Pith. sign in

REVIEW 3 major objections 4 minor 60 references

The $\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\frac{n}{2}$ and beyond

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves existence of complete conformal metrics solving the fully nonlinear Loewner-Nirenberg problem whenever the cone parameter exceeds $1-\delta$, covering the threshold case $k=n/2$.

desk verdict A genuine threshold result for the fully nonlinear Loewner-Nirenberg problem, with the main risk being the paper's reliance on an external gradient estimate whose uniformity at the critical case is asserted but not reproved. read the letter →

arxiv 2507.16394 v1 pith:4W5ADCM6 submitted 2025-07-22 math.AP math.DG

Jonah A. J. Duncan , Luc Nguyen This is my paper · ORCID
classification math.APmath.DG MSC 35J6035D4053C2158J05
keywords fullynonlinearLoewner-NirenbergproblemSchoutentensorsigma_k-curvatureadmissibleconeviscositysolutionconformalmetricellipticregularisationRiemannianmanifoldwithboundary
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 proves that the fully nonlinear Loewner-Nirenberg problem—the conformal uniformisation problem prescribing a symmetric function of the eigenvalues of the Schouten tensor—admits a complete conformal metric on any compact Riemannian manifold with non-empty boundary, provided the cone parameter $\mu_\Gamma^+$ exceeds $1-\delta$. The constant $\delta$ depends only on bounds for the Ricci curvature, injectivity radii, diameter, boundary mean curvature, and a H\"older norm. In particular, this settles the $\sigma_k$-Loewner-Nirenberg problem for all $k\le n/2$, including the borderline case $k=n/2$ that resisted earlier methods. A second group of results shows that a solution exists whenever any conformal metric already keeps $-g^{-1}A_g$ inside the cone, with no condition on $\mu_\Gamma^+$; the solution is smooth and unique when $(1,0,\dots,0)$ lies in the interior of $\Gamma$.

What carries the argument

Two devices carry the argument. The first is the cone parameter $\mu_\Gamma^+$, defined by $(-\mu_\Gamma^+,1,\dots,1)\in\partial\Gamma$, which measures how wide the cone $\Gamma$ is; previous methods worked only for $\mu_\Gamma^+>1$, while $\Gamma^+_{n/2}$ has $\mu_\Gamma^+=1$. The second is an explicit conformal deformation $g_N=e^{2e^{Nv}}g$ generated by a function $v\ge1$ with no critical points; a direct eigenvalue computation shows $\lambda(-g_N^{-1}A_{g_N})$ lies in $\Gamma$ once $\mu_\Gamma^+$ is close enough to $1$, and a term that earlier arguments dropped is retained to extend the range to $\mu_\Gamma^+\ge1$. Compactness of Riemannian manifolds with boundary under the uniform bounds, via harmonic-radius estimates and $C^{1,\sigma}$-precompactness, converts this qualitative statement into a uniform gap $\delta>0$. For the solution step, the $\tau$-regularisation $f^\tau(\lambda)=f(\tau\lambda+(1-\tau)\sigma_1(\lambda)e)/(\tau+n(1-\tau))$ and $\Gamma^\tau=\{\lambda:\tau\lambda+(1-\tau)\sigma_1(\lambda)e\in\Gamma\}$ makes the equation elliptic for $\tau<1$; the continuity method, two-sided $C^0$ estimates, and the imported local interior gradient estimate produce solutions, and an explicit annulus barrier controls the boundary gradient and asymptotic behaviour.

What would settle it

Look for a sequence of approximating problems at $\mu_\Gamma^+=1$ with two-sided $C^0$ bounds but $|\nabla u|/u$ unbounded uniformly in the regularisation parameter; the paper itself records that one-sided variants fail at $\mu_\Gamma^+=1$, so the two-sided uniformity is exactly the load-bearing input. A cheaper computational check is to solve the radial equation on a Euclidean annulus for $\Gamma^+_{n/2}$ and test whether the $C^0$-normalised gradient bound is independent of $\tau$; a blow-up would falsify the central existence claim.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for any manifold in the compactness class $\mathcal{M}^n_\sigma$ with uniform bounds on Ricci curvature, injectivity radii, boundary mean curvature, and diameter, there exists $\delta>0$ such that whenever $\mu_\Gamma^+>1-\delta$, the equation $f(\lambda(-g_u^{-1}A_{g_u}))=1/2$ with $u=0$ on $\partial M$ has a maximal locally Lipschitz viscosity solution $g_u=u^{-2}g_0$ satisfying $u/d_{g_0}(\cdot,\partial M)\to 1$. The proof splits into two independent statements. Theorem 1.6 constructs a conformal metric $g$ with $\lambda(-g^{-1}A_g)\in\Gamma$ when $\mu_\Gamma^+>1-\delta$, using a critical-point-free auxiliary function and $C^{1,\sigma}$-compactness of manifolds with boundary. Theorem 1.7 turns any admissible conformal metric into a solution of the boundary-value problem, using elliptic regularisation and stability of two-sided interior gradient estimates as the regularisation parameter tends to $1$. Consequently the $\sigma_k$ problem is solved for all $k\le n/2$, and the Dirichlet version with positive boundary data is solved under the same admissible-metric hypothesis.

Load-bearing premise

The load-bearing premise is that the local interior gradient estimate imported from the literature remains valid uniformly as the regularisation parameter approaches $1$ and as the cone approaches the critical case $\mu_\Gamma^+=1$; if this uniformity fails, the limiting argument that produces the solution collapses.

Editorial extensions

If this is right

  • For every $k\le n/2$, the $\sigma_k$-Loewner-Nirenberg problem on any compact Riemannian manifold with boundary in the class $\mathcal{M}^n_\sigma$ admits a complete locally Lipschitz viscosity solution with $u/d_{g_0}(\cdot,\partial M)\to1$ at the boundary.
  • If $(1,0,\dots,0)\in\Gamma$, the solution is smooth and is the unique continuous viscosity solution with $u=0$ on $\partial M$.
  • The Dirichlet problem with positive boundary data is solvable whenever an admissible conformal metric exists; the solution is Lipschitz, and smooth and unique when $(1,0,\dots,0)\in\Gamma$.
  • Existence follows from the mere existence of a conformal metric $g$ with $\lambda(-g^{-1}A_g)\in\Gamma$, with no condition on $\mu_\Gamma^+$; this covers cases with $(1,0,\dots,0)\in\partial\Gamma$ that were previously open.
  • In dimension three, the fully nonlinear Loewner-Nirenberg problem is now solved for all admissible cones, combining Theorem 1.7 with the existing three-dimensional existence result [60].

Reading between the lines

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

  • Beyond the paper: the uniform gap $\delta$ in Theorem 1.2 is obtained by contradiction and is non-constructive; the explicit eigenvalue computation in Lemma 2.1 suggests a quantitative lower bound in terms of the $C^1$ size of the metric and an upper bound on the Schouten tensor, which would indicate exactly how far past $\mu=1$ the method reaches.
  • Beyond the paper: the same compactness argument shows that the existence statement is stable under $C^{1,\sigma}$ perturbations of the background metric, so the admissible-metric hypothesis in Theorem 1.7 is effectively an open condition in that topology.
  • Beyond the paper: at the threshold $\mu_\Gamma^+=1$ with $(1,0,\dots,0)\in\partial\Gamma$, the solutions should be expected to be merely Lipschitz rather than smooth, consistent with the known non-differentiability examples on annuli; the paper does not claim otherwise.
  • Beyond the paper: a natural test of sharpness is whether $\delta$ in Theorem 1.2 can be replaced by the full range $\mu_\Gamma^+>0$ in dimensions $n\ge4$, or whether the known failure of one-sided boundary gradient estimates at $\mu_\Gamma^+\le1$ marks a genuine obstruction.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper studies the fully nonlinear Loewner-Nirenberg problem on compact Riemannian manifolds with non-empty boundary. The main result, Theorem 1.2, asserts that for cones Γ with μ_Γ^+ > 1−δ, where δ depends on geometric bounds, there exists a maximal locally Lipschitz viscosity solution with boundary asymptotics u/d→1; in particular, it covers the threshold case k=n/2 for the σ_k problem. The proof is split into Theorem 1.6, which constructs an admissible conformal metric when μ_Γ^+ is close to 1 via a barrier/compactness argument, and Theorem 1.7, which produces solutions from the existence of any admissible metric. A Dirichlet version with positive boundary data is also proved. The genuinely new analytical content is the treatment of μ_Γ^+=1 by elliptic regularization τ→1.

Significance. If the proof is completed, the results are significant: they settle the σ_k-Loewner-Nirenberg problem for all k≤n/2, including the borderline case k=n/2, and provide a general existence criterion based on the presence of an admissible metric, without any assumption on μ_Γ^+. The paper contains several well-executed new tools: Lemma 2.1 (construction of admissible metrics for μ_Γ^+≥1 by keeping a term dropped by Yuan), Proposition 3.8 (an explicit annulus barrier), and the C^{1,σ}-compactness argument with boundary. The main caveat is that the threshold case imports its key interior gradient estimate from the preprint [10] and several convergence/asymptotics arguments from the authors' earlier work [14,48], rather than proving them in the present setting.

major comments (3)
  1. [§3.5, proof of Theorem 1.7′; see also §1] The proof of the threshold case τ0=1 relies on [10, Theorem 7.1], a local interior gradient estimate depending on two-sided C^0 bounds, asserted to hold uniformly for all τ≤1 and under perturbations of (f,Γ). This uniformity is what allows the subsequential convergence to u∈C^{0,1}_{loc} in the limit τ→1; without it the limit construction collapses. The estimate is imported from an arXiv preprint and is not proved or stated in the paper. Please either include a proof or state the precise theorem and verify that the family (f^τ,Γ^τ) satisfies its hypotheses uniformly, including at τ=1. This is load-bearing for Theorems 1.7 and 1.2.
  2. [§3.2, Proposition 3.3 and Lemma 3.5] From f(λ)≤σ1(λ)/n (proved in Proposition 3.3) and f=1/2, one obtains σ1(−g_u^{−1}A_{g_u})≥n/2. Since σ1(−g^{−1}A_g)=−R_g/(2(n−1)), this yields R_{g_u}≤−n(n−1), not the stated R_{g_u}≤−2n(n−1). The comparison solution v in Lemma 3.5 solves R_{g_v}=−2n(n−1), so the comparison in Proposition 3.3 is in the wrong direction as written; the lower bounds in Propositions 3.3 and 3.6, and the lower boundary gradient estimate in Proposition 3.7, depend on this comparison. The factor in (3.3) and the displayed conformal-transformation formula for R_{g_v} need to be corrected.
  3. [§3.5, proof of Theorem 1.7′; §3.4] The boundary asymptotics and maximality are transferred verbatim from [14, Section 4], and the viscosity-convergence argument is transferred from [48, Theorems 1.3 and 1.4]. The present setting includes μ_Γ^+=1 and (1,0,…,0)∈∂Γ, where the original hypotheses of [14] (μ_Γ^+>1) are not satisfied. Please spell out why the arguments of [14, Section 4] apply to the regularized family (f^τ,Γ^τ) uniformly as τ→1, or provide the necessary adaptations.
minor comments (4)
  1. [§2.1, equation (2.6)] The displayed conformal transformation formula is missing the positive factor e^{−2e^{Nv}} on the first three terms; since Γ is a cone this does not affect the cone-membership conclusion of Lemma 2.1, but the identity as written is not correct.
  2. [§2.3, proof of Theorem 1.6′ and §2.4] There are typographical errors: 'satifying' in (2.1) and a double '∈∈' in the proof of Theorem 2.8.
  3. [§3.3, Proposition 3.8] After defining v(r) in (3.6), the boundary conditions are written as v(x)=δ on {r=r1}; the notation v(r1)=δ and v(r2)=m would be clearer.
  4. [§3.4, proof of Theorem 1.10′] The statement that the viscosity-convergence argument in the τ0=1 case is identical to [48, Theorem 1.4] is terse; since the present cone is fully nonlinear and not restricted to σ_k on annuli, a brief outline of the stability argument would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the threshold case is built from an external gradient estimate plus the authors' prior work as proof templates, not from the conclusion itself.

full rationale

The paper's central result, Theorem 1.2, is assembled from two independent ingredients: Theorem 1.6 constructs an admissible conformal metric when mu_Gamma^+ > 1 - delta using Lemma 2.1 and C^{1,sigma} compactness, and Theorem 1.7 converts the existence of any admissible metric into a solution of the Loewner-Nirenberg problem. Neither ingredient is assumed as the conclusion. The genuinely new threshold case mu_Gamma^+ = 1 is treated in Section 3.5 by taking tau -> 1, and the limit is justified by the interior gradient estimate of Chu, Li and Li [10, Theorem 7.1]. That estimate is external to this paper: its authors are not the present authors, and the paper does not reprove it. Depending on an unproved external estimate is a correctness or completeness risk, not circularity. The paper's self-citations, mainly to [14, Section 4] and [48, Theorems 1.3 and 1.4], are used as proof templates for boundary asymptotics, maximality, and viscosity limits; they do not contain the threshold result, since [14] handled only mu_Gamma^+ > 1. No fitted parameter is renamed as a prediction, and no quantity is defined in terms of the conclusion. Thus the derivation is not circular, though it does lean on prior work that is not fully reproduced here.

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

No numerical constants are fitted to data. The constants delta, epsilon, eta and N are existential constants chosen from geometric bounds, not fitted values. The paper proposes no new physical or geometric entities, so the invented-entities ledger is empty. The main external inputs are analytic estimates from prior papers, listed as axioms.

assumptions (6)
  • domain assumption Chu-Li-Li local interior gradient estimate depending on two-sided C^0 bounds, uniform for all tau<=1 and stable under perturbations of (f,Gamma) [10, Theorem 7.1]
    Imported in Sections 3.3 to 3.5 to pass from C^0 estimates to gradient estimates at mu_Gamma^+=1; not proved in this paper.
  • domain assumption Comparison principle for viscosity solutions of (3.2) and (3.1) in the non-Euclidean setting (cited to [14, Proposition 3.7] and [49])
    Used in Propositions 3.3, 3.6 and 3.7 and in the uniqueness claims.
  • domain assumption Anderson-Katsuda-Kurylev-Lassas-Taylor compactness and harmonic radius estimates for manifolds with boundary [2], including the dichotomy that pointed limits are R^n or R^n_+
    Used in the proof of Theorem 2.4 in Appendix A to obtain uniform harmonic coordinates.
  • domain assumption Transferability of boundary asymptotics, maximality and viscosity-convergence arguments from [14, Section 4] and [48, Theorems 1.3 and 1.4] to the present setting
    The proof of Theorem 1.7' states that the arguments apply verbatim.
  • domain assumption Aviles-McOwen existence of complete conformal metrics with negative scalar curvature [6], used in Lemma 3.5 and Proposition 3.6
    Provides sub- and supersolutions and comparison for the C^0 estimates.
  • standard math Existence of a smooth function v>=1 with no critical points on any compact manifold with non-empty boundary
    Used in Lemma 2.1 as the auxiliary function; standard but unproved in the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The $\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\frac{n}{2}$ and beyond." pith.science (2026). https://pith.science/paper/4W5ADCM6

@misc{pith2026250716394,
  author       = {Pith},
  title        = {Pith review of: The $\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k=\fracn2$ and beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4W5ADCM6}},
  note         = {Machine review of arXiv:2507.16394}
}
abstract

Let $(M^n,g_0)$ be a smooth compact Riemannian manifold of dimension $n\geq 3$ with smooth non-empty boundary $\partial M$. Let $\Gamma\subset\mathbb{R}^n$ be a symmetric convex cone and $f$ a symmetric defining function for $\Gamma$ satisfying standard assumptions. Denoting by $A_{g_u}$ the Schouten tensor of a conformal metric $g_u = u^{-2}g_0$, we show that the associated fully nonlinear Loewner-Nirenberg problem \begin{align*} \begin{cases} f(\lambda(-g_u^{-1}A_{g_u})) = \frac{1}{2}, \quad \lambda(-g_u^{-1}A_{g_u})\in\Gamma & \text{on }M\backslash \partial M \newline u = 0 & \text{on }\partial M \end{cases} \end{align*} admits a solution if $\mu_\Gamma^+ > 1-\delta$, where $\mu_\Gamma^+$ is defined by $(-\mu_\Gamma^+,1,\dots,1)\in\partial\Gamma$ and $\delta>0$ is a constant depending on certain geometric data. In particular, we solve the $\sigma_k$-Loewner-Nirenberg problem for all $k\leq \frac{n}{2}$, which extends recent work of the authors to include the important threshold case $k=\frac{n}{2}$. In the process, we establish that the fully nonlinear Loewner-Nirenberg problem and corresponding Dirichlet boundary value problem with positive boundary data admit solutions if there exists a conformal metric $g\in[g_0]$ such that $\lambda(-g^{-1}A_g)\in\Gamma$ on $M$; these latter results require no assumption on $\mu_\Gamma^+$ and are new when $(1,0,\dots,0)\in\partial\Gamma$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 59 canonical work pages

  1. [14]

    The $\sigma_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k<\frac{n}{2}$

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

    work page Pith review arXiv 2023

  2. [48]

    , Solutions to the σk-Loewner-Nirenberg problem on annuli are locally Lipschitz and not differen- tiable, J. Math. Study, 54 (2021), pp. 123–141

    work page 2021

  3. [10]

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

    work page Pith review arXiv 2023

  4. [1]

    P. T. Allen, J. Isenberg, J. M. Lee, and I. S. Allen, Weakly asymptotically hyperbolic manifolds, Comm. Anal. Geom., 26 (2018), pp. 1–61

    work page 2018

  5. [2]

    Anderson, A

    M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas, and M. Taylor , Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem , Invent. Math., 158 (2004), pp. 261–321

    work page 2004

  6. [3]

    M. T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds , Invent. Math., 102 (1990), pp. 429–445. 34

    work page 1990

  7. [4]

    Andersson, P

    L. Andersson, P. T. Chru ´sciel, and H. Friedrich , On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein ’s field equations , Comm. Math. Phys., 149 (1992), pp. 587–612

    work page 1992

  8. [5]

    A viles, A study of the singularities of solutions of a class of nonlinear elliptic partial differential equations, Comm

    P. A viles, A study of the singularities of solutions of a class of nonlinear elliptic partial differential equations, Comm. Partial Differential Equations, 7 (1982), pp. 609–643

    work page 1982

Show all 60 references
  1. [6]

    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

  2. [7]

    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

  3. [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

  4. [9]

    , An equation of Monge-Amp` ere type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2), 155 (2002), pp. 709–787

    2002

  5. [11]

    D. M. DeTurck and J. L. Kazdan , Some regularity theorems in Riemannian geometry , Ann. Sci. ´Ecole Norm. Sup. (4), 14 (1981), pp. 249–260

    1981

  6. [12]

    J. A. J. Duncan, Augmented Hessian equations on Riemannian manifolds: from integral to pointwise local second derivative estimates , Nonlinear Anal., 238 (2024), pp. Paper No. 113396, 22

    2024

  7. [13]

    J. A. J. Duncan and L. Nguyen , Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry , Adv. Math., 432 (2023), p. Paper No. 109263

    2023

  8. [15]

    , The fully nonlinear Loewner-Nirenberg problem: Liouville theorems and counterexamples to local boundary estimates, (submitted 2025)

    2025

  9. [16]

    L. C. Evans , Classical solutions of fully nonlinear, convex, second-order elliptic equations , Comm. Pure Appl. Math., 35 (1982), pp. 333–363

    1982

  10. [17]

    D. L. Finn , Existence of positive solutions to ∆gu = uq + Su with prescribed singularities and their geometric implications, Comm. Partial Differential Equations, 23 (1998), pp. 1795–1814

    1998

  11. [18]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger , Elliptic partial differential equations of second order , Classics in Mathematics, Springer-Verlag, Berlin, 2001

    2001

  12. [19]

    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

  13. [20]

    A. R. Gover and A. W aldron, Renormalized volume, Comm. Math. Phys., 354 (2017), pp. 1205– 1244

    2017

  14. [21]

    C. R. Graham , Volume renormalization for singular Yamabe metrics , Proc. Amer. Math. Soc., 145 (2017), pp. 1781–1792

    2017

  15. [22]

    C. R. Graham and M. J. Gursky , Chern-Gauss-Bonnet formula for singular Yamabe metrics in dimension four , Indiana Univ. Math. J., 70 (2021), pp. 1131–1166. 35

    2021

  16. [23]

    Guan, Complete conformal metrics of negative Ricci curvature on compact manifolds with boundary, Int

    B. Guan, Complete conformal metrics of negative Ricci curvature on compact manifolds with boundary, Int. Math. Res. Not. IMRN, (2008). Art. ID rnn 105, 25pp

    2008

  17. [24]

    J., 163 (2014), pp

    , Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian man- ifolds, Duke Math. J., 163 (2014), pp. 1491–1524

    2014

  18. [25]

    Math., 415 (2023), pp

    , The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds , Adv. Math., 415 (2023), pp. Paper No. 108899, 32

    2023

  19. [26]

    Guan and H

    B. Guan and H. Jiao , Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds , Calc. Var. Partial Differential Equations, 54 (2015), pp. 2693–2712

    2015

  20. [27]

    , The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds , Discrete Contin. Dyn. Syst., 36 (2016), pp. 701–714

    2016

  21. [28]

    P. Guan, J. Viaclovsky, and G. W ang, Some properties of the Schouten tensor and applications to conformal geometry , Trans. Amer. Math. Soc., 355 (2003), pp. 925–933

    2003

  22. [29]

    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

  23. [30]

    M. J. Gursky and J. Viaclovsky , Convexity and singularities of curvature equations in conformal geometry, Int. Math. Res. Not., (2006), pp. Art. ID 96890, 43

    2006

  24. [31]

    M. J. Gursky and J. A. Viaclovsky , Fully nonlinear equations on Riemannian manifolds with negative curvature, Indiana Univ. Math. J., 52 (2003), pp. 399–419

    2003

  25. [32]

    Q. Han, X. Jiang, and W. Shen , The Loewner-Nirenberg problem in cones , J. Funct. Anal., 287 (2024), p. 110566

    2024

  26. [33]

    Han and W

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

    2020

  27. [34]

    Hogg and L

    J. Hogg and L. Nguyen, Existence and uniqueness for the non-compact Yamabe problem of negative curvature type, Anal. Theory Appl., 40 (2024), pp. 57–91

    2024

  28. [35]

    Jiang and N

    F. Jiang and N. S. Trudinger , Oblique boundary value problems for augmented Hessian equations II, Nonlinear Anal., 154 (2017), pp. 148–173

    2017

  29. [36]

    , Oblique boundary value problems for augmented Hessian equations I , Bull. Math. Sci., 8 (2018), pp. 353–411

    2018

  30. [37]

    Partial Differential Equations, 44 (2019), pp

    , Oblique boundary value problems for augmented Hessian equations III, Comm. Partial Differential Equations, 44 (2019), pp. 708–748

    2019

  31. [38]

    Jiang, Boundary expansion for the Loewner-Nirenberg problem in domains with conic singularities , J

    X. Jiang, Boundary expansion for the Loewner-Nirenberg problem in domains with conic singularities , J. Funct. Anal., 281 (2021), pp. Paper No. 109122, 41

    2021

  32. [39]

    Khomrutai, Regularity of singular solutions to sigma(k)-Yamabe problems , ProQuest LLC, Ann Arbor, MI, 2009

    S. Khomrutai, Regularity of singular solutions to sigma(k)-Yamabe problems , ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–University of Notre Dame

    2009

  33. [40]

    N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations , Izv. Akad. Nauk SSSR Ser. Mat., 46 (1982), pp. 487–523, 670

    1982

  34. [41]

    , Boundedly inhomogeneous elliptic and parabolic equations in a domain , Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), pp. 75–108. 36

    1983

  35. [42]

    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

  36. [43]

    Li, Two flow approaches to the Loewner-Nirenberg problem on manifolds, J

    G. Li, Two flow approaches to the Loewner-Nirenberg problem on manifolds, J. Geom. Anal., 32 (2022), pp. Paper No. 7, 30

    2022

  37. [44]

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

    2021

  38. [45]

    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

  39. [46]

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

    2014

  40. [47]

    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

  41. [49]

    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

  42. [50]

    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

  43. [51]

    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

  44. [52]

    X.-N. Ma, N. S. Trudinger, and X.-J. W ang , Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech. Anal., 177 (2005), pp. 151–183

    2005

  45. [53]

    Mazzeo, Regularity for the singular Yamabe problem , Indiana Univ

    R. Mazzeo, Regularity for the singular Yamabe problem , Indiana Univ. Math. J., 40 (1991), pp. 1277– 1299

    1991

  46. [54]

    Mazzeo and F

    R. Mazzeo and F. Pacard , Poincar´ e-Einstein metrics and the Schouten tensor, Pacific J. Math., 212 (2003), pp. 169–185

    2003

  47. [55]

    Petersen , Riemannian geometry , vol

    P. Petersen , Riemannian geometry , vol. 171 of Graduate Texts in Mathematics, Springer, Cham, third ed., 2016

    2016

  48. [56]

    V´eron, Singularit´ es ´ eliminables d’´ equations elliptiques non lin´ eaires, J

    L. V´eron, Singularit´ es ´ eliminables d’´ equations elliptiques non lin´ eaires, J. Differential Equations, 41 (1981), pp. 87–95

    1981

  49. [57]

    J. A. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations , Duke Math. J., 101 (2000), pp. 283–316

    2000

  50. [58]

    Wu , Regularity of viscosity solutions of the σk-Yamabe-type Problem for k > n/ 2, https://arxiv.org/abs/2407.08300, (2024)

    J. Wu , Regularity of viscosity solutions of the σk-Yamabe-type Problem for k > n/ 2, https://arxiv.org/abs/2407.08300, (2024)

    2024 arXiv

  51. [59]

    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

  52. [60]

    , Notes on conformal metrics of negative curvature on manifolds with boundary , J. Math. Study, 57 (2024), pp. 373–378. 37

    2024

Pith tools

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