$, Poznyak's classical theorem identifies the minimal target dimension for smooth metrics: every smooth Riemannian metric on the closure of a bounded simply connected planar domain has a smooth isometric immersion into $\mathbb{R}^4$, and the dimension $4$ cannot be lowered, since Poznyak also gave a smooth disk metric with no $C^2$ isometric immersion into $\mathbb{R}^3$. Poznyak's proof reduces the problem to a scalar curvature equation for a conformal factor, i.e. a scalar function $w$ rescaling the metric by $e^{-2w}$: writing $\tilde g=e^{-2w}g-2\epsilon^2\nabla w\otimes\nabla w$ with $\epsilon>0$, one seeks $w$ such that $\tilde g$ is positive definite and has zero Gaussian curvature $\kappa(\tilde g)=0$ (the Gaussian curvature being the intrinsic curvature of a two-dimensional metric); a flat metric $\tilde g$ is then parametrized by $v$ with $(\nabla v)^T\nabla v=\tilde g$, and $u(x)=\epsilon e^{w(x)}(\cos(v^1(x)/\epsilon),\sin(v^1(x)/\epsilon),\cos(v^2(x)/\epsilon),\sin(v^2(x)/\epsilon))$ is the desired immersion. For low-regularity metrics, Hartman and Wintner observed that an isometric immersion of a $C^{r,\beta}$ metric can be no smoother than $C^{r,\beta}$, since its second fundamental form determines the Gaussian curvature; thus for $g\in C^2$ the critical regularity window ends exactly at $C^{1,1}$.

Lewicka (arXiv:2511.16305) proved the full-flexibility counterpart of Poznyak's theorem in this window: for $g\in C^{r,\beta}$ with $0<r+\beta\le2$ on a planar domain diffeomorphic to a disk, every strictly short immersion $u\in C^1(\bar\omega,\mathbb{R}^4)$ is the uniform limit of exact isometric immersions $\bar u\in C^{1,\alpha}(\bar\omega,\mathbb{R}^4)$ for every $\alpha<\min\{(r+\beta)/2,1\}$. In particular, every two-dimensional $C^2$ metric admits isometric immersions into $\mathbb{R}^4$ of every regularity $C^{1,\alpha}$ with $\alpha<1$, arbitrarily close to any short immersion, so no rigidity phenomenon survives in codimension two just below the critical regularity. The proof is a convex integration scheme combining Kuiper corrugations in the two normal directions with a staggered multi-frequency cascade, transferring the strategy developed by Inauen and Lewicka for the two-dimensional Monge-Ampère system (the equation $\det\nabla^2 v=f$ and its vectorial generalizations, whose flexibility theory closely parallels the immersion system). What the theorem does not deliver is the endpoint exponent $\alpha=1$ itself, a global statement on compact surfaces, and any derivation through Poznyak's scalar curvature equation. Of these three gaps, the source itself poses only the last one as an open question, in the closing sentence of its Appendix; that question is the subject of this problem, while the strict inequality $\alpha<1$ and the restriction to planar domains are recorded here solely as limitations of the proved theorem.

Problem Statement

The objective is the open question left by Lewicka's full-flexibility theorem (arXiv:2511.16305, Theorem 1.1), stated in the closing sentence of Section 9 of the source ('Appendix: Poznyak's theorem'): 'It would be interesting to see whether the main result of this paper could be recovered by applying convex integration directly to find $w$ that solves $\kappa(\tilde g)=0$.'

The result to be recovered is Theorem 1.1, in the setting of metrics $g\in C^{r,\beta}$ with $0<r+\beta\le2$ on the closure $\bar\omega$ of a bounded simply connected planar domain $\omega$ diffeomorphic to a disk, with the target dimension $\mathbb{R}^4$ of Poznyak's theorem: every strictly short immersion $u\in C^1(\bar\omega,\mathbb{R}^4)$, i.e. every $u$ with $(\nabla u)^T\nabla u<g$, is the uniform limit of exact isometric immersions $\bar u\in C^{1,\alpha}(\bar\omega,\mathbb{R}^4)$ for every $\alpha<\min\{(r+\beta)/2,1\}$.

The question is whether this theorem can be obtained by applying convex integration directly to the scalar Gaussian-curvature equation $\kappa(\tilde g)=0$ for the conformal factor $w$ in Poznyak's proof scheme, where $\tilde g=e^{-2w}g-2\epsilon^2\nabla w\otimes\nabla w$ (with $\epsilon>0$) must remain a positive definite metric, rather than by the corrugation scheme applied to the full first-order system $(\nabla u)^T\nabla u=g$. In Poznyak's scheme, once $\tilde g>0$ and $\kappa(\tilde g)=0$ hold, the flat metric $\tilde g$ is parametrized by $v$ with $(\nabla v)^T\nabla v=\tilde g$, and $u(x)=\epsilon e^{w(x)}(\cos(v^1(x)/\epsilon),\sin(v^1(x)/\epsilon),\cos(v^2(x)/\epsilon),\sin(v^2(x)/\epsilon))$ is the desired immersion.

References

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