A geometric–analytic proof of the reverse isoperimetric inequality in Einstein gravity
Abstract
We develop a geometric–analytic proof of the reverse isoperimetric inequality for compact AdS black hole horizons in Einstein gravity. At fixed thermodynamic volume, a York slice/Jacobi variation of the gravitational action shows that the spherical Schwarzschild–AdS solution is a strict local entropy maximum against generic non-spherical deformations. The finite global extension is formulated within the hypotheses of the Sherif–Dunsby (SD) rigidity theorem: compactness and spherical topology, the required intrinsic Ricci decomposition Ric h = α e ⊗ e + β N with α ≠ β , a proper conformal transformation of the hypersurface metric given by h ˜ a b = e 2 φ h a b that preserves its intrinsic scalar curvature, R ˜ ≡ R [ h ˜ ] = R [ h ] ≡ R > 0 , and negative sheet expansion. Compactness and S 3 topology are built into the class considered here. In the static spherical sector such as Schwarzschild–AdS and RN–AdS, the SD Ricci form is naturally realized while it is an additional restriction for Kerr–AdS case. For the S 3 topology, a positive Yamabe type conformal class exists for which the Yamabe theorem provides a representative of positive and constant scalar-curvature. We further restrict to the scalar-curvature preserving conformal transformation required by the SD theorem. The negative sheet expansion is obtained using gravitational focusing. Within this admissible class, a non-round Kerr–AdS horizon cannot maximize the entropy, while fixed thermodynamic volume selects the corresponding round Schwarzschild–AdS comparator. The analysis thus establishes a generic local entropy maximum and a finite global result within the stated geometric rigidity class, identifying the role of gravity in reversing the ordinary isoperimetric problem.
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BibTeX
@article{kumar2026geometricanalytic,
title = {{A geometric–analytic proof of the reverse isoperimetric inequality in Einstein gravity}},
author = {Naman Kumar},
journal = {Physics Letters B},
year = {2026},
month = oct,
doi = {10.1016/j.physletb.2026.141111},
url = {https://linkinghub.elsevier.com/retrieve/pii/S0370269326009627},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A geometric–analytic proof of the reverse isoperimetric inequality in Einstein gravity
AU - Naman Kumar
PY - 2026
DA - 2026/10/01
JO - Physics Letters B
DO - 10.1016/j.physletb.2026.141111
UR - https://linkinghub.elsevier.com/retrieve/pii/S0370269326009627
AB - We develop a geometric–analytic proof of the reverse isoperimetric inequality for compact AdS black hole horizons in Einstein gravity. At fixed thermodynamic volume, a York slice/Jacobi variation of the gravitational action shows that the spherical Schwarzschild–AdS solution is a strict local entropy maximum against generic non-spherical deformations. The finite global extension is formulated within the hypotheses of the Sherif–Dunsby (SD) rigidity theorem: compactness and spherical topology, the required intrinsic Ricci decomposition Ric h = α e ⊗ e + β N with α ≠ β , a proper conformal transformation of the hypersurface metric given by h ˜ a b = e 2 φ h a b that preserves its intrinsic scalar curvature, R ˜ ≡ R [ h ˜ ] = R [ h ] ≡ R > 0 , and negative sheet expansion. Compactness and S 3 topology are built into the class considered here. In the static spherical sector such as Schwarzschild–AdS and RN–AdS, the SD Ricci form is naturally realized while it is an additional restriction for Kerr–AdS case. For the S 3 topology, a positive Yamabe type conformal class exists for which the Yamabe theorem provides a representative of positive and constant scalar-curvature. We further restrict to the scalar-curvature preserving conformal transformation required by the SD theorem. The negative sheet expansion is obtained using gravitational focusing. Within this admissible class, a non-round Kerr–AdS horizon cannot maximize the entropy, while fixed thermodynamic volume selects the corresponding round Schwarzschild–AdS comparator. The analysis thus establishes a generic local entropy maximum and a finite global result within the stated geometric rigidity class, identifying the role of gravity in reversing the ordinary isoperimetric problem.
SN - 0370-2693
SN - 1873-2445
ER -