Fully Nonlinear Loewner–Nirenberg Problem

• The fully nonlinear Loewner–Nirenberg problem is a boundary blow-up challenge seeking complete conformal metrics with curvature defined by nonlinear symmetric functions of geometric tensors.
• It generalizes the classical singular Yamabe problem by replacing constant negative scalar curvature with functions of the eigenvalues of the Schouten tensor or negative Ricci tensor, constrained by admissibility conditions.
• Methodologies such as viscosity solutions, barrier constructions, and asymptotic analysis are central in proving existence, uniqueness, and regularity on both Euclidean domains and compact manifolds.

Searching arXiv for recent and foundational papers on the fully nonlinear Loewner–Nirenberg problem. The fully nonlinear Loewner–Nirenberg problem is the conformally invariant boundary blow-up problem of constructing complete conformal metrics whose curvature is prescribed by a nonlinear symmetric function of a curvature tensor, rather than by scalar curvature alone. In the classical case, one prescribes constant negative scalar curvature and seeks a conformal metric that is complete toward the boundary. In the fully nonlinear setting, the prescribed quantity is typically a symmetric function $f$ of the eigenvalues of either the Schouten tensor or, in a distinct variant, the negative Ricci tensor of the conformal metric. The problem is formulated on domains in $\mathbb R^n$ or on compact Riemannian manifolds with boundary, with the analytic boundary condition $u\to+\infty$ or equivalently $u=0$ in a reciprocal conformal factor variable, encoding completeness of the resulting metric. Existence, uniqueness, regularity, and asymptotic behavior depend sharply on admissibility cones, codimension of singular sets, and the structure of the nonlinear operator (González et al., 2018, Duncan et al., 2023, Sui, 2023, Duncan et al., 22 Jul 2025).

1. Classical origin and geometric reformulation

The classical Loewner–Nirenberg problem, also called the singular Yamabe problem, asks for a conformal metric on a domain that is complete in the interior and has constant negative scalar curvature. In Euclidean form, this corresponds to solving

$\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$

with boundary blow-up, or equivalently constructing a metric $u^{4/(n-2)}dx^2$ of scalar curvature $-n(n-1)$ that is complete toward the boundary. The foundational geometric point is that completeness is analytically encoded by the singular boundary condition $u\to+\infty$ (Kichenassamy, 3 Jul 2025).

Fully nonlinear generalizations replace scalar curvature by a symmetric curvature function. One important line uses the Schouten tensor $A_g$, so that the equation prescribes $f(\lambda(-g_u^{-1}A_{g_u}))$, with eigenvalues constrained to lie in an admissible cone $\mathbb R^n$0. Another line, developed for Ricci curvature, replaces the Schouten tensor by the negative Ricci tensor and studies $\mathbb R^n$1 (Duncan et al., 2023, Sui, 2023). In both cases the geometric objective remains the same: produce a complete conformal metric with a prescribed negative curvature-type quantity.

A related but distinct boundary formulation treats an embedded hypersurface as conformal infinity and seeks a conformally compact metric of constant scalar curvature on the complement. In that setting the governing equation is written in terms of a defining density $\mathbb R^n$2 satisfying $\mathbb R^n$3, where $\mathbb R^n$4 is the scale tractor. This boundary Loewner–Nirenberg–Yamabe problem links the singular metric asymptotics to extrinsic conformal invariants of the hypersurface (Gover et al., 2015).

2. PDE formulations and admissibility structures

For Schouten-type equations on a manifold $\mathbb R^n$5, the unknown conformal metric is typically written as

$\mathbb R^n$6

depending on normalization. The governing problem takes the form

$\mathbb R^n$7

The Schouten tensor transforms conformally by a fully nonlinear second-order law involving the Hessian and quadratic gradient terms, which is why the resulting PDE is of augmented Hessian type (Duncan et al., 2023, Duncan et al., 22 Jul 2025).

For the $\mathbb R^n$8-Loewner–Nirenberg problem in Euclidean space, the equation is usually written as

$\mathbb R^n$9

where $u\to+\infty$0 is the conformal Hessian. In the negative formulation used for complete metrics, admissibility means that the eigenvalues lie in the negative cone $u\to+\infty$1, which is the elliptic regime for the singular problem (Li et al., 2022, Fang et al., 14 Jun 2026).

For the Ricci-curvature analogue on a Euclidean domain $u\to+\infty$2, $u\to+\infty$3, one seeks

$u\to+\infty$4

with $u\to+\infty$5 and $u\to+\infty$6 on $u\to+\infty$7. Under the conformal change formula for Ricci curvature, this becomes

$u\to+\infty$8

where $u\to+\infty$9 is the explicit second-order operator given in the source formulation, and admissibility requires $u=0$0 pointwise (Sui, 2023).

A central algebraic invariant in the Schouten formulation is

$u=0$1

For $u=0$2, one has

$u=0$3

This parameter sharply governs existence and rigidity. In particular, $u=0$4 is equivalent to $u=0$5 for the Gårding cones and underlies the first general manifold existence theorem in the subcritical range (Duncan et al., 2023). Later work extends existence to the threshold and slightly beyond, under geometric compactness hypotheses, for $u=0$6, thereby covering all $u=0$7 in the $u=0$8 case (Duncan et al., 22 Jul 2025).

3. Existence theories on Euclidean domains and manifolds

A major Euclidean-domain existence theorem for general fully nonlinear conformal equations proves that if $u=0$9 is bounded with smooth boundary and $\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$0 satisfy the structural assumptions of symmetry, positivity in the cone, vanishing on the boundary of the cone, monotonicity under addition of positive matrices, and homogeneity, then

$\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$1

has a unique continuous viscosity solution, locally Lipschitz in $\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$2, with sharp boundary asymptotics (González et al., 2018). That work also constructs maximal solutions on general domains by exhaustion and identifies a sharp codimension criterion for higher-codimension singular sets in terms of the model eigenvalue vector

$\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$3

On compact Riemannian manifolds with boundary, the Schouten-based theory was developed first for $\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$4. Under the standard assumptions on $\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$5 and the algebraic condition $\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$6, there exists a locally Lipschitz viscosity solution to the singular problem, with sharp asymptotic

$\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$7

The solution is maximal among continuous viscosity solutions, and if $\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$8, then it is smooth and unique among continuous viscosity solutions (Duncan et al., 2023).

This result was later extended to the threshold regime $\Delta u=\frac{n(n-2)}{4}u^{\frac{n+2}{n-2}}$9 and slightly beyond. If $u^{4/(n-2)}dx^2$0 lies in the geometric compactness class $u^{4/(n-2)}dx^2$1, then for $u^{4/(n-2)}dx^2$2 there exists a maximal locally Lipschitz viscosity solution of

$u^{4/(n-2)}dx^2$3

with

$u^{4/(n-2)}dx^2$4

If $u^{4/(n-2)}dx^2$5, the solution is smooth and unique (Duncan et al., 22 Jul 2025). This theorem relies on first constructing a conformal metric in the class $u^{4/(n-2)}dx^2$6 with $u^{4/(n-2)}dx^2$7, and then solving the infinite-boundary-value problem from that admissible background.

A broader structural theory considers equations of the form

$u^{4/(n-2)}dx^2$8

on compact manifolds with boundary. Under concavity, growth, and cone assumptions, together with the geometric condition $u^{4/(n-2)}dx^2$9, one obtains a smooth complete admissible metric solving the fully nonlinear Loewner–Nirenberg problem. That framework emphasizes “partial uniform ellipticity” and uses Morse theory to construct admissible metrics under weaker assumptions on the background geometry (Yuan, 2022, Yuan, 24 Mar 2025).

For the Ricci version, if $-n(n-1)$0 has boundary consisting of finitely many disjoint smooth compact hypersurfaces, then there exists a smooth positive admissible solution of

$-n(n-1)$1

equivalently a smooth complete conformal metric $-n(n-1)$2 satisfying

$-n(n-1)$3

Uniqueness holds when $-n(n-1)$4 is bounded or when $-n(n-1)$5 (Sui, 2023).

4. Boundary asymptotics, regularity, and singular-set criteria

The blow-up rate near a smooth hypersurface boundary is sharp and universal in the conformal class: in the classical and many fully nonlinear settings,

$-n(n-1)$6

or equivalently, in logarithmic variables,

$-n(n-1)$7

For the Ricci-curvature problem, the asymptotic growth is

$-n(n-1)$8

and the blow-up order is exactly $-n(n-1)$9 in the hypersurface case (Sui, 2023).

Near-boundary regularity in the $u\to+\infty$0-Loewner–Nirenberg problem is substantially subtler than in the scalar case. For $u\to+\infty$1, the viscosity solution on a bounded smooth Euclidean domain is smooth in a collar neighborhood $u\to+\infty$2, and

$u\to+\infty$3

for every $u\to+\infty$4. The expansion of $u\to+\infty$5 contains a first logarithmic obstruction coefficient $u\to+\infty$6, and

$u\to+\infty$7

in a collar neighborhood. The coefficient $u\to+\infty$8 is a polynomial in the principal curvatures of $u\to+\infty$9 and their covariant derivatives and is Möbius invariant (Li et al., 2022).

The possibility of blow-up along higher-codimension boundary pieces is governed by an explicit model vector. For the Schouten-type Euclidean problem, regularity of a codimension-$A_g$0 singular set is determined by whether

$A_g$1

lies in the admissible cone $A_g$2 (González et al., 2018). For the Ricci analogue, the corresponding criterion uses

$A_g$3

If every boundary component has codimension $A_g$4 with $A_g$5, a complete solution exists; if some component has $A_g$6, no complete conformal metric solving the equation exists. The borderline case $A_g$7 remains open (Sui, 2023).

A frequent misconception is that completeness alone should force blow-up regardless of boundary codimension. The codimension criteria show otherwise: completeness is compatible with the equation only when the model eigenvalue vector lies in the admissible cone. This is a sharp phenomenon in both Schouten and Ricci formulations (González et al., 2018, Sui, 2023).

5. Regularity breakdown, model classification, and boundary rigidity

Once $A_g$8, smoothness can fail even when existence holds. On annuli $A_g$9, $f(\lambda(-g_u^{-1}A_{g_u}))$0, $f(\lambda(-g_u^{-1}A_{g_u}))$1, the unique viscosity solution of the $f(\lambda(-g_u^{-1}A_{g_u}))$2-Loewner–Nirenberg problem is radially symmetric, smooth away from the sphere $f(\lambda(-g_u^{-1}A_{g_u}))$3, and only

$f(\lambda(-g_u^{-1}A_{g_u}))$4

on either side. The radial derivative has a jump across $f(\lambda(-g_u^{-1}A_{g_u}))$5, and the solution is not $f(\lambda(-g_u^{-1}A_{g_u}))$6 for any $f(\lambda(-g_u^{-1}A_{g_u}))$7 (Li et al., 2020). This provides an explicit mechanism for interior non-differentiability.

More generally, if a bounded smooth domain has disconnected boundary, then for $f(\lambda(-g_u^{-1}A_{g_u}))$8 the $f(\lambda(-g_u^{-1}A_{g_u}))$9-Loewner–Nirenberg problem admits no positive $\mathbb R^n$00 solution. Consequently, the unique locally Lipschitz viscosity solution is not $\mathbb R^n$01 (Li et al., 2022). The proof uses a minimal hypersurface obstruction in the conformal metric and shows that disconnected boundary is incompatible with global differentiability.

A complementary rigidity theory concerns the upper half-space model. For the boundary value problem

$\mathbb R^n$02

with metric $\mathbb R^n$03, all positive viscosity solutions are functions of $\mathbb R^n$04. If $\mathbb R^n$05, then the unique solution is the hyperbolic one

$\mathbb R^n$06

If $\mathbb R^n$07, there exists a monotonically increasing one-parameter family $\mathbb R^n$08, with $\mathbb R^n$09 minimal (Duncan et al., 22 Jul 2025). Thus the threshold $\mathbb R^n$10 separates rigidity from non-uniqueness.

This classification has a further implication: when $\mathbb R^n$11, local boundary $\mathbb R^n$12 and gradient estimates fail in general, because the family $\mathbb R^n$13 produces counterexamples with arbitrarily large size near the boundary (Duncan et al., 22 Jul 2025). A plausible implication is that the analytic mechanisms underpinning compactness in the subcritical range cannot be extended naively past the threshold.

6. Methods, extensions, and topological obstructions

The analytic toolbox of the fully nonlinear Loewner–Nirenberg problem combines viscosity-solution methods, comparison principles, barriers, finite-boundary-data approximation, and nonlinear regularity theory. In the Schouten-based manifold theory, one solves finite Dirichlet problems with positive boundary data, obtains uniform $\mathbb R^n$14 bounds, and passes to the singular limit. For $\mathbb R^n$15 in the cone-regularized problems, continuity methods and Hessian estimates yield smooth solutions; the degenerate limit $\mathbb R^n$16 is then obtained via stability of viscosity solutions (Duncan et al., 2023, Duncan et al., 22 Jul 2025).

For the Ricci formulation, the presence of an explicit Laplacian term in the conformal Ricci tensor formula is analytically advantageous. The existence theory uses explicit radial barriers on balls and exterior domains, comparison principles for $\mathbb R^n$17-admissible sub- and supersolutions, interior and boundary $\mathbb R^n$18 estimates, Evans–Krylov regularity, monotone exhaustion of general domains, and degree theory for the generalized equation with the $\mathbb R^n$19 term (Sui, 2023). The paper explicitly emphasizes that the Ricci-curvature version is in some sense smoother than the Schouten-tensor analogue studied earlier by González–Li–Nguyen, because one can obtain second-order estimates directly and thus smooth solutions (Sui, 2023).

There are also flow approaches. For the generalized $\mathbb R^n$20-Ricci equation on a compact manifold with boundary, one can evolve by

$\mathbb R^n$21

with time-dependent Dirichlet data diverging at the boundary. Starting from a $\mathbb R^n$22 subsolution, the flow exists globally and converges in $\mathbb R^n$23 to the stationary solution of the generalized Loewner–Nirenberg problem (Li, 2021). In the scalar case, analogous direct and Yamabe flows converge to the classical Loewner–Nirenberg solution under suitable assumptions (Li, 2021).

Topology can obstruct admissibility. For the negatively admissible $\mathbb R^n$24-Loewner–Nirenberg problem,

$\mathbb R^n$25

a nondegenerate solution on $\mathbb R^n$26 forces

$\mathbb R^n$27

In orientable embedded-submanifold settings with $\mathbb R^n$28, this implies that the singular set $\mathbb R^n$29 must be connected (Fang et al., 14 Jun 2026). Related structural work also shows topological obstruction phenomena for more general fully nonlinear conformal problems and uses them to argue that certain cone conditions are optimal (Yuan, 2022, Yuan, 24 Mar 2025).

A plausible synthesis is that the fully nonlinear Loewner–Nirenberg problem is governed by three interacting layers: cone geometry, analytic degeneracy, and topology. Cone parameters such as $\mathbb R^n$30 determine ellipticity and rigidity; boundary geometry controls asymptotics and regularity; and topology restricts the very existence of negatively admissible complete conformal metrics (Duncan et al., 2023, Duncan et al., 22 Jul 2025, Fang et al., 14 Jun 2026).