Conformal Continuation Methods

• Conformal continuation is a family of methods that uses conformal maps to extend analytic, geometric, or categorical data in various mathematical and physical settings.
• It finds applications in complex analysis, celestial conformal field theory, AdS geometry, and numerical spectral methods, each employing unique continuation strategies.
• The approach leverages principles such as boundary regularity, monodromy projection, and conformal invariance to solve extension problems across diverse domains.

Searching arXiv for recent and foundational uses of “conformal continuation” across fields. Across the literature surveyed here, conformal continuation is not a single standardized construction but a family of procedures in which conformal structure is used to extend, transport, or uniquely determine analytic, geometric, or categorical data. In celestial conformal field theory it denotes the analytic continuation of Appell $F_1$ between different OPE regions together with monodromy projection (Fan, 2023). In complex analysis it denotes continuous extension of inverse Riemann maps to the boundary, or conformal extension of maps from subsets of Riemann surfaces (Qiu, 2013, Gauthier et al., 2014). In asymptotically anti-de Sitter geometry it denotes unique continuation from the conformal boundary (McGill et al., 2020, Guisset, 17 Jan 2025). In numerical and spectral settings it denotes analytic continuation implemented by conformal maps that transport a domain of analyticity to the unit disk (Abbasbandy et al., 2011, Bergamaschi et al., 2023). This suggests a family resemblance centered on conformally natural extension rather than a single doctrine.

1. Semantic range and recurrent structure

A useful way to organize the term is by the type of object being continued. In the sources considered here, the relevant object may be a boundary value, a special function, a conformal block, a Green function, a vacuum solution near conformal infinity, or even a gauge group.

Context	Meaning of conformal continuation	Representative sources
Complex analysis	Continuous or conformal extension to a boundary or neighborhood	(Qiu, 2013, Gauthier et al., 2014, Luo et al., 2020)
Celestial and chiral CFT	Analytic continuation of conformal blocks across channels or configuration-space domains	(Fan, 2023, Moriwaki, 22 Jun 2026, Kravchuk et al., 2018, Dotsenko, 2016)
Lorentzian PDE and AdS geometry	Unique continuation from a conformal boundary	(McGill et al., 2020, Guisset et al., 2023, Guisset, 17 Jan 2025, Chrusciel et al., 2011)
Geometric analysis	Unique continuation in conformally adapted gauges	(Lassas et al., 2019, Vita, 2024)
Numerical and spectral analysis	Conformal-map-based analytic continuation to a unit disk or slit domain	(Abbasbandy et al., 2011, Bergamaschi et al., 2023, Ghisa et al., 2015)
Gauge-theoretic gravity	Continuation of the conformal group $SO(2,4)\to SO(1,5)$	(Apolo et al., 2016)

The common structural feature is the use of a conformally natural representation in which a continuation problem becomes tractable. Depending on the setting, the operative mechanism is monodromy, boundary accessibility, Carleman estimates, conformal covariance of differential operators, or explicit conformal maps.

2. Boundary extension and conformal maps in complex analysis

In planar conformal mapping theory, a foundational use of the term concerns boundary extension. For a simply connected proper domain $G$, Qiu defines accessible boundary points $\partial_a G$, non-accessible boundary points $\partial_n G$, and semi-unreachable points $\partial_{sn}G$. The central theorem states that for a Riemann map $\psi:D\to G$ with inverse $\phi=\psi^{-1}$, one has

$$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$

so continuous boundary extension is characterized by the absence of semi-unreachable points (Qiu, 2013). In this usage, the paper explicitly distinguishes conformal continuation from analytic continuation: the issue is continuity up to $\partial G$, not holomorphic extension across $SO(2,4)\to SO(1,5)$0 (Qiu, 2013).

This criterion yields a short proof of the Osgood conjecture in the Jordan-domain case. If $SO(2,4)\to SO(1,5)$1 is a Jordan domain, then $SO(2,4)\to SO(1,5)$2 extends to a homeomorphism $SO(2,4)\to SO(1,5)$3, because $SO(2,4)\to SO(1,5)$4 and injectivity on the boundary circle follows from a crosscut argument (Qiu, 2013). The same theme reappears in a much broader setting for infinitely connected planar domains. For a conformal homeomorphism $SO(2,4)\to SO(1,5)$5 from a generalized Jordan domain $SO(2,4)\to SO(1,5)$6, continuous extension to $SO(2,4)\to SO(1,5)$7 is equivalent to local connectedness of every boundary component of $SO(2,4)\to SO(1,5)$8, provided two finiteness hypotheses hold: the non-degenerate boundary components of $SO(2,4)\to SO(1,5)$9 are at most countable with finite sum of diameters, and either the degenerate boundary components of $G$0 or those of $G$1 form a set of sigma-finite linear measure (Luo et al., 2020). The same paper identifies this boundary condition with $G$2 being a Peano compactum and proves the equivalence with Property $G$3, local accessibility, local sequential accessibility, finite connectivity at the boundary, and compactness of the Mazurkiewicz completion (Luo et al., 2020).

A related but distinct extension problem arises for maps defined on arbitrary subsets of Riemann surfaces. If $G$4 and $G$5 locally agrees with holomorphic maps $G$6 having nonvanishing derivative at the base point, and if the cluster sets $G$7 are pairwise disjoint, then $G$8 admits a one-to-one conformal extension to a neighborhood of $G$9 (Gauthier et al., 2014). The same framework proves that a regular analytic arc is equivalent to a conformal arc, so the two common definitions of analytic arc coincide (Gauthier et al., 2014).

These results collectively place one major meaning of conformal continuation within the Carathéodory–prime-end tradition, but with elementary criteria such as semi-unreachable points, cluster-set separation, or local connectedness replacing prime-end language in key theorems.

3. Analytic continuation in conformal field theory

In conformal field theory, conformal continuation usually means analytic continuation of conformal data rather than boundary extension. A prominent example is celestial conformal field theory for four-dimensional massless scalars. There, celestial primaries have

$\partial_a G$0

and the Mellin transform of four-dimensional amplitudes produces celestial correlators. For the shadowed scalar four-point function, the block decomposition requires splitting the original real-line integral into the three channel contributions $\partial_a G$1, $\partial_a G$2, and $\partial_a G$3, each of which evaluates to an Appell hypergeometric function $\partial_a G$4 multiplied by a Beta function (Fan, 2023). The $\partial_a G$5- and $\partial_a G$6-channel expressions are naturally written with arguments $\partial_a G$7 or $\partial_a G$8, so analytic continuation of $\partial_a G$9 is required to return them to $\partial_n G$0. Because continuation of a two-variable Appell function generates extra mixed-argument terms and $\partial_n G$1 terms, the paper combines analytic continuation with monodromy projection, retaining only the terms with the correct monodromy in the chosen OPE channel (Fan, 2023). The resulting decomposition is consistent with crossing symmetry at the correlator level and at the level of each explicit block. In the conformal soft limit, the scalar case differs from the celestial gluon case: $\partial_n G$2 does not reduce to a Gauss hypergeometric function because $\partial_n G$3 remains nonzero (Fan, 2023).

A second CFT use concerns analytic continuation in spin. Light-ray operators furnish continuous-spin operators, and for non-integer $\partial_n G$4 they are genuinely nonlocal. The light transform maps a primary $\partial_n G$5 to an operator transforming with quantum numbers $\partial_n G$6, and matrix elements of light-ray operators are expressed through integrals of double commutators against conformal blocks (Kravchuk et al., 2018). This gives a Lorentzian derivation of Caron-Huot’s inversion formula and realizes analytic continuation of OPE data in spin through intrinsically Lorentzian integral transforms (Kravchuk et al., 2018).

A third use appears in VOA theory. For a $\partial_n G$7-cofinite module category, spaces of conformal blocks form locally constant sheaves on configuration spaces, and analytic continuation along paths in those configuration spaces yields an action of the parenthesized braid operad on conformal blocks (Moriwaki, 22 Jun 2026). In the rational $\partial_n G$8-cofinite case this pseudo-braided structure is represented by tensor products. For the $\partial_n G$9 Virasoro VOA, all four-point blocks can be written explicitly in terms of hypergeometric functions, and analytic continuation determines the braiding and associator, identifying the resulting balanced braided tensor category with the Tambara–Yamagami category over $\partial_{sn}G$0 (Moriwaki, 22 Jun 2026).

The same analytic-continuation logic also underlies the relation between minimal models and Liouville theory. The general three-point function with continuous charges is obtained by successive analytic continuations starting from minimal-model three-point functions, first continuing the screening numbers and then continuing to the Liouville regime, where the DOZZ structure constants emerge (Dotsenko, 2016). In a different axiomatic setting, Minkowski Wightman four-point functions in CFT are shown to be analytic continuations of one another by combining the Wightman axioms, Jost’s theorem, and the edge-of-the-wedge theorem, and this yields the conformal bootstrap crossing equation (Maharana, 2021).

4. Unique continuation from conformal boundaries

In Lorentzian PDE and AdS geometry, conformal continuation denotes unique continuation from conformal infinity. For Klein–Gordon equations on asymptotically anti-de Sitter spacetimes in Fefferman–Graham gauge, McGill and Shao formulate a null convexity criterion on the boundary metric $\partial_{sn}G$1: $\partial_{sn}G$2 for all $\partial_{sn}G$3-null vectors $\partial_{sn}G$4 (McGill et al., 2020). This criterion yields pseudoconvex level sets for a Carleman weight $\partial_{sn}G$5, and the resulting Carleman estimate implies unique continuation from a boundary timeslab whose height is controlled by the near-boundary null-geodesic return times $\partial_{sn}G$6 (McGill et al., 2020). The same paper argues that the timespan assumption is essentially optimal by relating it to near-boundary null geodesics.

The sharpness mechanism is made explicit for critically singular wave equations. If

$\partial_{sn}G$7

and there exists a regular family of null geodesics trapped near $\partial_{sn}G$8, then one can construct smooth $\partial_{sn}G$9 that vanish to infinite order at the boundary and satisfy $\psi:D\to G$0, thereby producing counterexamples to unique continuation from the conformal boundary (Guisset et al., 2023). In pure AdS this realizes the critical time barrier $\psi:D\to G$1; in planar AdS it yields failures on arbitrarily long time slabs (Guisset et al., 2023).

The Einstein–Maxwell system admits an analogous boundary continuation theory. In Fefferman–Graham gauge, the near-boundary data consist of

$\psi:D\to G$2

together with the universal boundary constraints. If two AdS–Einstein–Maxwell solutions have gauge-equivalent holographic data on a domain $\psi:D\to G$3 satisfying the generalized null convexity criterion, then they are related near $\psi:D\to G$4 by a boundary-preserving diffeomorphism (Guisset, 17 Jan 2025). A notable structural point is that the geometric condition required for unique continuation is identical to the vacuum GNCC, despite the presence of Maxwell fields (Guisset, 17 Jan 2025).

A related continuation statement appears for stationary vacuum spacetimes. If two stationary Lorentzian Einstein metrics coincide up to order one along a timelike hypersurface, then they are locally isometric near that hypersurface. In the AdS-type setting, equality of the conformal infinity and of the undetermined Fefferman–Graham term implies local isometry near conformal infinity, and conformal Killing vectors of the boundary that preserve the undetermined term extend to bulk Killing vectors (Chrusciel et al., 2011).

5. Conformally adapted gauges and boundary-to-interior reduction

A more differential-geometric use of conformal continuation is built from conformally natural gauges. Conformal harmonic coordinates are defined by

$\psi:D\to G$5

where $\psi:D\to G$6, $\psi:D\to G$7, and

$\psi:D\to G$8

These coordinates are conformally invariant, exist under general conditions, and become a close conformal analogue of harmonic coordinates (Lassas et al., 2019). Once one normalizes the determinant by $\psi:D\to G$9, the Weyl tensor, Cotton tensor, Bach tensor, and Fefferman–Graham obstruction tensor become elliptic or overdetermined elliptic operators in conformal harmonic coordinates. In particular, for even $\phi=\psi^{-1}$0,

$\phi=\psi^{-1}$1

and obstruction-flat metrics become real-analytic in this gauge (Lassas et al., 2019). The same framework yields unique continuation results for conformal mappings in both Riemannian and Lorentzian signatures, and for local conformal flatness on Bach-flat or obstruction-flat manifolds (Lassas et al., 2019).

In planar potential theory, a different conformal reduction turns a boundary problem into an interior one. For a harmonic function $\phi=\psi^{-1}$2 vanishing continuously on a boundary arc, one constructs a positive harmonic function $\phi=\psi^{-1}$3 with the same vanishing boundary set, takes its harmonic conjugate $\phi=\psi^{-1}$4, and defines the hodograph map

$\phi=\psi^{-1}$5

Its Jacobian satisfies $\phi=\psi^{-1}$6, and $\phi=\psi^{-1}$7 maps the relevant boundary portion into a straight segment $\phi=\psi^{-1}$8 (Vita, 2024). The transported function $\phi=\psi^{-1}$9 vanishes on that interior line, so odd reflection across $$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$0 produces a harmonic function in a full ball. This boundary-to-interior continuation yields two sharp consequences: in chord-arc domains the boundary critical set $$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$1 has zero arclength measure, and in $$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$2 domains the full critical set in $$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$3 is finite (Vita, 2024).

6. Numerical, spectral, algebraic, and gauge-theoretic continuations

In numerical analysis, conformal continuation often means replacing a divergent or slowly convergent local expansion by a convergent expansion in a conformally mapped variable. For nonlinear ODE two-point boundary value problems, one starts from a Taylor series $$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$4 and composes with

$$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$5

which sends a sector of analyticity to the unit disk and typically maps $$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$6 to $$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$7 (Abbasbandy et al., 2011). Re-expanding in $$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$8 yields a series convergent up to the second boundary, and the paper emphasizes that the geometry of movable singularities controls the efficiency of this method relative to Padé and Padé–Hankel approximants (Abbasbandy et al., 2011).

A closely related conformal-map continuation appears in spectral reconstruction from Euclidean data. Retarded Green functions $$\phi \text{ extends continuously to } \overline G \iff \partial_{sn}G=\emptyset,$$9, analytic in the upper half-plane, are mapped to Schur-class functions on the disk by the Cayley transform $\partial G$0 in the fermionic case and by the composition of $\partial G$1 with $\partial G$2 in the bosonic case (Bergamaschi et al., 2023). The continuation problem becomes a Schur/Nevanlinna–Pick interpolation problem, and the full family of admissible interpolants at a target point is a disk $\partial G$3 in the unit disk. Mapping that disk back to the upper half-plane yields rigorous pointwise uncertainty regions for $\partial G$4, and

$\partial G$5

is interpreted as a Poisson-smeared spectral function (Bergamaschi et al., 2023).

For general Dirichlet series, Ghisa uses conformal mapping geometry of the $\partial G$6-plane itself. The plane is partitioned into strips $\partial G$7 and fundamental domains $\partial G$8 on which the Dirichlet-series function is univalent and maps onto slit planes. Continuation is then performed by lifting curves across the slit boundaries from one fundamental domain to another (Ghisa et al., 2015). Within that framework, Ghisa formulates and proves a “Great Riemann Hypothesis” for a class of general Dirichlet series satisfying his continuation and functional-equation hypotheses (Ghisa et al., 2015).

The expression also appears in gauge-theoretic gravity in a more algebraic sense. In a non-linear theory of interacting spin-2 fields, the conformal group $\partial G$9 is replaced by $SO(2,4)\to SO(1,5)$00, and the dilation generator $SO(2,4)\to SO(1,5)$01 is continued to the generator of a local $SO(2,4)\to SO(1,5)$02 acting on a pair of vielbeins (Apolo et al., 2016). In perturbation theory around de Sitter space, this local $SO(2,4)\to SO(1,5)$03 symmetry transmutes into the partially massless symmetry of a massive spin-2 field, with the vector gauge field $SO(2,4)\to SO(1,5)$04 playing a crucial role (Apolo et al., 2016).

Finally, in the Einstein constraint equations, analytic and numerical continuation techniques are used to explore the far-from-CMC regime of the conformal method. Continuation in parameters such as the mean-curvature amplitude and the TT-tensor scale reveals folds, multiplicity, disconnected loops, and nonexistence regions, showing that the good CMC and near-CMC behavior does not persist uniformly in the far-from-CMC case (Dilts et al., 2017).

Across these domains, conformal continuation consistently denotes an extension principle that becomes available only after the problem has been recast in conformally natural variables: Appell functions and monodromy sectors in celestial CFT, crosscuts and accessible boundary points in planar mapping theory, Fefferman–Graham gauges and Carleman weights near conformal infinity, determinant-normalized conformal harmonic gauges in geometric analysis, or explicit disk-valued conformal maps in numerical continuation.