Existence of bulk solutions realizing generalized power-law boundary conditions

Establish the existence of an actual bulk solution of f(R) gravity in the enlarged solution space associated with the generalized power-law boundary condition, together with a boundary surface satisfying the specified extrinsic-curvature structure and the corresponding variation constraint.

Background

The paper constructs generalized Gibbons-Hawking-York boundary terms for f(R) gravity under power-law conditions of the form δR|∂M = αKrδK. These conditions formally enlarge the class of admissible variations beyond the usual δR|∂M = 0 case and the Madsen–Barrow condition. Their compatibility with the Gauss–Codazzi equation requires the boundary extrinsic curvature to satisfy a prescribed decomposition and an additional constraint on its variations.

The authors explicitly caution that, for generic values of α, the resulting larger solution space may be empty. Consequently, an unresolved problem is to exhibit a bulk solution and a compatible boundary satisfying the geometric requirements used in the construction, thereby demonstrating that the generalized variational principle is realized by an actual gravitational configuration rather than only formally.

References

Strictly speaking, the larger space of solutions allowed by, say, Eq. (\ref{eq3: power law condition for the variation of the scalar curvature on the boundary}) could be empty for $\alpha \neq 0, -2\epsilon \frac{D+1}{D}$. It would be important, therefore, to find an actual bulk solution lying in this larger space, along with a boundary surface satisfying Eqs. (\ref{eq3: structure of the extrinsic curvature to allow the power-law condition}) and (\ref{eq3: condition to realize power-law boundary condition from GC equation}).

— On more general Gibbons-Hawking-York boundary terms for f(R) gravity  (2609.37692 - Scali et al., 29 Sep 2026) in Conclusions