Sharp rigidity and classification of constant-scalar-curvature hypersurfaces under weak curvature hypotheses and extensions of the Cheng–Yau operator beyond space forms

Cheng and Yau (Math. Ann. 1977) introduced the self-adjoint second-order operator $\square f=\sum_{i,j}\phi_{ij}f_{;ij}$ associated to a divergence-free symmetric tensor — for a hypersurface of constant scalar curvature in a space form, the Newton tensor $\phi=nH\delta-h$ of the second fundamental form, which is divergence-free precisely because $h$ satisfies the Codazzi equation — and used it, together with an eigenvalue-comparison principle for possibly degenerate elliptic operators, to classify compact hypersurfaces with nonnegative sectional curvature and constant scalar curvature in space forms (totally umbilical — all principal curvatures equal — or a product of two totally umbilical factors, or flat) and complete noncompact convex hypersurfaces of Euclidean space with constant scalar curvature (generalized cylinders). They closed the paper with the remark that these operators and estimates 'should have more applications'. The open core of that invitation is to determine the sharp conditions under which rigidity of constant-scalar-curvature ($\sigma_2$) hypersurfaces persists when the nonnegative-sectional-curvature and space-form hypotheses are weakened — via pinching or traceless-second-fundamental-form bounds, integral conditions, embeddedness, or generalized Codazzi equations in non-space-form ambients — and to construct the non-rigid examples that live beyond that boundary. Later literature (Li's pinching theorems of 1996, Ros's embedded congruence theorem of 1988, the maximum-principle and gap-theorem school of Alías, Rigoli and de Lima, the sharp $\Phi$-constraint and non-isoparametric (principal curvatures not all constant) rotational examples of Alías–Meléndez–Palmas 2018, and the active constant-mean-and-scalar-curvature program of Núñez 2017 and Ge–Tao 2025) resolves many constrained cases but leaves the general classification open.

Background

Let $M^n$ be an immersed hypersurface of a Riemannian manifold $\tilde M^{n+1}$. Its extrinsic geometry is encoded by the second fundamental form $h=(h_{ij})$, a symmetric tensor whose eigenvalues $\kappa_1,\dots,\kappa_n$ are the principal curvatures; the mean curvature is $H=\frac{1}{n}\sum_i\kappa_i$ and the normalized scalar curvature is $R=\frac{1}{n(n-1)}\operatorname{Scal}$, the scalar curvature of $(M,g)$ normalized by $\frac{1}{n(n-1)}$ (which in a space form of curvature $c$ equals $c+\frac{2}{n(n-1)}\sum_{i<j}\kappa_i\kappa_j$), so that, in a space form, 'constant scalar curvature' means that the second elementary symmetric function $\sigma_2(\kappa_1,\dots,\kappa_n)=\sum_{i<j}\kappa_i\kappa_j$ is constant — a Weingarten-type condition (a fixed functional relation among the curvature functions of the principal curvatures) one order weaker than the hypersurface Einstein condition (proportionality of the intrinsic Ricci tensor to the metric). When $\tilde M$ is a space form, that is, a complete Riemannian manifold of constant sectional curvature $c$, the Gauss equation gives $n(n-1)(R-c)=n^2H^2-|h|^2$ and the second fundamental form satisfies the Codazzi equation $h_{ij;k}=h_{ik;j}$, making $h$ a Codazzi tensor. For complete surfaces in $\mathbb{R}^3$ the classification under constant Gauss curvature is classical (nonzero constant curvature forces the sphere; zero constant curvature forces a plane or cylinder, the Hartman–Nirenberg theorem), and the analogous higher-dimensional statements at the level of scalar curvature were open before 1977 except for Thomas's local theorem on Einstein hypersurfaces. Cheng and Yau filled this gap by an operator method. For a symmetric tensor $\phi=(\phi_{ij})$ they defined the second-order operator $\square f=\sum_{i,j}\phi_{ij}f_{;ij}$ (their eq. (1.7)) and proved that $\square$ is formally self-adjoint on a compact orientable manifold exactly when $\phi$ is divergence-free; divergence-free symmetric tensors arise structurally — the Einstein tensor $R_{ij}-\frac{R}{2}g_{ij}$ by the second Bianchi identity (the standard differential identity satisfied by the Riemann curvature tensor), and, for any Codazzi tensor $\psi$, the Newton-type tensor $\operatorname{tr}(\psi)g-\psi$. On a space-form hypersurface with constant scalar curvature this yields the hypersurface operator $\square f=\sum_{i,j}(nH\delta_{ij}-h_{ij})f_{;ij}$, the linearization of the scalar curvature functional, now called the Cheng–Yau operator. Section 2 of their paper computes the Laplacian of a Codazzi tensor in the tradition of Simons' classical computation of the Laplacian of the second fundamental form of a minimal hypersurface and yields Theorem 1: a Codazzi tensor with constant nonnegative second elementary symmetric function on a compact manifold of nonnegative sectional curvature is pointwise diagonalizable with at most two constant-multiplicity eigenvalues, splitting the manifold locally into a product. Section 3 converts this into hypersurface classifications (Theorem 2: a compact hypersurface with nonnegative sectional curvature and constant normalized scalar curvature $R\geq c$ in a space form of curvature $c$ is totally umbilical (i.e. $h=Hg$, all principal curvatures equal), a product of two totally umbilical constantly curved submanifolds, or flat, the flat case only when $c=0$, and in Euclidean space compactness then forces the round sphere; Theorem 3: a mean-curvature estimate under positive sectional curvature). Section 4 treats the complete noncompact case with Proposition 2, an eigenvalue-comparison principle for formally self-adjoint, possibly degenerate elliptic operators on compact manifolds with boundary, combined with the volume growth of the slabs $D_r=\{x:\langle x,a\rangle\leq r\}$ and a spherical-image argument (control of the image of the Gauss map) for convex hypersurfaces; Theorem 4: a complete noncompact hypersurface of $\mathbb{R}^{n+1}$ with nonnegative sectional curvature and constant scalar curvature is a generalized cylinder $S^p\times\mathbb{R}^{n-p}$ (the zero-scalar-curvature case reducing to the Hartman–Nirenberg flat-cylinder theorem). The paper ends its introduction with: 'The operators and the estimates that we have in this paper should have more applications. We hope we can come back to this again.' The machinery rests on two structural inputs — the Codazzi equation for $h$, specific to space-form ambients, and the sign of the sectional curvature, which controls the degenerate ellipticity of $\square$ and forces the Bochner-type integrand to vanish — and the natural question is what survives when either input is weakened.

Problem Statement

Determine how far the Cheng–Yau classification of hypersurfaces with constant scalar curvature extends when its two structural hypotheses are weakened — the space-form assumption on the ambient, which is what makes $h$ a Codazzi tensor and $\square$ self-adjoint, and the nonnegative-sectional-curvature assumption on the hypersurface. Concretely, with $\Phi=h-Hg$ the traceless second fundamental form: (i) compact case in space forms — classify compact immersed hypersurfaces with constant scalar curvature under hypotheses strictly weaker than nonnegative sectional curvature (pinching or bounds on $|h|^2$ or $\sup|\Phi|$, integral curvature conditions, or embeddedness), and locate the sharp boundary between the rigidity conclusions (totally umbilical, i.e. $h=Hg$; isoparametric, i.e. with all principal curvatures constant, having two distinct values) and the known non-spherical immersed examples; (ii) complete noncompact case in $\mathbb{R}^{n+1}$ (and the other space forms) — determine the sharp conditions (convexity-type, $\sup|\Phi|$ bounds, or other weak constraints) under which the generalized-cylinder conclusion of the convex theory persists, and describe the complete non-cylindrical examples on the other side of the boundary; (iii) ambients beyond space forms — extend the operator method to hypersurfaces in locally symmetric ambients (ambients whose curvature tensor is parallel), product ambients and homogeneous ambients, where $h$ satisfies only the generalized Codazzi equation with ambient-curvature terms, establishing the corresponding rigidity statements (including for linear Weingarten hypersurfaces, i.e. those satisfying $R=aH+b$ for constants $a$ and $b$) under correspondingly weak ambient hypotheses. Items (i)–(iii) are three settings of this single question, and the parenthetical lists of hypotheses are alternative weakenings — any one of which may replace the corresponding strict hypothesis — not conditions to be imposed jointly. The target is to determine, for each setting, the exact hypothesis boundary between the rigid models and the non-rigid examples.

References

1. S.-Y. Cheng and S.-T. Yau, 'Hypersurfaces with constant scalar curvature', Mathematische Annalen 225 (1977), 195–204. DOI: 10.1007/BF01425237, https://doi.org/10.1007/BF01425237
2. H. Li, 'Hypersurfaces with constant scalar curvature in space forms', Mathematische Annalen 305 (1996), 665–672. DOI: 10.1007/BF01444243, https://doi.org/10.1007/BF01444243
3. P. Hartman and L. Nirenberg, 'On spherical image maps whose Jacobians do not change sign', American Journal of Mathematics 81 (1959), 901–920. DOI: 10.2307/2372995, https://doi.org/10.2307/2372995
4. J. Simons, 'Minimal varieties in Riemannian manifolds', Annals of Mathematics 88 (1968), 62–105. DOI: 10.2307/1970556, https://doi.org/10.2307/1970556
5. A. Ros, 'Compact hypersurfaces with constant scalar curvature and a congruence theorem', Journal of Differential Geometry 27 (1988), 215–220. DOI: 10.4310/jdg/1214441779, https://doi.org/10.4310/jdg/1214441779
6. T. Hasanis and T. Vlachos, 'Hypersurfaces with constant scalar curvature and constant mean curvature', Annals of Global Analysis and Geometry 13 (1995), 69–77. DOI: 10.1007/BF00774569, https://doi.org/10.1007/BF00774569
7. T. Okayasu, 'On compact hypersurfaces with constant scalar curvature in the Euclidean space', Kodai Mathematical Journal 28 (2005), 577–585. DOI: 10.2996/kmj/1134397770, https://doi.org/10.2996/kmj/1134397770
8. T. Okayasu, 'New examples of complete hypersurfaces with constant positive scalar curvature in the Euclidean space', Journal of the Mathematical Society of Japan 62 (2010), 1137–1166. DOI: 10.2969/jmsj/06241137, https://doi.org/10.2969/jmsj/06241137
9. M. L. Leite, 'Rotational hypersurfaces of space forms with constant scalar curvature', Manuscripta Mathematica 67 (1990), 285–304. DOI: 10.1007/BF02568434, https://doi.org/10.1007/bf02568434
10. H. Alencar, M. do Carmo and W. Santos, 'A gap theorem for hypersurfaces of the sphere with constant scalar curvature one', Commentarii Mathematici Helvetici 77 (2002), 549–562. DOI: 10.1007/s00014-002-8351-1, https://doi.org/10.1007/s00014-002-8351-1
11. L. J. Alías, S. C. García-Martínez and M. Rigoli, 'A maximum principle for hypersurfaces with constant scalar curvature and applications', Annals of Global Analysis and Geometry 41 (2011), 307–320. DOI: 10.1007/s10455-011-9284-y, https://doi.org/10.1007/s10455-011-9284-y
12. A. Brasil, A. G. Colares and O. Palmas, 'Complete hypersurfaces with constant scalar curvature in spheres', Monatshefte für Mathematik 161 (2009), 369–380. DOI: 10.1007/s00605-009-0128-9, https://doi.org/10.1007/s00605-009-0128-9
13. Y. Li, X. Xu and J. Zhou, 'The complete hyper-surfaces with zero scalar curvature in $\mathbb{R}^{n+1}$', Annals of Global Analysis and Geometry 44 (2013), 401–416. DOI: 10.1007/s10455-013-9373-1, https://doi.org/10.1007/s10455-013-9373-1
14. S. Zhang and B. Wu, 'Rigidity theorems of hypersurfaces in locally symmetric Riemannian manifold', Proceedings of the American Mathematical Society 141 (2013), 4015–4025. DOI: 10.1090/s0002-9939-2013-11780-6, https://doi.org/10.1090/s0002-9939-2013-11780-6
15. R. A. Núñez, 'On complete hypersurfaces with constant mean and scalar curvatures in Euclidean spaces', Proceedings of the American Mathematical Society 145 (2017), 2677–2688. DOI: 10.1090/proc/13525, https://doi.org/10.1090/proc/13525
16. L. J. Alías, J. Meléndez and O. Palmas, 'Hypersurfaces with constant scalar curvature in space forms', Differential Geometry and its Applications 58 (2018), 65–82. DOI: 10.1016/j.difgeo.2018.01.001, https://doi.org/10.1016/j.difgeo.2018.01.001
17. E. L. de Lima and H. F. de Lima, 'A gap theorem for constant scalar curvature hypersurfaces', Collectanea Mathematica 73 (2022), 1–11 (online 2020). DOI: 10.1007/s13348-020-00304-3, https://doi.org/10.1007/s13348-020-00304-3
18. J. Ge and Y. Tao, 'Complete hypersurfaces in $\mathbb{R}^{n+1}$ with constant mean and scalar curvature', arXiv:2512.22598 (2025). https://arxiv.org/abs/2512.22598
19. D.-S. Kim, J. R. Kim and Y. H. Kim, 'Cheng–Yau operator and Gauss map of surfaces of revolution', Bulletin of the Malaysian Mathematical Sciences Society 39 (2016), 1319–1327 (online 2015). DOI: 10.1007/s40840-015-0234-x, https://doi.org/10.1007/s40840-015-0234-x
20. L. J. Alías, P. Mastrolia and M. Rigoli, 'Maximum Principles and Geometric Applications', Springer Monographs in Mathematics, Springer, Cham, 2016. ISBN 978-3-319-24335-1. DOI: 10.1007/978-3-319-24337-5, https://doi.org/10.1007/978-3-319-24337-5
21. A. Caminha and H. de Lima, 'A generalized maximum principle for Yau's square operator, with applications to the steady state space', arXiv:math/0610943 (2006). https://arxiv.org/abs/math/0610943