Existence of a smooth isometric immersion of the entire hyperbolic plane into $\mathbb{R}^4$

Does the complete hyperbolic plane $\mathbb{H}^2$ (constant Gaussian curvature $-1$) admit a $C^\infty$ isometric immersion into the Euclidean space $\mathbb{R}^4$? Hilbert's theorem (1901) rules this out in codimension one ($\mathbb{R}^3$), Blanuša (1955) constructed a smooth isometric embedding of the whole $\mathbb{H}^2$ into $\mathbb{R}^6$ and Rozendorn (1960) a smooth isometric immersion into $\mathbb{R}^5$, and low-regularity ($C^1$, $C^{1,1}$, $W^{2,2}$) immersions into $\mathbb{R}^3$ exist — but for smooth immersions in codimension two neither existence nor non-existence is known. The source paper (Dione–Diop–Niang, 2021) reduced the question to first-order PDE systems, immersed certain strips, proved non-existence only under additional restrictions, and explicitly left the global problem open; the unrestricted global question remains open.

Background

An isometric immersion of a Riemannian surface $(M,g)$ into Euclidean space is a smooth map $f:M\to\mathbb{R}^n$ whose derivative preserves the metric, $\langle\partial_i f,\partial_j f\rangle=g_{ij}$, so that all intrinsic lengths and angles — in particular the Gaussian curvature — are unchanged. The hyperbolic plane $\mathbb{H}^2$ is the unique complete, simply connected surface of constant Gaussian curvature $-1$; a concrete model is the upper half-plane $\{(x,y):y>0\}$ with metric $g=(dx^2+dy^2)/y^2$. A classical theorem of Hilbert (1901) states that there is no complete $C^2$ isometric immersion of $\mathbb{H}^2$ into $\mathbb{R}^3$, and Efimov strengthened this to every surface whose curvature is bounded above by a negative constant. These are codimension-one results: they depend on the asymptotic coordinates (curves along which the second fundamental form — the quadratic form encoding how the surface bends in the ambient space — vanishes) that negative curvature forces on a surface in $\mathbb{R}^3$. At low regularity the obstruction disappears: by the Nash–Kuiper $C^1$ isometric embedding theorem, $C^1$ isometric embeddings of $\mathbb{H}^2$ into $\mathbb{R}^3$ exist, and $C^{1,1}$ immersions (one continuous derivative whose derivative is Lipschitz) of the standard hyperbolic plane into $\mathbb{R}^3$ were constructed by Li (2020); the finite-bending ($W^{2,2}$, i.e. Sobolev regularity with two weak square-integrable derivatives) theory of negatively curved surfaces remains active (Derfner–Kupferman–Maor, 2026). For smooth immersions in higher codimension, Blanuša (1955) built an explicit smooth isometric embedding of the entire $\mathbb{H}^2$ into $\mathbb{R}^6$ by gluing together pieces modeled on the pseudosphere (the surface of revolution of constant Gaussian curvature $-1$); Rozendorn (1960), adapting Blanuša's method, constructed a $C^\infty$ isometric immersion of the entire Lobachevskiĭ plane into $\mathbb{R}^5$ (an immersion with self-tangency lines, whose regularity and induced complete hyperbolic metric are verified in Aminov (2009); Gromov gave another proof). Thus five is the least Euclidean dimension in which a complete smooth isometric immersion of $\mathbb{H}^2$ is known to exist, and six the least in which a complete smooth isometric embedding is known. The borderline case $\mathbb{R}^4$ (codimension two) has resisted both approaches. For a surface in $\mathbb{R}^4$ the Gauss–Codazzi–Ricci compatibility system — the linear-algebraic and differential equations that the second fundamental form and the normal connection (the covariant rotation of the normal directions as one moves along the surface) must satisfy — is underdetermined, and the second fundamental form may have rank 2, so the Hilbert–Efimov asymptotic-coordinate mechanism no longer applies; on the other hand, no construction is known to close the gap between five and four ambient dimensions. Two results delimit what is possible in $\mathbb{R}^4$: Niang (2013) proved that for any negatively curved surface immersed in $\mathbb{R}^4$, having second fundamental form of rank 1 everywhere is equivalent to possessing local asymptotic coordinates and to lying locally inside some $\mathbb{R}^3\subset\mathbb{R}^4$ — hence, by Hilbert's theorem, no immersion of the whole $\mathbb{H}^2$ with rank-1 second fundamental form exists — and constructed explicit isometric immersions of the half-planes $\mathbb{R}\times(c,+\infty)\subset\mathbb{H}^2$ with rank-2 second fundamental form. Brander (2008), from the loop-group perspective on generalizations of Hilbert's theorem, showed that within the class of immersions with flat normal bundle (vanishing curvature of the normal connection) the only possible non-compact generalizations are the flat-normal-bundle and the Lagrangian cases (a Lagrangian immersion into $\mathbb{C}^n$ is one along which the ambient symplectic form vanishes), where local solutions exist but global existence is unknown. The source paper (Dione–Diop–Niang, Journal of Mathematics, 2021) gave a PDE reformulation: it reduced the immersion equations for open subsets of $\mathbb{H}^2$ into $\mathbb{R}^4$ to first-order linear systems coupled by an algebraic constraint on a quadruple $(\lambda,\mu_1,\mu_2,\theta)$, obtained necessary and sufficient conditions, immersed strips $\mathbb{R}\times(c,+\infty)$, showed that certain other strips and degenerate cases admit no global solutions, and proved that under natural additional conditions there are no complete solutions — while stating that it is not yet known whether the hyperbolic plane can be immersed isometrically in $\mathbb{R}^4$, and describing the paper as a first step toward coming works on the existence or nonexistence of isometric immersions of particular open subsets of $\mathbb{H}^2$ into $\mathbb{R}^4$. General surveys of isometric embedding (e.g. Gromov's Bulletin of the AMS survey (2017) of the descendants of Nash's theorems) frame such existence questions as part of the current boundary of knowledge in the field.

Problem Statement

Let $\mathbb{H}^2$ be the complete hyperbolic plane, realized as $\mathbb{R}\times\mathbb{R}_{>0}$ with the metric $g=(dx^2+dy^2)/y^2$ of constant Gaussian curvature $-1$. Determine whether there exists a $C^\infty$ isometric immersion $f:\mathbb{H}^2\to\mathbb{R}^4$, i.e. a smooth map defined on all of $\mathbb{H}^2$ with $\langle\partial_i f,\partial_j f\rangle=g_{ij}$ everywhere: either construct such an immersion, or prove that none exists. In the source paper's formulation, this is equivalently the question whether their first-order PDE systems with the algebraic constraint on $(\lambda,\mu_1,\mu_2,\theta)$ can be satisfied globally on $\mathbb{H}^2$. The answer settles whether a complete immersed surface of constant curvature $-1$ in $\mathbb{R}^4$ free of singularities exists, and is the essential step toward the source paper's stated expectation of coming works on the existence or nonexistence of isometric immersions of particular open subsets of $\mathbb{H}^2$ into $\mathbb{R}^4$.

References

1. Mor Dione, Bira Gaye Diop, Athoumane Niang, 'A PDE for Smooth Isometric Immersion of an Open Set of the Hyperbolic Plane H^2 into R^4', Journal of Mathematics, vol. 2021, Article ID 2012354 (2021), pages 1-7, DOI: 10.1155/2021/2012354, https://doi.org/10.1155/2021/2012354
2. Athoumane Niang, 'About the existence of isometric immersions of the hyperbolic plane H^2 in R^4' (Autour de l'existence d'immersion(s) isométrique(s) du plan hyperbolique H^2 dans R^4), Afrika Matematika 24 (2013), 457-464, DOI: 10.1007/s13370-012-0071-0, https://doi.org/10.1007/s13370-012-0071-0
3. David Hilbert, 'Ueber Flächen von constanter Gaussscher Krümmung', Transactions of the American Mathematical Society 2 (1901), 87-99, DOI: 10.1090/S0002-9947-1901-1500557-5, https://doi.org/10.1090/S0002-9947-1901-1500557-5
4. N. V. Efimov, 'Appearance of singularities on surfaces of negative curvature' (English translation; Russian original Uspekhi Matematicheskikh Nauk 21 (1966)), American Mathematical Society Translations, Series 2, vol. 66 (1968), 154-190, DOI: 10.1090/trans2/066/07, https://doi.org/10.1090/trans2/066/07
5. Danilo Blanuša, 'Über die Einbettung hyperbolischer Räume in euklidische Räume', Monatshefte für Mathematik 59 (1955), 217-229, DOI: 10.1007/BF01303796, https://doi.org/10.1007/bf01303796
6. John Nash, 'C^1 Isometric Imbeddings', The Annals of Mathematics 60 (1954), 383-396, DOI: 10.2307/1969840, https://doi.org/10.2307/1969840
7. Nicolaas H. Kuiper, 'On C^1-isometric imbeddings. I', Indagationes Mathematicae (Proceedings) 58 (1955), 545-556, DOI: 10.1016/S1385-7258(55)50075-8, https://doi.org/10.1016/s1385-7258(55)50075-8
8. David Brander, 'Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane', Electronic Research Announcements in Mathematical Sciences 15 (2008), 8-16, arXiv:0710.0507, https://arxiv.org/abs/0710.0507
9. Siran Li, 'On the Existence of C^{1,1}-Isometric Immersions of Several Classes of Negatively Curved Surfaces into R^3', Archive for Rational Mechanics and Analysis 236 (2020), 419-449, DOI: 10.1007/s00205-019-01472-8, https://doi.org/10.1007/s00205-019-01472-8
10. Gilad Derfner, Raz Kupferman, Cy Maor, 'Branch points and non-density for finite-bending isometric immersions of hyperbolic surfaces', arXiv:2608.10550 (2026), https://arxiv.org/abs/2608.10550
11. Misha Gromov, 'Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems', Bulletin of the American Mathematical Society 54 (2017), 173-245, DOI: 10.1090/bull/1551, https://doi.org/10.1090/bull/1551
12. Ernest R. Rozendorn, 'Realization of a metric ds^2 = du^2 + f^2(u) dv^2 in the 5-dimensional Euclidean space', Doklady Akademii Nauk Armyanskoy SSR 30:4 (1960), 197-199.
13. Yuri A. Aminov, 'Extrinsic geometric properties of the Rozendorn surface, an isometric immersion of the Lobachevskiĭ plane in E^5', Sbornik: Mathematics 200:11 (2009), 1575-1586, DOI: 10.1070/SM2009v200n11ABEH004051, https://doi.org/10.1070/sm2009v200n11abeh004051