Rigidity of extrinsic spheres in Einstein spacetimes and the Goddard conjecture

Mohammed Guediri; Mohammed Guediri

doi:10.3934/math.2026385

AIMS Mathematics

2026, Volume 11, Issue 4: 9319-9333. doi: 10.3934/math.2026385

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Research article

Rigidity of extrinsic spheres in Einstein spacetimes and the Goddard conjecture

Mohammed Guediri ^,

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 15 December 2025 Revised: 27 February 2026 Accepted: 10 March 2026 Published: 07 April 2026
MSC : 53C24, 53C25, 53C42, 53E20, 53Z05

Goddard's conjecture asserts that within de Sitter space, a spacelike hypersurface that is complete and has constant mean curvature must exhibit total umbilicity. Although counterexamples show that the conjecture does not hold in full generality, it remains valid under additional geometric assumptions, notably in the compact case. In this paper, we establish a broad extension of Goddard's conjecture to compact, spacelike hypersurfaces in Lorentzian manifolds that admit a timelike conformal vector field. Under a natural integral condition involving the ambient Ricci tensor and a mild assumption of the behavior of the mean curvature along the induced tangential flow, we prove that the hypersurface must be totally umbilical. As an application of our approach, we establish rigidity phenomena in Einstein Lorentzian manifolds and retrieve, as a special instance, the well-known classification of compact, spacelike hypersurfaces with constant mean curvature in de Sitter space as round spheres.
- Lorentzian manifolds,
- Einstein manifolds,
- spacelike hypersurfaces,
- Ricci and scalar curvatures,
- extrinsic spheres,
- Goddard conjecture
Citation: Mohammed Guediri. Rigidity of extrinsic spheres in Einstein spacetimes and the Goddard conjecture[J]. AIMS Mathematics, 2026, 11(4): 9319-9333. doi: 10.3934/math.2026385

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Abstract

Goddard's conjecture asserts that within de Sitter space, a spacelike hypersurface that is complete and has constant mean curvature must exhibit total umbilicity. Although counterexamples show that the conjecture does not hold in full generality, it remains valid under additional geometric assumptions, notably in the compact case. In this paper, we establish a broad extension of Goddard's conjecture to compact, spacelike hypersurfaces in Lorentzian manifolds that admit a timelike conformal vector field. Under a natural integral condition involving the ambient Ricci tensor and a mild assumption of the behavior of the mean curvature along the induced tangential flow, we prove that the hypersurface must be totally umbilical. As an application of our approach, we establish rigidity phenomena in Einstein Lorentzian manifolds and retrieve, as a special instance, the well-known classification of compact, spacelike hypersurfaces with constant mean curvature in de Sitter space as round spheres.

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