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The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates

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Pith's one-line read All half-space solutions to the fully nonlinear Loewner-Nirenberg problem are classified, and local boundary estimates fail when a cone parameter is at most one.

desk verdict A serious classification paper with a real phase transition; the main theorems hold up, but the relaxed-assumption symmetry claim and the imported comparison principle need referee attention. read the letter →

arxiv 2507.16383 v1 pith:MFW5XHUU submitted 2025-07-22 math.AP math.DG

Jonah A. J. Duncan , Luc Nguyen This is my paper · ORCID
classification math.APmath.DG MSC 35J6035J7035B5353C21
keywords LiouvilletheoremLoewner-NirenbergproblemfullynonlinearellipticequationsSchoutentensormethodofmovingspheresviscositysolutionsboundaryestimatesnegativecurvatureconformalmetrics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper settles the Liouville question for a wide class of conformally invariant equations $f(\lambda(-A_w)) = 1/2$ on the upper half-space, with $A_w$ the Schouten tensor of the metric $g_w = w^{-2}|dx|^2$ and $w=0$ on the boundary. The classification is decided by a single number $\mu_\Gamma^+$, defined by $(-\mu_\Gamma^+,1,\dots,1)\in\partial\Gamma$: if $\mu_\Gamma^+>1$ the hyperbolic solution $w^{(0)}=x_n$ is the unique positive viscosity solution, while if $\mu_\Gamma^+\leq 1$ the solutions are exactly the increasing one-parameter family $\{w^{(a)}(x_n)\}_{a\geq 0}$, with $w^{(0)}$ the minimal member. In the non-unique regime every solution depends only on the distance to the boundary, each $w^{(a)}$ gives a metric locally complete near the boundary, and only $w^{(0)}$ is globally complete. It follows that local boundary $C^0$ and gradient estimates fail when $\mu_\Gamma^+\leq 1$, which is the regime $\Gamma=\Gamma_k^+$ with $k\geq n/2$ for the Gårding cones.

What carries the argument

The proof rests on a variant of the method of moving spheres in which the spheres are centred in the lower half-space rather than on the hyperplane $\{x_n=0\}$, so the comparison domain is bounded and neither a Hopf lemma along the boundary nor a 'touching at infinity' argument is needed. Three ingredients make this work: the comparison principle imported as Propositions 2.3 and 2.4; the barriers giving $w\geq x_n$ globally and $w\leq x_n+C x_n^2$ near the boundary; and the small-perturbation theorem of [45] giving $C^{2,\alpha}$ regularity up to the boundary, following the scheme of [37]. After symmetry reduces the problem to an ODE in $t=x_n$, the key object is the function $\varphi$ defined by $f(\varphi(s),1,\dots,1)=1/(2s)$, which converts the PDE into $ww''=(w')^2(1-\varphi((w')^2/2))/2$; the first integral $G(s)=((s-\tfrac12)/s)^{1/n}e^{B(s)}$ and its inverse $K=G^{-1}$ control the solution, and the asymptotic $K(x)=x^{1+\mu_\Gamma^++o(1)}$ decides whether solutions blow up in finite time ($\mu_\Gamma^+>1$) or exist globally ($\mu_\Gamma^+\leq 1$).

What would settle it

Integrate the first-order system $w'=\sqrt{2K(bw)}$ (equivalently the ODE (3.9)) for a pair $(f,\Gamma)$ with $\mu_\Gamma^+\leq 1$: the paper predicts that for every $b>0$ the solution with $w(0)=0$ satisfies $w'(0)=1$ and $w(1)<\infty$. Finding any $b>0$ whose solution has $w'(0)\neq 1$, or a finite-time blow-up, would falsify Theorem 1.1; alternatively, exhibiting two bounded-domain viscosity solutions with boundary order $w_2<w_1$ but interior touching would falsify the comparison principle on which the symmetry step rests.

Watch

Extended reading notes

Core claim

The central discovery is that all continuous viscosity solutions of (1.2) are one-dimensional: under the relaxed assumptions (1.3′)–(1.5′), (1.6) and (1.7) any such solution satisfies $w=w(x_n)$, and under the standard assumptions together with the weak inequality (1.8′) it must coincide either with $w^{(0)}$ or with one of the $w^{(a)}$ from Theorem 1.1. The family has $w^{(a)}(1)=1+a$, $(w^{(a)})'\geq 1$, $(w^{(a)})''\geq 0$, and $(w^{(a)})'(0)=1$; the metrics $g_{w^{(a)}}$ are locally complete near the boundary but incomplete on the whole half-space for $a>0$. The same result implies uniqueness of the solution $u=x_n^{-(n-2)/2}$ to the semilinear Yamabe-type equation (1.9), and Theorem 1.5 shows that $\inf_{t\in[\varepsilon,E]}w^{(a)}(t)$ and $\inf_{t\in[\varepsilon,E]}(w^{(a)})'(t)$ diverge as $a\to\infty$, giving counterexamples to local boundary $C^0$ and gradient estimates.

Load-bearing premise

The argument depends on a comparison principle for viscosity solutions on bounded domains, assumed to hold even when the defining function is neither convex nor homogeneous; if that principle fails, the conclusion that every solution depends only on distance to the boundary collapses.

Editorial extensions

If this is right

  • For $\mu_\Gamma^+>1$, the hyperbolic metric $x_n^{-2}|dx|^2$ is the unique locally complete conformal metric of the prescribed curvature type on the half-space; in the semilinear case this recovers the uniqueness of $u=x_n^{-(n-2)/2}$ for (1.9).
  • For $\mu_\Gamma^+\leq 1$, every positive viscosity solution is one of the $w^{(a)}$; the hyperbolic solution is the minimal solution and is the only one whose metric is complete on all of $\mathbb{R}_+^n$.
  • For the Gårding cones $\Gamma_k^+$, the threshold $\mu_\Gamma^+\leq 1$ is exactly $k\geq n/2$, so non-uniqueness begins halfway up the cone hierarchy.
  • Local boundary $C^0$ and gradient estimates fail when $\mu_\Gamma^+\leq 1$: $\inf_{t\in[\varepsilon,E]}w^{(a)}(t)$ and $\inf_{t\in[\varepsilon,E]}(w^{(a)})'(t)$ blow up as $a\to\infty$.
  • When $\mu_\Gamma^+\leq 1$, the comparison principle fails on unbounded domains and no lower-semicontinuous admissible supersolution can blow up along a boundary patch (Propositions B.1 and B.3).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The dichotomy at $\mu_\Gamma^+=1$ may be a general phenomenon for conformally invariant Dirichlet problems modelled on the hyperbolic metric: the same parameter should govern compactness of solution families and the validity of boundary estimates for other fully nonlinear curvature equations.
  • Because the moving-spheres argument runs from centres below the boundary, it sidesteps the failure of the Hopf lemma; this variant may transfer to other degenerate-elliptic boundary problems where solutions touch at the boundary with equal normal derivatives.
  • The failure of local boundary $C^0$ estimates for $\mu_\Gamma^+\leq 1$ suggests that any existence theorem for the fully nonlinear Loewner-Nirenberg problem on compact manifolds in that range must use global geometric data rather than boundary-local barriers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The paper classifies positive viscosity solutions w of the fully nonlinear Loewner–Nirenberg problem f(λ(−A_w)) = 1/2, λ(−A_w) ∈ Γ in R^n_+, with w = 0 on ∂R^n_+. Under the standard hypotheses (1.3)–(1.8) the authors prove that the hyperbolic solution w^(0)(x)=x_n is the unique solution when μ_Γ^+ > 1, and that when μ_Γ^+ ≤ 1 the solution set is a one-parameter family {w^(a)(x_n)}_{a≥0} with w^(a)(1)=1+a, each solution depending only on x_n. The proof has three stages: C^0 bounds via barriers, C^{2,α} and C^1 regularity up to the boundary via Savin's small-perturbation theorem, and a moving-spheres argument with spheres centred in the lower half-space. The resulting one-dimensional ODE is analysed in Section 3, including existence, uniqueness, asymptotics and a first integral. As applications, the paper gives counterexamples to local boundary C^0 and gradient estimates when μ_Γ^+ ≤ 1 and a counterexample to the comparison principle on unbounded domains.

Significance. If the main classification is correct, it is a substantial contribution: it gives a complete Liouville theorem for a large class of conformally invariant fully nonlinear equations on the half-space, identifies the sharp parameter μ_Γ^+ as the criterion for uniqueness, and shows a surprising failure of local boundary estimates. The moving-spheres construction with centres below the boundary is a genuine novelty, and the ODE analysis is detailed and largely self-contained. The paper also honestly discusses limitations and connections to prior work. However, several load-bearing statements are made under relaxed assumptions (1.3′)–(1.5′) that the proofs do not actually support; the central classification under the full hypotheses (1.3)–(1.8) is plausible and likely repairable, but the manuscript in its current form overclaims its generality.

major comments (1)
  1. [§2.4, definition of G and Lemma 2.8] The regularity argument near ∂R^n_+ relies on the identity G(∇²û, ∇û, û, x) = x_n²(f(λ(−A_u)) − (1/2)e^{2u}) and on the normalization G(0,0,0,x) = 0. Both require f(tλ) = tf(λ), i.e. homogeneity. Under the stated relaxed assumptions (1.3′)–(1.5′), (1.6), (1.7), the rescaled operator G does not have this form, and the uniform ellipticity at (0,0,0,x) is not justified. Thus Lemma 2.8, and with it the C^1 up-to-boundary information used in Lemma 2.10, is not proved in the stated generality. This is a separate gap from the barrier issue, since even if one fixes the lower and upper C^0 bounds, the Savin-type perturbation argument in Step 1 of Lemma 2.8 requires the rescaling identity that homogeneity provides.
minor comments (2)
  1. [§2.4, near equation (2.6)] The sentence 'This follows from the fact that G is uniformly elliptic at (0,0,0,x) for x bounded away from 0 and infinity' should be expanded: one should explicitly note that (1/2)e ∈ Γ and that ∂f/∂λ_i > 0 at λ = e/2 by (1.6), since this is the actual hypothesis justifying the uniform ellipticity.
  2. [Throughout] There are a few minor typographical issues: in the line after (2.6), 'for e.g. r > 2δ^{-1}' appears to mean r > 2/δ; in the proof of Lemma 3.5 the interval notation in 'Dom(ψ) ∩ (0, ∞) = (0, ∞)' is redundant; and the displayed equation for G in §2.4 has a mismatch between the matrix argument and the preceding definition of  that would benefit from a brief explanatory sentence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the classification follows from a moving-spheres symmetry reduction plus ODE analysis; the imported comparison and regularity results are prior external theorems, and the one-parameter family is constructed from the ODE rather than fitted to the conclusion.

full rationale

After walking the derivation chain, I find no step in which a stated output is equivalent by construction to an input. The main classification (Theorem 1.3) rests on Proposition 2.1, which proves symmetry w = w(x_n) via a moving-spheres argument. That argument uses the comparison principle imported from [36, Theorem 3.2] through [22, Proposition 2.2] as Proposition 2.4. Although [36] shares an author with the present paper, the comparison principle is a separately published theorem used as standard machinery, not as a restatement of the desired Liouville theorem. The one-parameter family in Theorem 1.1 is generated from an ODE initial-value problem with w(0)=delta, w'(0)=p and w(1)=a, then taking delta to 0; this is a genuine construction from the equation, not an input-to-output identity. The ODE uniqueness in Section 3 uses classical ODE theory plus the imported comparison principle, but the comparison principle is not equivalent to the conclusion that every solution is one of the w(a). I also note two non-circular concerns: Proposition 2.4 is asserted for relaxed hypotheses (1.3')-(1.5') with no proof in this paper, and Lemma 2.6's hyperbolic-metric barrier step uses f((1/2)e)=1/2, which homogeneity (1.5) guarantees but the relaxed (1.5') does not; these are hypothesis-coverage issues, not circular reductions. Since no prediction is renamed from a fit and no uniqueness theorem is imported except a genuine earlier theorem, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No constants are fitted to data; the family parameter a is a genuine degree of freedom of the solution set, not an ad hoc fitting parameter. The central claim rests on standard PDE tools plus the paper-specific weak concavity condition and an ellipticity assertion.

assumptions (6)
  • domain assumption Standard structural assumptions (1.3)-(1.8): Γ is an open, convex, connected, symmetric cone with vertex at 0, Γ_n^+⊆Γ⊆Γ_1^+, and f is homogeneous of degree one, symmetric, positive with f=0 on ∂Γ and all partial derivatives positive.
    Defines the class of equations studied; Theorems 1.1, 1.3 and 1.5 are stated for (f,Γ) satisfying these assumptions.
  • domain assumption Weak concavity condition (1.8'): f(t,1,...,1) ≤ f(e) + ∂_{λ_1}f(e)(t-1) for t>-μ_Γ^+.
    Used in Lemma 3.6 to prove the key inequality A(s)≥1/(ns), which underpins the ODE analysis and the definition of the Hamiltonian G.
  • standard math Li-Nguyen-Wang comparison principle [36, Theorem 3.2], imported as Propositions 2.3-2.4.
    Foundational for the strict comparison in the moving spheres argument and for the ODE uniqueness arguments in Section 3.
  • standard math Savin's small perturbation theorem [45].
    Used in Lemma 2.8 to obtain C^{2,α} regularity near the boundary from C^0 bounds.
  • ad hoc to paper Uniform ellipticity of the rescaled operator G in the perturbation window Σ_{rδ}.
    Asserted in Section 2.4 ('G is uniformly elliptic in this range'); the verification is sketched but not fully written out.
  • standard math Classical regularity theory for viscosity solutions of uniformly elliptic equations (Caffarelli-Cabré [5]).
    Used in Lemma 3.4 to pass from viscosity to classical solutions of the ODE.

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Pith. "Pith review of The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates." pith.science (2026). https://pith.science/paper/MFW5XHUU

@misc{pith2026250716383,
  author       = {Pith},
  title        = {Pith review of: The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MFW5XHUU}},
  note         = {Machine review of arXiv:2507.16383}
}
abstract

In this paper we give a complete classification of positive viscosity solutions $w$ to conformally invariant equations of the form \begin{align}\label{ab}\tag{$*$} \begin{cases} f(\lambda(-A_w)) = \frac{1}{2}, \quad \lambda(-A_w)\in\Gamma & \text{in }\mathbb{R}_+^n \newline w = 0 & \text{on }\partial\mathbb{R}_+^n, \end{cases} \end{align} where $A_w$ is the Schouten tensor of the metric $g_w = w^{-2}|dx|^2$, $\Gamma\subset\mathbb{R}^n$ is a symmetric convex cone and $f$ is an associated defining function satisfying standard assumptions. Solutions to \eqref{ab} yield metrics $g_w$ of negative curvature-type which are locally complete near $\partial\mathbb{R}_+^n$. In particular, when $(f,\Gamma) = (\sigma_1,\Gamma_1^+)$, \eqref{ab} is the Loewner-Nirenberg problem in the upper half-space. More precisely, let $\mu_\Gamma^+$ denote the unique constant satisfying $(-\mu_\Gamma^+, 1,\dots,1)\in\partial\Gamma$. We show that when $\mu_\Gamma^+ >1$ (e.g. when $\Gamma = \Gamma_k^+$ for $k<\frac{n}{2}$), the hyperbolic solution $w^{(0)}(x) := x_n$ is the unique solution to \eqref{ab}. More surprisingly, we show that when $\mu_\Gamma^+ \leq 1$ (e.g. when $\Gamma = \Gamma_k^+$ for $k\geq \frac{n}{2}$), the solution set consists of a monotonically increasing one-parameter family $\{w^{(a)}(x_n)\}_{a\geq 0}$, of which the hyperbolic solution $w^{(0)}$ is the minimal solution. In either case, solutions of \eqref{ab} are functions of $x_n$. Our proof involves a novel application of the method of moving spheres for which we must establish new estimates and regularity near $\partial\mathbb{R}_+^n$, followed by a delicate ODE analysis. As an application, we give counterexamples to local boundary $C^0$ estimates on solutions to the fully nonlinear Loewner-Nirenberg problem when $\mu_\Gamma^+ \leq 1$.

Figures

Figures reproduced from arXiv: 2507.16383 by the authors.

Figure 1
Figure 1. The set-up for the method of moving spheres. Note that since Pν is the reflection of {xn = 0} across Hν, the two angles indicated in the figure are equal. Now for any two distinct points x = (x ′ , xn) and xe = (xe ′ , xn) in R n + with |x ′ | = |xe ′ |, for ν = x−xe |x−xe| we have x − 2(x · ν)ν = xe, and thus w(x) ≥ w(xe) by (2.9). Reversing the roles of x and xe, we therefore see that w(x) = w(xe). Finally, for an… view at source ↗
Figure 2
Figure 2. The cone with vertex at −Rv indicated in the figure is generated by vectors in ∂U. If s is sufficiently small and the vector µ satisfies dist(µ, ∂U) ≤ s, then Σs is the shaded region in the figure. As s decreases, the maximum height η(s) of a point in Σs decreases. Lemma 2.10. For any ν ∈ S n−1 + \{(0, . . . , 0, 1)} and R > 0, there exists s0 = s0(ν, R, w) > 0 such that w ≥ wν,R in Σs0 · ·= {x = −Rν + rµ ∈ Dν,R : µ… view at source ↗

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Works this paper leans on

52 extracted references · 35 canonical work pages

  1. [1]

    A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. V , Vestnik Leningrad. Univ., 13 (1958), pp. 5–8

    work page 1958

  2. [2]

    A viles and R

    P. A viles and R. C. McOwen, Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds , Duke Math. J., 56 (1988), pp. 395–398

    1988

  3. [3]

    Berestycki and L

    H. Berestycki and L. Nirenberg , On the method of moving planes and the sliding method , Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), pp. 1–37

    work page 1991

  4. [4]

    Caffarelli, L

    L. Caffarelli, L. Nirenberg, and J. Spruck , The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian , Acta Math., 155 (1985), pp. 261–301

    1985

  5. [5]

    L. A. Caffarelli and X. Cabr´e, Fully nonlinear elliptic equations, vol. 43 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 1995

    work page 1995

  6. [6]

    L. A. Caffarelli, B. Gidas, and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth , Comm. Pure Appl. Math., 42 (1989), pp. 271–297

    work page 1989

  7. [7]

    J. S. Case and Y. W ang, Boundary operators associated to the σk-curvature, Adv. Math., 337 (2018), pp. 83–106

    work page 2018

  8. [8]

    S.-Y. A. Chang, M. J. Gursky, and P. C. Yang, An a priori estimate for a fully nonlinear equation on four-manifolds , J. Anal. Math., 87 (2002), pp. 151–186

    2002

Show all 52 references
  1. [9]

    2 of New Stud

    , Entire solutions of a fully nonlinear equation , in Lectures on partial differential equations, vol. 2 of New Stud. Adv. Math., Int. Press, Somerville, MA, 2003, pp. 43–60

    2003

  2. [10]

    W. X. Chen and C. Li , Classification of solutions of some nonlinear elliptic equations , Duke Math. J., 63 (1991), pp. 615–622

    1991

  3. [11]

    Chipot, I

    M. Chipot, I. Shafrir, and M. Fila , On the solutions to some elliptic equations with nonlinear Neumann boundary conditions , Adv. Differential Equations, 1 (1996), pp. 91–110

    1996

  4. [12]

    B. Z. Chu, Y. Y. Li, and Z. Li, Liouville theorems for conformally invariant fully nonlinear equations I, arXiv:2311.07542 [math.AP], (2023)

    2023 arXiv

  5. [13]

    , Liouville Theorem with Boundary Conditions from Chern–Gauss–Bonnet Formula , https://arxiv.org/abs/2410.16384, (2024)

    2024 arXiv

  6. [14]

    I , https://arxiv.org/abs/2410.09683, (2024)

    , On the fully nonlinear Yamabe problem with constant boundary mean curvature. I , https://arxiv.org/abs/2410.09683, (2024)

    2024 arXiv

  7. [15]

    J. A. J. Duncan and L. Nguyen , The σk-Loewner-Nirenberg problem on Riemannian manifolds for k <n 2 , arxiv.org/abs/2310.01346, to appear in Anal. PDE, (2023). 40

    2023 arXiv

  8. [16]

    , The σk-Loewner-Nirenberg problem on Riemannian manifolds for k = n 2 and beyond, (2025)

    2025

  9. [17]

    J. F. Escobar , Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate , Comm. Pure Appl. Math., 43 (1990), pp. 857–883

    1990

  10. [18]

    F ang, B

    H. F ang, B. Ma, and W. Wei , A Liouville’s theorem for some Monge-Amp` ere type equations , J. Funct. Anal., 285 (2023), pp. Paper No. 109973, 47

    2023

  11. [19]

    Ge and G

    Y. Ge and G. W ang , On a fully nonlinear Yamabe problem , Ann. Sci. ´Ecole Norm. Sup. (4), 39 (2006), pp. 569–598

    2006

  12. [20]

    Gidas, W

    B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), pp. 209–243

    1979

  13. [21]

    7 of Adv

    , Symmetry of positive solutions of nonlinear elliptic equations in Rn, in Mathematical analysis and applications, Part A, vol. 7 of Adv. Math. Suppl. Stud., Academic Press, New York-London, 1981, pp. 369–402

    1981

  14. [22]

    M. d. M. Gonz ´alez, Y. Y. Li, and L. Nguyen , Existence and uniqueness to a fully nonlinear version of the Loewner-Nirenberg problem , Commun. Math. Stat., 6 (2018), pp. 269–288

    2018

  15. [23]

    M. J. Gursky, J. Streets, and M. W arren, Existence of complete conformal metrics of negative Ricci curvature on manifolds with boundary , Calc. Var. PDE, 41 (2011), pp. 21–43

    2011

  16. [24]

    M. J. Gursky and J. A. Viaclovsky , Prescribing symmetric functions of the eigenvalues of the Ricci tensor, Ann. of Math. (2), 166 (2007), pp. 475–531

    2007

  17. [25]

    Han and W

    Q. Han and W. Shen , The Loewner-Nirenberg problem in singular domains , J. Funct. Anal., 279 (2020), pp. 108604, 43

    2020

  18. [26]

    Q. Jin, Y. Y. Li, and H. Xu , Nonexistence of positive solutions for some fully nonlinear elliptic equations, Methods Appl. Anal., 12 (2005), pp. 441–449

    2005

  19. [27]

    Li and Y

    A. Li and Y. Y. Li , On some conformally invariant fully nonlinear equations , Comm. Pure Appl. Math., 56 (2003), pp. 1416–1464

    2003

  20. [28]

    , On some conformally invariant fully nonlinear equations. II. Liouville, Harnack and Yamabe , Acta Math., 195 (2005), pp. 117–154

    2005

  21. [29]

    , A fully nonlinear version of the Yamabe problem on manifolds with boundary , J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 295–316

    2006

  22. [30]

    Y. Y. Li , Degenerate conformally invariant fully nonlinear elliptic equations , Arch. Ration. Mech. Anal., 186 (2007), pp. 25–51

    2007

  23. [31]

    Pure Appl

    , Local gradient estimates of solutions to some conformally invariant fully nonlinear equations , Comm. Pure Appl. Math., 62 (2009), pp. 1293–1326

    2009

  24. [32]

    Y. Y. Li, H. Lu, and S. Lu , A Liouville Theorem for M¨ obius Invariant Equations, Peking Mathe- matical Journal (2021)

    2021

  25. [33]

    Y. Y. Li and L. Nguyen , A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary , https://arxiv.org/abs/0911.3366, (2009)

    2009 arXiv

  26. [34]

    , A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound, J. Funct. Anal., 266 (2014), pp. 3741–3771. 41

    2014

  27. [35]

    Pure Appl

    , Harnack inequalities and Bˆ ocher-type theorems for conformally invariant, fully nonlinear degen- erate elliptic equations , Comm. Pure Appl. Math., 67 (2014), pp. 1843–1876

    2014

  28. [36]

    Y. Y. Li, L. Nguyen, and B. W ang , Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations , Calc. Var. Partial Differential Equations, 57 (2018), pp. Paper No. 96, 29

    2018

  29. [37]

    Y. Y. Li, L. Nguyen, and J. Xiong , Regularity of viscosity solutions of the σk-Loewner-Nirenberg problem, Proc. Lond. Math. Soc. (3), 127 (2023), pp. 1–34

    2023

  30. [38]

    Y. Y. Li and L. Zhang , Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), pp. 27–87

    2003

  31. [39]

    Y. Y. Li and M. Zhu , Uniqueness theorems through the method of moving spheres , Duke Math. J., 80 (1995), pp. 383–417

    1995

  32. [40]

    Liouville , Sur l’´ equation aux diff´ erences partiellesd2 log λ dudv ± λ 2a2 = 0., Journal de Math´ ematiques Pures et Appliqu´ ees, (1853), pp

    J. Liouville , Sur l’´ equation aux diff´ erences partiellesd2 log λ dudv ± λ 2a2 = 0., Journal de Math´ ematiques Pures et Appliqu´ ees, (1853), pp. 71–72

  33. [41]

    Loewner and L

    C. Loewner and L. Nirenberg , Partial differential equations invariant under conformal or projec- tive transformations , in Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York-London, 1974, pp. 245–272

    1974

  34. [42]

    Lou and M

    Y. Lou and M. Zhu , Classifications of nonnegative solutions to some elliptic problems , Differential Integral Equations, 12 (1999), pp. 601–612

    1999

  35. [43]

    Obata, The conjectures on conformal transformations of Riemannian manifolds , J

    M. Obata, The conjectures on conformal transformations of Riemannian manifolds , J. Differential Geometry, 6 (1971/72), pp. 247–258

    1971

  36. [44]

    Osserman, On the inequality ∆u ≥ f (u), Pacific J

    R. Osserman, On the inequality ∆u ≥ f (u), Pacific J. Math., 7 (1957), pp. 1641–1647

    1957

  37. [45]

    Savin, Small perturbation solutions for elliptic equations , Comm

    O. Savin, Small perturbation solutions for elliptic equations , Comm. Partial Differential Equations, 32 (2007), pp. 557–578

    2007

  38. [46]

    R. M. Schoen and S.-T. Yau , Conformally flat manifolds, Kleinian groups and scalar curvature , Invent. Math., 92 (1988), pp. 47–71

    1988

  39. [47]

    Serrin, A symmetry problem in potential theory , Arch

    J. Serrin, A symmetry problem in potential theory , Arch. Rational Mech. Anal., 43 (1971), pp. 304– 318

    1971

  40. [48]

    S. Tang, L. W ang, and M. Zhu , Nonlinear elliptic equations on the upper half space , Commun. Contemp. Math., 24 (2022), pp. Paper No. 2050085, 20

    2022

  41. [49]

    J. A. Viaclovsky , Conformally invariant Monge-Amp` ere equations: global solutions , Trans. Amer. Math. Soc., 352 (2000), pp. 4371–4379

    2000

  42. [50]

    Wei , Liouville Theorem for k-curvature equation with fully nonlinear boundary in half space , https://arxiv.org/abs/2403.19268, (2024)

    W. Wei , Liouville Theorem for k-curvature equation with fully nonlinear boundary in half space , https://arxiv.org/abs/2403.19268, (2024)

    2024

  43. [51]

    Yuan, The partial uniform ellipticity and prescribed problems on the conformal classes of complete metrics, https://arxiv.org/abs/2203.13212, (2022)

    R. Yuan, The partial uniform ellipticity and prescribed problems on the conformal classes of complete metrics, https://arxiv.org/abs/2203.13212, (2022)

    2022 arXiv

  44. [52]

    , Notes on conformal metrics of negative curvature on manifolds with boundary , J. Math. Study, 57 (2024), pp. 373–378. 42

    2024

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Reviewed August 6, 2026 · model on record in the stance chip above.