Island and Hartman–Maldacena Surfaces in Holography

• Island surfaces are extremal codimension-two surfaces that arise in holographic entanglement entropy, characterizing phase transitions via the Page curve.
• The Hartman–Maldacena surface is a time-dependent, connected extremal surface traversing black hole wormholes, responsible for early-time linear entropy growth.
• Their competition, governed by generalized entropy extremization and homology constraints, offers insights into quantum unitarity restoration in gravitational settings.

An island surface is a class of extremal codimension-two surface arising in holographic entanglement entropy prescriptions for gravitational systems coupled to non-gravitating baths, notably in doubly holographic setups. The Hartman–Maldacena (HM) surface is the connected extremal surface traversing the black hole wormhole geometry, responsible for linear-in-time entropy growth in eternal AdS black holes. Together, these surfaces diagnose phase transitions in entanglement entropy—the so-called Page curve—by competing as candidate minimizers of generalized entropy functionals. The paper of their properties, phase transitions, and generalizations to non-static, higher-curvature, and cosmological backgrounds has illuminated mechanisms of unitarity restoration and resolved issues in the information paradox.

1. Definitions and Geometric Construction

Island surfaces are minimal—more generally, quantum extremal—codimension-two surfaces whose boundaries may partially lie on dynamical branes or defects ("Planck branes" or ETW branes) embedded in holographic bulk spacetimes. Hartman–Maldacena surfaces are time-dependent connected extremal surfaces that cross black hole horizons, characteristically responsible for early-time entropy growth. In AdS/CFT, the HM surface stretches between the two asymptotic boundaries of the eternal black hole, parameterized on a time slice $t$ by the Ryu–Takayanagi prescription (Neuenfeld, 2021).

In double holography, further generalizations involve braneworlds in higher-dimensional gravity theories. Bath regions, radiating to infinity, are glued to gravitational interiors via branes (codimension-one or codimension-two) (Chou et al., 2021, Hu et al., 2022). For $dS_2$ static patch or $AdS_2$ eternal black hole, surfaces are minimal curves in braneworld embeddings in $AdS_3$ bulk coordinates, with details fixed by matching conditions (e.g., $$z_{dS_2}(\omega^\pm) = (\epsilon/2)|1-\omega^+\omega^-|$$) (Jiang et al., 12 Feb 2025).

2. Generalized Entropy and Extremization

The key quantity is the generalized entropy,

$$S_{\mathrm{gen}}[\Gamma;I]=\frac{\mathrm{Area}(\Gamma)}{4G_N} + S_{\text{matter}}(I\cup R)$$

where $\Gamma$ is an extremal surface (either HM or island), and $I$ is an "island" region on the brane (or black hole interior), $R$ the radiation region in the bath (Neuenfeld, 2021). Extremization is carried over all candidate surfaces and island endpoints, subject to homology constraints reflecting the boundary conditions for each physical setup. For codimension-two branes, the quantum extremal surface equation is

$K^A(\partial I) = 0$

with $K^A$ the bulk extrinsic curvature of the surface (Hu et al., 2022). In higher-derivative theories (e.g., Gauss–Bonnet gravity), contributions from intrinsic Ricci scalar and extrinsic curvature of the surface boundary appear (Li et al., 2023).

3. Phase Structure and Critical Times

Competition between HM and island surfaces gives rise to sharp phase transitions in entanglement entropy,

$$S(t) = \min\left\{ S_{\mathrm{HM}}(t), S_{\mathrm{island}}\right\}$$

marking the so-called Page time $t_{\text{Page}}$, when the area functional of the HM saddle equals that of the island saddle (Neuenfeld, 2021, Jeong et al., 2023). The HM surface yields early-time linear growth (e.g., $S_{\mathrm{HM}}(t) = S_{\mathrm{HM}}(0) + v_E t$), while the island surface saturates at a finite value proportional to the black hole horizon area (Jeong et al., 2023). In many models the HM surface ceases to exist beyond a finite critical time $t_{\max}$, but nearly always $t_{\max}>t_{\text{Page}}$, so no information loss occurs (Hu et al., 2022, Li et al., 2023).

More intricate phase structures emerge. In $dS_2$ double holography, a third mixed saddle $S_2$ appears, involving geodesics connecting both baths and island endpoints, leading to three entropy functions $S_1,S_2,S_3$ and corresponding regimes; a generalized mutual information $I_{\text{gen}}$ efficiently diagnoses these transitions (Jiang et al., 12 Feb 2025).

4. Extensions: Brane Codimension, Higher-Derivatives, Rotation, and Cosmology

Islands have been constructed not only on codimension-one branes (braneworld holography) but also on codimension-two branes, which generate conical singularities and modify graviton spectra—massless modes are generically absent, but localized massive modes support static islands (Hu et al., 2022, Li et al., 2023). In higher-derivative gravity (Gauss–Bonnet), all causal GB couplings yield viable Page curves; $t_{\text{Page}}$ increases with both GB coupling and brane tension (Li et al., 2023).

Rotating cylindrical black holes exhibit three distinct island regimes, separated by two critical brane-angles ($\mu_c,\mu_e$) or degree-of-freedom ratios, with rotation ("boost parameter" $\Omega$) pushing islands away from the horizon and enlarging the entropy belt (Billiato et al., 3 Dec 2025). The universality of the island prescription extends to these dynamical and stationary backgrounds.

In cosmological (de Sitter) settings, analogous transitions occur. Island-like surfaces induce an entanglement wedge transition at half the horizon size; a "de Sitter version" of the HM prescription describes time-evolving connected extremal surfaces, saturating maximally at late times (Shaghoulian, 2021).

5. Homology and Boundary Constraints

The precise surface competition depends strongly on underlying homology constraints, which differ whether entropy is computed in a boundary BCFT or brane gravity perspective. In the boundary perspective, RT surfaces may end on the ETW brane, while in the brane (semi-classical gravity) perspective, only surfaces anchored on the asymptotic boundary or interior regions are allowed. This distinction underpins the "island rule" and realizes the mechanism by which coarse-grained and fine-grained entropies differ (Neuenfeld, 2021).

Toy models exhibiting island and wormhole effects reinforce that entanglement entropy calculations and operator transformations in double holography are sensitive to such homology and subsystem enlargement rules.

6. Negative Geodesic Lengths and Novel Saddles

A unique feature emerges in closed $dS_2$ models and their $AdS_2$ analogs: extremal surfaces whose bulk geodesics traverse regions inside the horizon acquire formally negative length contributions (due to signature flips in timelike-to-spacelike coordinates). Without further prescription, this would permit unphysical entropies (e.g., $-\infty$ for coincident multiple such geodesics). The resolution is to count each distinct geodesic only once; this ensures bounded entropy and correct phase structure (Jiang et al., 12 Feb 2025).

A plausible implication is that in gravitational path integrals, only non-degenerate extremal surface configurations with distinct endpoints should be summed over, reflecting physical causality and preventing overcounting of wormhole-like contributions.

7. Universality and Consistency Across Models

Across codimension, static/dynamic backgrounds, higher-curvature corrections, and even massless gravity, the island prescription is robust: competing extremal surfaces reproduce the unitary Page curve, safeguarding late-time entropy saturation and unitarity (Li et al., 2023, Li et al., 2023). The HM surface is universally identified with the linear growth regime, while the island governs saturation. Subtleties—such as finite HM lifetimes, negative length contributions, and nontrivial parameter windows for island existence—do not disrupt the core mechanism; indeed, they illustrate the breadth and adaptability of the prescription in modern gravitational entropy calculations.