Gubser's Potential Criterion

• Gubser’s potential criterion is a condition in Einstein–scalar systems that requires the on-shell scalar potential to remain bounded along singular flows, excluding solutions where it diverges to +∞.
• The criterion is used to differentiate acceptable singularities by comparing bounded potential values with alternative tests like the horizon and Maldacena–Nuñez criteria across examples such as moduli-space flows and KT/KS solutions.
• The geometrized refinement recasts the criterion into a Ricci-scalar bound linked to field-space distance, allowing for controlled ultraviolet completions in cases like EFT strings and D7-branes despite divergent potentials.

Gubser’s potential criterion is a necessary condition for the acceptability of codimension-one curvature singularities in Einstein–scalar systems. In the formulation used for end-of-the-world singularities and dynamical cobordisms, the criterion requires that, along a singular flow, the scalar potential evaluated on the solution remain bounded above as the singular locus is approached: if $r \to r_*$ denotes the singular point and $\phi(r)$ the running scalars, then

$\sup_{r\to r_*} V(\phi(r)) < +\infty .$

Potentials that diverge to $+\infty$ near the singularity are therefore excluded. In the analysis of codimension-one flows, this criterion is treated as necessary but not generally sufficient, and its scope is reassessed by comparison with Gubser’s horizon criterion, the Maldacena–Nuñez criterion, and a geometrized refinement based on field-space distance and Ricci-scalar growth (Calderón-Infante et al., 18 Mar 2026).

1. Formal definition in Einstein–scalar systems

The criterion is studied in a $(d+1)$-dimensional Einstein–scalar theory. In one normalization, the action is

$$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$

The paper adopts instead the $D$-dimensional normalization

$$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$

with $D=d+1$. The difference is a factor of $2$ in the kinetic and potential normalizations, but the acceptance criterion depends only on boundedness and is therefore invariant under these rescalings (Calderón-Infante et al., 18 Mar 2026).

For zero-temperature codimension-one flows with flat slicing $\phi(r)$0, the metric ansatz is

$\phi(r)$1

with $\phi(r)$2. An equivalent domain-wall gauge is

$\phi(r)$3

At finite temperature, a blackening factor is introduced: $\phi(r)$4 The corresponding one-dimensional effective Lagrangian is

$\phi(r)$5

where $\phi(r)$6, together with the constraint

$\phi(r)$7

and

$\phi(r)$8

where $\phi(r)$9 is related to the temperature by

$\sup_{r\to r_*} V(\phi(r)) < +\infty .$0

Within this framework, Gubser’s potential criterion is applied directly to the potential evaluated along the singular solution, not to the off-shell scalar potential viewed abstractly over field space. Its operational content is local: one follows the actual flow to the singular locus and excludes solutions for which $\sup_{r\to r_*} V(\phi(r)) < +\infty .$1.

2. Relation to horizon cloaking and the Maldacena–Nuñez test

The criterion is motivated by Gubser’s original expectation that a good singularity should admit a near-extremal deformation whose regular horizon cloaks the singular region. In the working formulation used here, the horizon criterion is sufficient: a singular solution is acceptable if there exists a one-parameter near-extremal family with regular horizon for every finite $\sup_{r\to r_*} V(\phi(r)) < +\infty .$2, continuously approaching the singular solution as $\sup_{r\to r_*} V(\phi(r)) < +\infty .$3. In this setup, such deformations correspond to $\sup_{r\to r_*} V(\phi(r)) < +\infty .$4, or equivalently $\sup_{r\to r_*} V(\phi(r)) < +\infty .$5 in the finite-temperature equations. In asymptotically AdS settings, existence of this near-extremal family implies that the potential along the flow is bounded above, so the horizon criterion implies the potential criterion (Calderón-Infante et al., 18 Mar 2026).

The converse does not hold in general. The paper emphasizes that the potential criterion is necessary but not sufficient, especially outside the AdS context. This distinction is sharpened by moduli-space flows, where $\sup_{r\to r_*} V(\phi(r)) < +\infty .$6 is constant. In such cases the potential criterion is trivially satisfied, yet a smooth near-extremal horizon generalization does not exist. Near a putative horizon,

$\sup_{r\to r_*} V(\phi(r)) < +\infty .$7

with $\sup_{r\to r_*} V(\phi(r)) < +\infty .$8 the geodesic kinetic constant. As $\sup_{r\to r_*} V(\phi(r)) < +\infty .$9, the scalar diverges, so the horizon becomes a null singularity rather than a regular blackening of the original solution. This makes the horizon criterion strictly stronger than the potential criterion.

The Maldacena–Nuñez criterion provides a different necessary test, formulated geometrically in ten dimensions. In the version used here, a singularity is acceptable if the ten-dimensional Einstein-frame time-time component does not grow toward the singularity. Its strong form requires $+\infty$0 to be non-increasing; its weak form permits growth so long as $+\infty$1 remains bounded above. Unlike Gubser’s criterion, this test is frame-sensitive and does not require an explicit scalar potential. In explicit uplifts of moduli-space flows, the MN criterion can be satisfied even when the horizon criterion fails. For the AdS$+\infty$2 “Gubser flow,” one finds $+\infty$3 in five-dimensional Einstein frame near the singularity, so $+\infty$4. For AdS$+\infty$5 flows, explicit ten-dimensional uplift formulas likewise yield decreasing $+\infty$6 in suitable duality frames.

A general argument based on the Emergent String Conjecture is then invoked: in the duality frame where the internal space decompactifies, the ten-dimensional Einstein-frame $+\infty$7 acquires an extra factor $+\infty$8 with $+\infty$9. Since $(d+1)$0 at the singularity, $(d+1)$1. This suggests that there exist singular flows acceptable by MN yet not “heatable up” in the sense required by Gubser’s horizon criterion.

3. Field-space distance and the geometrized refinement

The reappraisal of the potential criterion is tied to infinite-distance limits in scalar field space. The field-space distance along a radial flow is

$(d+1)$2

where $(d+1)$3 is a proper radial coordinate. For end-of-the-world singularities, the near-singularity region often lies at finite proper distance in spacetime but at infinite distance in field space (Calderón-Infante et al., 18 Mar 2026).

In codimension-one ETW flows, the Ricci scalar is found to diverge exponentially with a canonically normalized scalar $(d+1)$4 that runs to infinite distance: $(d+1)$5 with critical exponent $(d+1)$6. A broad family of string-motivated examples is parameterized by

$(d+1)$7

with $(d+1)$8 and $(d+1)$9. Kination corresponds to $$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$0, where the potential is subdominant.

In this language, Gubser’s potential criterion translates into a bound on the allowed exponential growth: $$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$1 This motivates a local geometric reformulation: $$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$2 or equivalently

$$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$3

This geometrized criterion is presented as a necessary local criterion. It captures all examples accepted by the potential test, but it is weaker: it also accepts flows whose asymptotic potentials grow positively, provided that growth does not exceed the kination-critical exponential. This is the crucial point behind its acceptance of EFT strings and D7-branes reduced to codimension one, despite their failure of the original potential criterion. A plausible implication is that the potential criterion over-penalizes radion-driven positive potentials in situations where the geometry remains compatible with a controlled ultraviolet completion.

4. Representative flows and diagnostic outcomes

The criterion is illustrated across several classes of singular solutions. The examples show that boundedness of $$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$4 is a strong discriminator, but not one perfectly aligned with ultraviolet completion or ten-dimensional regularity (Calderón-Infante et al., 18 Mar 2026).

Example	Potential criterion	Related outcome
Moduli-space flows	Satisfied	Horizon criterion fails; MN can be satisfied
Klebanov–Tseytlin	Violated	MN violated; geometrized bound violated
Klebanov–Strassler	Satisfied	Geometrized bound satisfied
EFT strings	Violated	Geometrized criterion accepts
D7-branes	Violated	Geometrized criterion accepts
Massive Type IIA strong coupling ETW	Violated	Geometrized criterion rejects

For moduli-space flows with $$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$5, the potential criterion is trivially satisfied since $$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$6 is finite everywhere. In gauge $$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$7, the domain-wall constraint with $$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$8 is

$$S = \int d^{d+1}x \sqrt{-g}\,\Big[ R - \frac12 G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - V(\phi)\Big].$$9

Near the singularity $D$0,

$D$1

while the geodesic scalar diverges logarithmically,

$D$2

Thus $D$3 as $D$4 despite finite proper distance. These flows pass the potential criterion but fail the horizon criterion because $D$5 diverges at any putative horizon.

The Klebanov–Tseytlin solution provides a canonical failure mode. In the five-dimensional $D$6 truncation with scalar set $D$7, the KT solution sets $D$8. Near $D$9, the RR five-form contribution dominates and

$$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$0

KT therefore violates Gubser’s potential criterion. In ten dimensions it also violates the MN criterion, since

$$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$1

In the cobordism parameterization, it has $$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$2 and $$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$3, which is supercritical and violates the geometrized bound.

The Klebanov–Strassler solution behaves differently. With all fields non-trivial and $$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$4, the singularity occurs at $$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$5, where $$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$6. The potential is then dominated by the internal curvature term and tends to $$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$7, which is acceptable by Gubser’s potential test. In cobordism variables it has $$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$8 and $$L = R - G_{ij}(\phi)\partial_\mu \phi^i \partial^\mu \phi^j - 2V(\phi),$$9, satisfying the geometrized bound. The comparison between KT and KS is used to argue that altering the infinite-distance target of the flow is not a ultraviolet resolution of the original singularity. KS modifies the field excursion and the infinite-distance endpoint relative to KT, even if the far asymptotics agree.

EFT strings furnish a counterexample to the sufficiency of the potential criterion as a physically informative diagnostic. In the four-dimensional $D=d+1$0 setup with single modulus $D=d+1$1 and

$D=d+1$2

reduction on the transverse $D=d+1$3 yields the three-dimensional kinetic terms and potential

$D=d+1$4

$D=d+1$5

Near $D=d+1$6, the radion-driven potential diverges, so $D=d+1$7 and the potential criterion fails. Yet these ETW flows have $D=d+1$8 and $D=d+1$9, so they satisfy the geometrized bound.

An analogous pattern appears for D7-branes. After reduction on the transverse circle to nine dimensions,

$2$0

$2$1

Again $2$2 near the singularity, so Gubser’s potential criterion fails. However, the ETW data are $2$3, $2$4, and for $2$5 one has $2$6, so the geometrized criterion accepts the solution.

By contrast, the massive Type IIA strong-coupling ETW singularity is rejected by both tests. The solution is

$2$7

defined only for $2$8. At $2$9, one has $\phi(r)$00 and

$\phi(r)$01

Its exponent $\phi(r)$02 exceeds $\phi(r)$03 for $\phi(r)$04, so it violates both the potential criterion and the geometrized bound.

5. Near-extremal deformations and finite-temperature scaling

The significance of Gubser’s criterion is tied to the question of whether a singular flow can be “heated up,” namely embedded into a one-parameter family of finite-temperature solutions with a regular horizon. The analysis of black D$\phi(r)$05-branes reduced to codimension one provides explicit representatives of flows that do admit near-extremal generalizations (Calderón-Infante et al., 18 Mar 2026).

Starting from ten-dimensional black D$\phi(r)$06-brane solutions in string frame, one compactifies the transverse $\phi(r)$07 as

$\phi(r)$08

with

$\phi(r)$09

and radion profile

$\phi(r)$10

Evaluated at the horizon $\phi(r)$11, and taking $\phi(r)$12 at fixed charge $\phi(r)$13, the field-space distance from $\phi(r)$14 to a reference point $\phi(r)$15 is, for $\phi(r)$16,

$\phi(r)$17

The temperature and entropy scale as

$\phi(r)$18

After eliminating $\phi(r)$19 in favor of the field-space distance and normalizing by the horizon Planck scale, one obtains

$\phi(r)$20

with

$\phi(r)$21

Thus

$\phi(r)$22

when $\phi(r)$23, as in $\phi(r)$24, whereas $\phi(r)$25 grows exponentially when $\phi(r)$26, as in $\phi(r)$27. A direct ten-dimensional analysis yields a similar relation and shows that only the D6 case has temperature blowing up relative to the ten-dimensional species cutoff.

This motivates a finite-temperature extension of the Distance Conjecture: $\phi(r)$28 with $\phi(r)$29 at zero temperature and $\phi(r)$30 at finite temperature, where $\phi(r)$31 is an $\phi(r)$32 constant. The implication is not that every acceptable singularity admits such a thermal interpretation, but rather that, when near-extremal horizons exist, thermal scales themselves can exhibit the same exponential sensitivity to infinite field-space distance that ordinarily characterizes towers of light states.

6. Scope, limitations, and conceptual status

Within the codimension-one framework, Gubser’s potential criterion remains a sharp necessary local test: singularities for which the on-shell scalar potential diverges to $\phi(r)$33 are excluded. In asymptotically AdS settings, this criterion is backed by the stronger statement that a smooth near-extremal cloaking implies boundedness of the potential. The analysis also makes clear, however, that the criterion should not be conflated with a universal characterization of physically acceptable singularities (Calderón-Infante et al., 18 Mar 2026).

Several limitations are explicit. The horizon criterion is sufficient but not necessary; moduli-space flows and Coulomb-branch-like flows may be acceptable without admitting near-extremal cloaking. The potential criterion itself can be too strong outside AdS, because radion-induced positive potentials in EFT strings and D7-branes lead to rejection despite sensible ultraviolet completions. The geometrized refinement is proposed only for flat slicings $\phi(r)$34; curved slicings can alter the scaling structure. The Maldacena–Nuñez criterion is frame-sensitive in ten dimensions, and local criteria of any type remain insensitive to the existence of a global embedding.

A further conceptual point concerns “resolution.” If a purported resolution changes the infinite-distance endpoint probed by the effective flow, then it does not resolve the original singularity in the sense relevant for the criterion. The KT-to-KS comparison is used precisely to support this distinction.

Taken together, these results place Gubser’s potential criterion in a more differentiated hierarchy. It is a robust necessary condition in the class for which it was designed, but not the final arbiter of acceptability across all end-of-the-world singularities. The proposed Ricci-scalar bound

$\phi(r)$35

is presented as a geometrization of Gubser’s idea: it retains all examples already accepted by the potential criterion, admits additional ultraviolet-complete cases such as EFT strings and D7-branes, and still rejects supercritical singularities such as the massive Type IIA strong-coupling endpoint. This suggests that, for infinite-distance codimension-one flows, the decisive datum may be not simply the sign of $\phi(r)$36 near the singularity, but the rate at which curvature diverges relative to field-space distance.