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Phys-74 · physics · mathematics · Physics

The nonmaximal anti-de Sitter enclosing-area inequality

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For smooth complete one-ended orientable three-dimensional asymptotically hyperbolic exteriors with cosmological constant $-3$, compact smooth boundary, the stated weighted-integrable DEC and $O_{2,\alpha }/O_{1,\alpha }$ hyperbolic-norm decay of rate $3/2<\tau <3$, future-timelike metric mass covector and $H+\operatorname{tr}_S$ $K\leq 0$ on every boundary component, is $m_AH\geq (r_A+r_A^3)/2$ for $r_A=\operatorname{sqrt}(A_min/(4\pi ))$? No maximality, compact-topology, boundary-genus or outermostness restriction is imposed; ask for the stated connected nondegenerate original-data Schwarzschild–AdS equality rigidity.

Short English statement

For smooth complete one-ended orientable three-dimensional asymptotically hyperbolic exteriors with cosmological constant $-3$, compact smooth boundary, the stated weighted-integrable DEC and $O_{2,\alpha }/O_{1,\alpha }$ hyperbolic-norm decay of rate $3/2<\tau <3$, future-timelike metric mass covector and $H+\operatorname{tr}_S$ $K\leq 0$ on every boundary component, is $m_AH\geq (r_A+r_A^3)/2$ for $r_A=\operatorname{sqrt}(A_min/(4\pi ))$? No maximality, compact-topology, boundary-genus or outermostness restriction is imposed; ask for the stated connected nondegenerate original-data Schwarzschild–AdS equality rigidity.

Sources

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Phys-74-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-74

Only small conformal perturbation branches of fixed Schwarzschild–AdS backgrounds and TT seeds are covered; general DEC nonmaximal exteriors and full original-data rigidity are outside this theorem. The theorem assumes tr_g K=0 and gives the numerical bound. It leaves the nonmaximal data class and general original-data equality classification outside. The corresponding spherical anti-de Sitter bound and original-data equality rigidity are claimed without maximality for the specified timelike four-flux class. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The nonmaximal anti-de Sitter enclosing-area inequality

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Small conformal changes of Schwarzschild--anti-de Sitter data within the maximal vacuum class are claimed to satisfy the horizon-area bound, with radial equality and nonradial strictness; Maximal asymptotically hyperbolic three-dimensional data are claimed to satisfy the sharp enclosing-area bound under a future-timelike mass covector and the stated energy and decay assumptions; The corresponding spherical anti-de Sitter bound and original-data equality rigidity are claimed without maximality for the specified timelike four-flux class. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
Fix $m_0>0$ and the Schwarzschild--anti-de Sitter background $g_0=ds^2+r(s)^2\sigma$ on $[0,\infty)\times\mathbb S^2$, where $r(0)=a>0$, $1+a^2-2m_0/a=0$, and $(r')^2=1+r^2-2m_0/r$, $r'>0$ for $s>0$. Choose a fixed smooth symmetric transverse-traceless seed $q$ with $\operatorname{tr}_{g_0}q=\operatorname{div}_{g_0}q=0$, $q\in C^{1,\alpha}_\tau$ for $0<\alpha<1$, $3/2<\tau<3$, using hyperbolic tensor decay. Suppose a smooth positive branch $\phi_\epsilon$ on $0\le\epsilon<\epsilon_*$ satisfies \[ \Delta_{g_0}\phi_\epsilon=\tfrac34(\phi_\epsilon^5-\phi_\epsilon) -\tfrac{\epsilon^2}8|q|^2\phi_\epsilon^{-7},\qquad \partial_s\phi_\epsilon|_{s=0}=\tfrac\epsilon4q_{ss}\phi_\epsilon^{-3}, \] \[ \phi_0=1,\qquad\phi_\epsilon-1\in C^{2,\alpha}_\tau,\qquad \|\phi_\epsilon-1\|_{C^{2,\alpha}_\tau}\to0. \] Set $g_\epsilon=\phi_\epsilon^4g_0$, $K_\epsilon=\epsilon\phi_\epsilon^{-2}q$, and $A_\epsilon=|\{0\}\times\mathbb S^2|_{g_\epsilon}$. The metric mass uses the four hyperbolic fluxes $p_a=(16\pi)^{-1}\lim_R\int_{r=R}[V_a(\operatorname{div}_be-d\operatorname{tr}_be)+(\operatorname{tr}_be)dV_a-e(\nabla_bV_a,\cdot)](\nu_b)dA_b$, with $b=(1+r^2)^{-1}dr^2+r^2\sigma$, $e=g_\epsilon-b$, $V_0=\sqrt{1+r^2}$, $V_i=r\omega_i$. Assume finite fluxes and a future-timelike covector for small positive $\epsilon$, and let $m_{\mathrm{AH}}(\epsilon)=\sqrt{p_0^2-\sum_ip_i^2}$. There exists $0<\epsilon_0\le\epsilon_*$ with \[ m_{\mathrm{AH}}(\epsilon)\ge\sqrt{A_\epsilon/(16\pi)}\left(1+\frac{A_\epsilon}{4\pi}\right) \quad(0\le\epsilon<\epsilon_0). \] For sufficiently small $\epsilon>0$, equality holds for rotation-invariant seeds and the inequality is strict for every other seed. This local branch theorem uses the prescribed boundary's own area; identifying it with an enclosing infimum additionally requires outer area minimization and outermostness. The constructed boundary is a future MOTS, and the vacuum maximal constraints hold.
Result 2.
Let $(\Omega^3,g,K)$ be smooth and orientable, complete including nonempty compact smooth boundary $S$, with one spherical hyperbolic end outside a compact set. Set $b=(1+r^2)^{-1}dr^2+r^2\sigma$, $\eta=\operatorname{arsinh}r$, and assume $g-b\in C^{2,\alpha}_\tau$, $K\in C^{1,\alpha}_\tau$ for $0<\alpha<1$, $3/2<\tau<3$. These weighted spaces control $e^{\tau\eta}$ times background covariant derivatives and local Holder seminorms on hyperbolic unit balls. With $e=g-b$, $V_0=\sqrt{1+r^2}$ and $V_i=r\omega_i$, define \[ p_a=\frac1{16\pi}\lim_{R\to\infty}\int_{r=R}\mathbb U_b(V_a,e)(\nu_b)dA_b,\quad \mathbb U_b(V,e)=V(\operatorname{div}_be-d\operatorname{tr}_be) +(\operatorname{tr}_be)dV-e(\nabla_bV,\cdot). \] Assume these fluxes are finite and $p_0>\sqrt{\sum_{i=1}^3p_i^2}$; put $m_{\mathrm{AH}}=\sqrt{p_0^2-\sum_{i=1}^3p_i^2}$. Choose a connected smooth end-side region $D\subset\Omega$, closed in $\Omega$, containing the distant end and having its manifold interior in $\operatorname{int}\Omega$. A full cut consists of all of its intrinsic boundary $\Gamma=\partial D$; require this boundary to be compact, smoothly embedded and two-sided. All components and portions coinciding with $S=\partial\Omega$ count. Put $a_g(S)=\inf_\Gamma\operatorname{Area}_g(\Gamma)$. For cosmological constant $-3$, impose \[ \operatorname{tr}_gK=0,\quad16\pi\mu=R_g+6-|K|^2,\quad8\pi J=\operatorname{div}_gK,\quad\mu\ge|J|, \qquad \int_{\mathrm{end}}V_0(\mu+|J|+|K|^2)dV_g<\infty. \] If $H+\operatorname{tr}_SK\le0$ on every boundary component, then \[ m_{\mathrm{AH}}\ge\frac12(r_A+r_A^3),\qquad r_A=\sqrt{a_g(S)/(4\pi)}. \] The coefficient is attained by Schwarzschild--anti-de Sitter exteriors of arbitrary positive mass on their $K=0$ slices at its minimal horizon. This paper's numerical theorem does not classify all equality data.
Result 3.
Let $(\Omega^3,g,K)$ be smooth and orientable, complete including nonempty compact smooth boundary $S$, with one spherical hyperbolic end outside a compact set. Set $b=(1+r^2)^{-1}dr^2+r^2\sigma$, $\eta=\operatorname{arsinh}r$, and assume $g-b\in C^{2,\alpha}_\tau$, $K\in C^{1,\alpha}_\tau$ for $0<\alpha<1$, $3/2<\tau<3$. These weighted spaces control $e^{\tau\eta}$ times background covariant derivatives and local Holder seminorms on hyperbolic unit balls. With $e=g-b$, $V_0=\sqrt{1+r^2}$ and $V_i=r\omega_i$, define \[ p_a=\frac1{16\pi}\lim_{R\to\infty}\int_{r=R}\mathbb U_b(V_a,e)(\nu_b)dA_b,\quad \mathbb U_b(V,e)=V(\operatorname{div}_be-d\operatorname{tr}_be) +(\operatorname{tr}_be)dV-e(\nabla_bV,\cdot). \] Assume these fluxes are finite and $p_0>\sqrt{\sum_{i=1}^3p_i^2}$; put $m_{\mathrm{AH}}=\sqrt{p_0^2-\sum_{i=1}^3p_i^2}$. Choose a connected smooth end-side region $D\subset\Omega$, closed in $\Omega$, containing the distant end and having its manifold interior in $\operatorname{int}\Omega$. A full cut consists of all of its intrinsic boundary $\Gamma=\partial D$; require this boundary to be compact, smoothly embedded and two-sided. All components and portions coinciding with $S=\partial\Omega$ count. Put $a_g(S)=\inf_\Gamma\operatorname{Area}_g(\Gamma)$. For cosmological constant $-3$, now permit arbitrary $\operatorname{tr}_gK$ and impose \[ 16\pi\mu=R_g+6+(\operatorname{tr}_gK)^2-|K|^2,\quad 8\pi J=\operatorname{div}_g(K-(\operatorname{tr}_gK)g),\quad\mu\ge|J|, \] with $\int_\Omega V_0(\mu+|J|+|K|^2)dV_g<\infty$, extending the positive weight smoothly over the compact part. If $H+\operatorname{tr}_SK\le0$ on every component, then \[ m_{\mathrm{AH}}\ge\sqrt{a_g(S)/(16\pi)}\left(1+\frac{a_g(S)}{4\pi}\right)=\tfrac12(r_A+r_A^3). \] For the connected future MOTS subclass, require outer area minimization and absence of any wholly exterior smooth compact embedded two-sided full enclosing surface with $\theta_+\le0$, including disconnected surfaces. Equality then holds exactly when the original $g,K$ arise from a proper smooth spacelike embedding into maximally extended Schwarzschild--anti-de Sitter with parameter $M=m_{\mathrm{AH}}$, interior in one exterior, boundary a full future-horizon section or bifurcation sphere, and end reaching that exterior's conformal infinity. The computed mass match is part of the condition, and disconnected equality is not classified.

Outline of the argument

Weighted expansions of the actual conformal solution are compared with CMC-sphere Hawking-mass jets. A quadratic inequality identifies its kernel, fourth-order analysis handles that kernel, and radial seed calculations classify equality without requiring an outer-minimizing hypothesis.

Static graph identities and finite-time area transfer reduce the trapped boundary to a minimal one. Electrostatic stress permits an asymptotically flat attachment with controlled mass, and the flat Riemannian comparison returns the Schwarzschild--anti-de Sitter bound.

A saturating conserved stress reduces the end to a flat comparison problem. Rank-one trace regularity and floor separation solve the nonmaximal deformation while protecting enclosure area; equality of the mass identities reconstructs the original Schwarzschild--anti-de Sitter slice.

Formal verification

For family 260, the only selected family entry is the CKS end-replacement construction and Schwarzschild equality examples. A connected-boundary inequality is conditional on the flat Penrose input; the general Bondi or spacetime inequalities are not formalized there. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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