Stability of rapidly rotating supermassive stars

Determine the linear stability of rapidly rotating supermassive stars with adiabatic exponent \(\gamma=4/3\), including the effects of rigid-rotation bifurcation points such as Maclaurin-like sequences where additional neutral modes may develop.

Background

The paper establishes spectral stability for sufficiently slowly and uniformly rotating supermassive stars under axisymmetric perturbations. Its argument relies on the reduced quadratic form and on the positivity generated by a small Rayleigh-stable rotational contribution.

Rapid rotation lies outside the perturbative regime treated by the paper. The authors note that rapidly rotating supermassive stars exist through variational methods, but their stability has not been resolved. In this regime, rigidly rotating sequences may encounter bifurcation points, including Maclaurin-like sequences, at which the linearized operator can acquire additional neutral modes, creating further analytical difficulties.

References

Several questions remain open. These include the asymptotic behavior of the unstable eigenvalues as the viscosity tends to zero, nonlinear secular evolution beyond the turning point, the center dynamics at the critical parameter, and nonaxisymmetric perturbations.

— Secular Instability and the Turning-Point Principle for Rigidly Rotating Viscous Stars  (2609.10098 - Cheng et al., 9 Sep 2026) in Introduction, concluding open-questions paragraph

Several questions remain open. These include the asymptotic behavior of the unstable eigenvalues as the viscosity tends to zero, nonlinear secular evolution beyond the turning point, the center dynamics at the critical parameter, and nonaxisymmetric perturbations.

— Secular Instability and the Turning-Point Principle for Rigidly Rotating Viscous Stars  (2609.10098 - Cheng et al., 9 Sep 2026) in Introduction, concluding open-questions paragraph

Several questions remain open. These include the asymptotic behavior of the unstable eigenvalues as the viscosity tends to zero, nonlinear secular evolution beyond the turning point, the center dynamics at the critical parameter, and nonaxisymmetric perturbations.

— Secular Instability and the Turning-Point Principle for Rigidly Rotating Viscous Stars  (2609.10098 - Cheng et al., 9 Sep 2026) in Introduction, concluding open-questions paragraph

Several questions remain open. These include the asymptotic behavior of the unstable eigenvalues as the viscosity tends to zero, nonlinear secular evolution beyond the turning point, the center dynamics at the critical parameter, and nonaxisymmetric perturbations.

— Secular Instability and the Turning-Point Principle for Rigidly Rotating Viscous Stars  (2609.10098 - Cheng et al., 9 Sep 2026) in Introduction, concluding open-questions paragraph

For rapidly rotating supermassive stars (whose existence for \gamma=4/3 was established by variational methods in ), the stability problem is open; the rigid rotation may reach bifurcation points (e.g. the Maclaurin-like sequences) where the linearized operator develops additional neutral modes.

— Slow uniform rotation reduces the critical adiabatic exponent for the stability of gaseous stars  (2608.20764 - Wang, 21 Aug 2026) in Section 6.4, “Open problems” (subsection “Rapid rotation”)