Global smooth isometric immersions of two-dimensional Riemannian manifolds into $\mathbb{R}^4$

Poznyak (1973) proved that every smooth two-dimensional Riemannian metric on a disk admits a smooth isometric immersion into $\mathbb{R}^4$, and that 4 is the minimal target dimension with this local property. The global counterpart is open: no characterization is known of which surfaces and metrics admit global $C^\infty$ isometric immersions into $\mathbb{R}^4$. The Nash–Gromov–Rokhlin–Günther theory guarantees only dimension 10 for arbitrary compact surface metrics; the complete hyperbolic plane $\mathbb{H}^2$ is smoothly immersible in $\mathbb{R}^5$ (non-injectively) but in $\mathbb{R}^4$ only a piecewise-analytic Lipschitz ($C^{0,1}$) immersion is known, and whether $\mathbb{H}^2$ immerses smoothly in $\mathbb{R}^4$ remains a recognized open problem, reconfirmed as such by Sabitov's 2023 review; the analogous question for an arbitrary smooth metric on a closed surface is likewise open, in the sense of Mirandola–Vitório's lowest-possible-codimension problem, and the recent low-regularity programs of Cao–Székelyhidi (global $C^{1,\theta}$ embeddings of compact surfaces into $\mathbb{R}^4$, with $\theta<1/3$) and Lewicka ($C^{1,\alpha}$ flexibility on planar domains, $\alpha<1$) stop below $C^2$. The problem asks whether every smooth two-dimensional Riemannian metric admits a global $C^\infty$ isometric immersion into $\mathbb{R}^4$, settling in particular the compact-surface existence question and the hyperbolic plane.

Background

An isometric immersion of a Riemannian manifold $(M,g)$ into Euclidean space is a map $u:M\to\mathbb{R}^n$ preserving the lengths of all curves; in local coordinates it solves the first-order system $(\nabla u)^T\nabla u=g$, i.e. the pullback metric of $u$ equals $g$. If $u$ is also injective and a homeomorphism onto its image, it is an isometric embedding. A short immersion is one whose induced metric is strictly smaller than $g$, in the sense that $g-(\nabla u)^T\nabla u$ is positive definite; convex integration is a technique that builds solutions of underdetermined PDE systems as limits of highly oscillatory perturbations of such approximate (short) solutions, typically at low regularity.

For surfaces, the target dimension controls the theory. Locally, the Janet–Cartan–Burstin theorem realizes any analytic 2-dimensional metric analytically in $\mathbb{R}^3$; for merely smooth metrics $\mathbb{R}^3$ fails in general, but Poznyak (1973) proved that any smooth 2-dimensional metric on the closure of a bounded simply connected planar domain admits a $C^\infty$ isometric immersion into $\mathbb{R}^4$, and 4 is minimal: Poznyak exhibited a smooth disk metric with no $C^2$ isometric immersion into $\mathbb{R}^3$ (an analytic positive-curvature example of Gromov–Rokhlin is recalled in Lewicka 2025). Globally, the general existence theory — Nash's theorem refined by Gromov–Rokhlin and Günther — guarantees every smooth surface metric on a compact surface a smooth isometric embedding into $\mathbb{R}^{10}$ (the bound $c(2)=\max\{7,10\}=10$; accounts in Mirandola–Vitório 2015 and Lewicka 2025), and Mirandola–Vitório recall that finding the lowest possible codimension is one of the major open problems of immersion theory. Between local dimension 4 and global dimension 10, no general smooth result is known, and dimension 4 is the natural threshold.

Curvature gives the sharpest known obstructions in codimension one: Hilbert (1901) proved that the hyperbolic plane $\mathbb{H}^2$ (the complete simply connected surface of constant Gaussian curvature $-1$) admits no $C^2$ isometric immersion into $\mathbb{R}^3$, while metrics of positive Gaussian curvature on the 2-sphere embed smoothly in $\mathbb{R}^3$ by the classical solution of Weyl's problem, and flat tori embed in $\mathbb{R}^4$ as products of circles. In higher codimension, Blanuša (1955) constructed an injective smooth isometric immersion of $\mathbb{H}^2$ into $\mathbb{R}^6$, later improved to non-injective immersions into $\mathbb{R}^5$ (surveyed in Mirandola–Vitório 2015 and Sabitov 2023). In $\mathbb{R}^4$ itself, Sabitov (1989) realized $\mathbb{H}^2$ only as a piecewise-analytic $C^{0,1}$ (Lipschitz) surface; whether a smooth immersion exists — and whether $\mathbb{H}^2$ embeds injectively even in $\mathbb{R}^5$ — is explicitly stated as open (Mirandola–Vitório 2015).

The regularity landscape sharpened recently. In $\mathbb{R}^3$, flexibility by convex integration holds up to Hölder exponents well below 1, while rigidity returns above $C^{1,2/3}$ (see the account in Lewicka 2025). In $\mathbb{R}^4$, Cao–Székelyhidi (2025) proved an isometric version of Whitney's embedding theorem: every compact smooth surface admits global isometric embeddings into $\mathbb{R}^4$ of class $C^{1,\theta}$, $\theta<1/3$, and Lewicka (2025) achieved full flexibility — approximation of any short immersion by $C^{1,\alpha}$ isometric immersions into $\mathbb{R}^4$ for all $\alpha<\min\{(r+\beta)/2,1\}$ — for any $C^{r,\beta}$ surface metric on a simply connected planar domain. These results show that rigidity disappears in codimension two below $C^2$, but no technique currently reaches $C^2$ or $C^\infty$: the $C^{1,1}$ endpoint is already open, and the globalization of Lewicka's construction to compact or complete surfaces is explicitly not carried out. The smooth global question therefore remains exactly where Poznyak's local theorem left it.

Problem Statement

Determine whether every smooth two-dimensional Riemannian metric $(M^2,g)$ admits a global smooth isometric immersion into $\mathbb{R}^4$: a map $u\in C^\infty(M,\mathbb{R}^4)$ with $(\nabla u)^T\nabla u=g$ on all of $M$. Since Poznyak's theorem guarantees such immersions locally on every disk and 4 is the minimal target dimension for a general smooth theory, this is precisely the local-to-global passage, the two-dimensional instance of the below-Günther-bound ('lowest possible codimension') problem. The central instances are (a) whether every smooth Riemannian metric on a closed (compact, boundaryless) surface — arbitrary metrics on $S^2$, $T^2$, and closed hyperbolic surfaces — admits a global $C^\infty$ isometric immersion into $\mathbb{R}^4$, or else exhibit a specific smooth surface metric together with a rigorous proof that it admits no $C^\infty$ isometric immersion into $\mathbb{R}^4$ (an obstruction at the $C^2$ level a fortiori excludes all $C^\infty$ immersions as well); and (b) whether the complete hyperbolic plane $\mathbb{H}^2$ admits a $C^\infty$ isometric immersion into $\mathbb{R}^4$. These benchmark instances and the general existence question are one linked objective: determining the exact reach of global smooth immersion theory for surfaces at the minimal dimension 4. The objective is not a general characterization of the immersible pairs $(M^2,g)$. Answers below the $C^2$ regularity threshold (e.g. of class $C^{1,\alpha}$) do not settle the question, since the codimension-one rigidity theory shows the threshold is essential; answers covering only restricted ansatz classes or only planar domains do not settle it either.

References

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