de Sitter Extremal Surfaces

• De Sitter extremal surfaces are codimension-2 (and quantum extremal) surfaces that probe the causal structure, horizon entropy, and entanglement-like measures in cosmological settings.
• They are classified into timelike, real, complex, and quantum types, each offering unique insights into analytic continuation, stability, and semiclassical wavefunction calculations.
• Their holographic interpretation links the regulated area of these surfaces to pseudo-entanglement entropy and plays a crucial role in understanding entropy measures and the Page curve in dS contexts.

De Sitter extremal surfaces are codimension-2 (and more generally, "quantum extremal") surfaces in de Sitter (dS) spacetimes that arise in multiple contexts: as classical probes of causal and horizon structure, as geometric regulators of entanglement-like measures in proposed dS holography, and as essential ingredients in semiclassical cosmological wavefunction calculations and quantum gravity consistency conditions. Their paper has spanned the construction and classification of real and complex extremal surfaces, the analysis of uniqueness and stability properties in gravitational theories with positive cosmological constant, the holographic interpretation of their areas as (pseudo-)entanglement entropies, and the adaptation of replica and cosmic brane techniques to cosmological settings.

1. Classical and Quantum Extremal Surfaces in de Sitter Spacetime

Extremal surfaces in de Sitter spacetime come in several distinct classes depending on physical and geometric context:

• Timelike extremal surfaces (codimension-2) stretching from the future boundary $\mathcal{I}^+$ to the past boundary $\mathcal{I}^-$ in the static (or global) patch of dS. Their area, upon suitable regularization, scales with the de Sitter entropy $S_{dS} = \frac{\pi l^2}{G_4}$. These surfaces are realized as solutions of variational problems for the area functional on slices of dS with a fixed Euclidean boundary configuration, and in appropriate limits, pass through the bifurcation surface at $\tau = 1$ (the intersection of future and past horizons) (Narayan, 2017, Narayan, 2019, Fernandes et al., 2019).
• Real extremal surfaces in the Poincaré patch, when anchored on a spatial subregion at $\mathcal{I}^+$ (constant boundary Euclidean time), reduce to boundaries of the past lightcone wedges of those subregions—i.e., they are null surfaces with vanishing area. Thus these surfaces do not encode an entanglement structure analogous to AdS/CFT minimal surfaces (Narayan, 2015).
• Complex extremal surfaces arise as analytic continuations in the extremization problem (typically by taking the bulk time coordinate $\tau$ into the complex plane to preserve real-valued boundary anchoring data). For example, in dS${}_4$ the area of such a complex extremal surface, calculated via a Ryu-Takayanagi-like formula with analytic continuation, is real and negative (Narayan, 2015), and its divergent and finite pieces structurally match holographic entanglement entropy in dual non-unitary Euclidean CFTs (Narayan, 2015).
• Quantum extremal surfaces and islands become crucial in black hole evaporation contexts (e.g., for small Schwarzschild–de Sitter black holes). The generalized entropy functional (the area of the QES plus bulk matter entropy) determines the entropy of radiation, leading to a Page curve consistent with unitary evolution—mirroring results in asymptotically flat space (Goswami et al., 2022).
• No-boundary extremal surfaces are constructed by gluing Lorentzian and Euclidean segments (the "top" being timelike in dS, the "bottom" a spacelike hemisphere), yielding complex-valued area measures. In slow-roll inflationary cosmologies and Schwarzschild–de Sitter spaces with small mass, leading finite corrections in both real and imaginary parts of the area reproduce those of the semiclassical cosmological action, reinforcing a "cosmic brane" interpretation (Goswami et al., 21 Sep 2024).

2. Mathematical Structure and Classification

Area Functional and Extremal Equations: The area functional for a codimension-2 surface in Poincaré/dS is

$$S_{\mathrm{dS}} = \frac{R_{\mathrm{dS}}^{d-1} V_{d-2}}{4G_{d+1}} \int \frac{d\tau}{\tau^{d-1}} \frac{1}{\sqrt{1 - (dx/d\tau)^2}}$$

with boundary anchoring conditions at $\tau \to 0$ (future boundary $\mathcal{I}^+$). The Euler–Lagrange equation leads to a conserved parameter $B$ and, for complex surfaces, requires continuation to $\tau = iT$ for real boundary separation.

Generalized Uniqueness Theorems: For extremal (degenerate) Killing horizons with compact cross-sections in Einstein (and Einstein–Maxwell) gravity with a positive cosmological constant, uniqueness results establish that the only analytic near-horizon geometries are the (charged) Nariai solution (dS${}_2 \times S^2$) for large horizons, the $\mathbb{R}^{1,1} \times S^2$ case at a critical area, or AdS${}_2 \times S^2$ geometries for "cold" solutions with smaller areas. In minimal five-dimensional de Sitter supergravity, pseudo-supersymmetric near-horizon geometries are uniquely fixed to be those of the dS BMPV solution; no black ring-like or other horizon topologies are admitted (Grover et al., 2010, Katona et al., 2023, Katona, 13 Mar 2024).

Replica Construction and Cosmic Branes: The calculation of (pseudo-)Renyi entropies in dS/CFT employs bulk geometries with replica boundary conditions—typically realized by constructing quotient spaces (e.g., branched $S^2$ for $d=3$ or $S^1 \times H^2$ for $d=4$) and requiring bulk regularity (e.g., smooth Euclidean completion or absence of conical singularities off branes). The area of the cosmic brane in the replica geometry matches the pseudo-extremal surface area, which is complex in general (Nanda et al., 2 Sep 2025).

3. Holography, Pseudo-Entanglement, and Entropy Measures

In dS/CFT, the semiclassical de Sitter wavefunction is identified with the partition function of a dual Euclidean CFT: $Z_{\text{CFT}} = \Psi_{\text{dS}}$. The area of the complex extremal surfaces in the dS bulk, computed via analytic continuation from AdS, structurally mimics the Ryu-Takayanagi formula. For even dimensional boundaries, the universal logarithmic divergence in the area corresponds precisely (including numerical coefficients) to the anomaly (e.g., a-type central charge) as in standard CFTs (Narayan, 2015).

Because the modular Hamiltonian and density matrices appropriate to Lorentzian (time-dependent) dS are not positive-definite (and the dual CFT is ghost-like), these entropy-like quantities are generally pseudo-entropies—complex-valued and built from transition matrix elements rather than strictly from Hermitian density matrices (Narayan, 2023, Nanda et al., 2 Sep 2025).

Key relations satisfied or modified in the dS context:

• Vanishing mutual information for disconnected subregions anchored on $\mathcal{I}^+$ and their conjugates on $\mathcal{I}^-$ (for top–bottom symmetric surfaces) (Narayan, 2020).
• Saturated strong subadditivity for adjacent intervals.
• Replica pseudo-entropy: With bulk action $I_n$ on the $n$-replica geometry, $S_n = (i(I_n-nI_1))/(1-n)$, where the real part of the $n \to 1$ limit is often half the de Sitter entropy for maximal subregions (Nanda et al., 2 Sep 2025).

4. Extremal Surfaces in Black Hole and Cosmological Settings

• Extremal and near-extremal horizons: For static or rotating black holes in dS, the event and cosmological horizons admit unique extremal (Nariai in the charged case, or BMPV in some 5D supergravity theories) near-horizon geometries, and indeed, any analytic perturbation away from these is ruled out by uniqueness theorems (Katona et al., 2023, Katona, 13 Mar 2024).
• Decay and stability: The probability for decay of extremal charged dS black holes via emission of charged shells is $P \sim \exp(\Delta S_b)$, where $\Delta S_b$ is the change in Bekenstein-Hawking entropy. The "Festina Lente" bound $m_s^2 \gg M_p H q$ is recognized as precisely the suppression of unsuppressed decay ($\Delta S_b \leq -1$) in the Nariai and probe limit. Backreaction can lead to "violations" of this bound, but unless the near-horizon energy $m_s$ becomes complex, the geometry does not evolve to a big crunch, and the extremal surface geometry remains stable (Aalsma et al., 2023).
• Quantum extremal surfaces and the Page curve: For small Schwarzschild–de Sitter black holes in a frozen dS background, quantum extremal surfaces ("islands") emerge near the black hole horizon at late times, leading to the saturation of the generalized entropy and a Page curve indicative of unitary evaporation—paralleling the situation in flat space (Goswami et al., 2022).

5. Analytic Continuation, Space-Time Rotation, and Time-Entanglement

The analytic continuation from AdS to dS (e.g., $L \to -il$, $r \to -i\tau$) realizes a spacetime "rotation" that maps minimal surfaces on constant–time AdS slices to extremal surfaces on constant boundary Euclidean slices in dS. The resulting surfaces no longer return to the original boundary but extend from I$^+$ to I$^-$, displaying a deep geometric and holographic connection between space-like and time-like entanglement structures (Narayan, 2023).

This leads to a framework of "time-entanglement" or "pseudo-entanglement" wedges in dS:

• Future–past thermofield double states: These are entangled states pairing configurations at I$^+$ and I$^-$, motivated by the bulk existence of future–past extremal surfaces and analogies to Hartman–Maldacena/Rindler walled surfaces in AdS. Such states have positive entanglement entropies, despite arising from a ghost-like (nonunitary) theory (Narayan, 2017, Narayan, 2019, Narayan, 2022).
• Reduced time evolution operators and weak values: These quantum mechanical constructions generalize the notion of partial trace to "tracing over" temporal segments, leading to complex-valued (pseudo-)entropies and reflecting the analytic properties of transition amplitudes rather than eigenvalues of Hermitian operators (Narayan, 2023, Nanda et al., 2 Sep 2025).

6. Extensions to Cosmological Spacetimes and Holographic Implications

• No-boundary surfaces in slow-roll inflation and perturbed cosmologies: The calculation of extremal surface areas in slow-roll backgrounds indicates that the real finite part exactly matches the semiclassical weight of the wavefunction of the universe; this is interpreted as the creation amplitude for "cosmic branes" along the extremal surface. These results extend to dS$_3$, slow-roll FRW, and Schwarzschild–de Sitter with small mass, suggesting broad applicability in semiclassical quantum cosmology (Goswami et al., 21 Sep 2024).
• Boundary Renyi entropies and replica smoothing: Explicit replica geometries for computing boundary Renyi entropies in dS/CFT have been constructed—smoothly for n integer via branched covers or hyperbolic slicing—correctly reproducing pseudo-Renyi entropies in both analytic and quotient variables. These methods confirm and extend earlier Lewkowycz–Maldacena-style arguments by tying the entire calculation to smooth geometry with regularity criteria corresponding to cosmic brane prescriptions (Nanda et al., 2 Sep 2025).
• Entropy inequalities and nonunitarity: Real and imaginary parts of mutual and tripartite information, computed on analytic continuations of extremal surface areas, respect generalized monotonicity and purity, even though the entropies are complex—reflecting the fundamental nonunitary or "ghost-like" nature of the dual field theory (Narayan, 2023).

The paper of de Sitter extremal surfaces thereby bridges classical general relativity, semiclassical quantum gravity, holography in spacetimes with cosmological horizons, and the emergence of quantum information-theoretic quantities in cosmological settings, with foundational implications for microscopic models of dS entropy and the structure of "pseudo-entanglement" in nonunitary dual theories.