Continuity of one-brane and two-brane solution branches

Determine whether the one-brane and two-brane branches of extra-dimensional solutions form continuous families for the fixed quadratic-curvature gravity action, and characterize the stability conditions near a possible bifurcation point connecting the branches.

Background

The paper presents numerical examples suggesting that varying an auxiliary parameter, specifically the initial derivative of the Ricci scalar, can change the number of regular zeros of the extra-dimensional radius function r(u). This change distinguishes a two-brane geometry from a one-brane geometry and indicates the possible existence of distinct topological branches for the same action and physical parameters.

The examples are not shown to satisfy the radion stability, ghost-freedom, and no-tachyon conditions. The unresolved issue is whether each topological branch independently forms a continuous family and how the stability conditions behave at a bifurcation point where the branches may be connected.

References

We conjecture that each of these branches forms a continuous family in the sense discussed above. A detailed analysis of the transition, including the behaviour of the stability conditions extr, Kpos, and Vpppos near a possible bifurcation point, is beyond the scope of the present work.

extr:

VE′(β=0)=0.V_E'(\beta=0)=0.

Kpos:

KE(β=0)=ΔK>0,K_E(\beta=0)=\Delta_K>0,

Vpppos:

VE′′(β=0)=ΔV>0,V_E''(\beta=0)=\Delta_V>0,

— Continuum Landscape of stable extra-dimensional metrics  (2610.08006 - Rubin, 6 Oct 2026) in Section 6, “Remark on a topological transition”

Several questions remain open. A complete spectral analysis including the massive modes present in theories with quadratic curvature invariants would go beyond the radion sector considered here. The behaviour of the continuum near arbitrary stable solution, and the stability conditions at the bifurcation point of the topological transition, also deserve a dedicated study.

— Continuum Landscape of stable extra-dimensional metrics  (2610.08006 - Rubin, 6 Oct 2026) in Section 7, “Conclusion”

Several questions remain open. A complete spectral analysis including the massive modes present in theories with quadratic curvature invariants would go beyond the radion sector considered here. The behaviour of the continuum near arbitrary stable solution, and the stability conditions at the bifurcation point of the topological transition, also deserve a dedicated study.

— Continuum Landscape of stable extra-dimensional metrics  (2610.08006 - Rubin, 6 Oct 2026) in Section 7, “Conclusion”