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Conformally compact metrics and the Lovelock tensors

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Pith's one-line read Conformally compact Lovelock metrics admit polyhomogeneous boundary expansions, generalizing the Einstein asymptotics.

desk verdict Real extension of BH14 to Lovelock metrics, but Theorem 1.1 is stated without the A1 nonzero hypothesis the proof actually needs; fixing that should be the referee's first demand. read the letter →

arxiv 2505.24188 v1 pith:4QBDV4DV submitted 2025-05-30 math.DG

Xinran Yu This is my paper
classification math.DG MSC 53C1853C2153C2558J20
keywords conformallycompactmetricsLovelocktensorspolyhomogeneousexpansionsFefferman-GrahamexpansionambientobstructiontensorsingularYamabeproblemDiracindexAdS/CFTcorrespondence
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 establishes that conformally compact metrics satisfying the Lovelock equations—a family of higher-curvature generalizations of the Einstein equation—admit polyhomogeneous expansions near their conformal boundary. That means the compactified metric can be written in powers of a defining function $x$ and, where needed, powers of $\log x$, so the formal Fefferman–Graham expansion is genuinely realized rather than being a formal device. In even boundary dimensions the expansion develops a logarithmic term whose leading coefficient is a trace-free tensor built from the boundary Schouten tensor, generalizing the ambient obstruction tensor of the Einstein case. The same methods produce a formal solution to the singular Yamabe-$(2q)$ problem and a Dirac-index obstruction to filling a spin manifold by such metrics when the scalar curvature is bounded below.

What carries the argument

The engine is the Lovelock tensor $F_\alpha(g)=\sum_q \alpha_q\big[(\operatorname{Ric}_g^{(2q)}-\lambda_{(2q)}g)-\frac{\alpha_q}{2q}(\operatorname{scal}_g^{(2q)}-(n+1)\lambda_{(2q)})g\big]$, a symmetric divergence-free polynomial in the Riemann tensor whose constants $\lambda_{(2q)}$ make hyperbolic space a solution. To get ellipticity, the paper uses the modified tensor $Q_\alpha(g,t)=F_\alpha(g)-\Phi_\alpha(g,t)$, where the gauge-fixing term $\Phi_\alpha$ cancels the Bianchi terms in the linearization; the resulting principal operator is $A_1(\alpha)/4[-(n-1)(\Delta+2n)(C_g(r)g)+2(\Delta-2)r_0]+O(x^{N+1})$. The coefficient $A_1(\alpha)=\sum_q \alpha_q(-\tfrac12)^{q-1}\frac{(n-2)!(2q)!}{2(n-2q)!}$ is what makes the operator elliptic. The argument then uses the known indicial roots of $\Delta+2n$ and $\Delta-2$ on trace and trace-free symmetric two-tensors, together with Green's operators $G_\infty$ and $G_0$ adapted to weighted Hölder and polyhomogeneous spaces, to construct corrections at an increasing sequence of weights $\mu_++a_k$; when an indicial root is hit, a logarithmic term is inserted exactly as in the Einstein case.

What would settle it

Choose a Lovelock coupling vector $\alpha$ with $A_1(\alpha)=0$ that still admits an asymptotically hyperbolic solution, and solve the formal Fefferman–Graham recursion to order $n-1$; if the recursion is underdetermined there or forces a power $x^\beta$ outside the monoid generated by the indicial roots, then the polyhomogeneity theorem as stated fails for that $\alpha$. The paper's hypothesis $1\in\operatorname{LimSec}(\alpha)$ excludes this case by definition, so the open question is whether the stated theorem can be freed from it.

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Extended reading notes

Core claim

The central claim, Theorem 6.1, is that any asymptotically hyperbolic Lovelock metric with a smooth conformal infinity is polyhomogeneous in a collar neighborhood of the boundary. Concretely, with $g=(dx^2+h_x)/x^2$, the family $h_x$ has an expansion $h_x=h_0+h_2x^2+\cdots+h_{n-1}x^{n-1}+h_nx^n+\cdots$ when $n$ is odd, and $h_x=h_0+h_2x^2+\cdots+h_{n,1}x^{n-1}\log x+h_nx^n+\cdots$ when $n$ is even. The paper proves that every such metric actually has this expansion, not merely that a formal expansion can be written down. The argument fixes a gauge by modifying the Lovelock tensor with a Bianchi-gauge term so that its linearization at an asymptotically hyperbolic metric is a Laplace-type operator with principal part $A_1(\alpha)[-(n-1)(\Delta+2n)(C_g(r)g)+2(\Delta-2)r_0]/4$; invertibility of the two Laplace pieces, control of indicial roots, and Green's operators on weighted spaces then build an approximating sequence whose remainder decays to all orders. The paper also computes the leading-order term of the even-dimensional obstruction tensor and derives formal solutions and filling obstructions from the same expansion machinery.

Load-bearing premise

The proof needs the nondegeneracy condition $A_1(\alpha)\neq 0$ (equivalently $1\in \operatorname{LimSec}(\alpha)$), which makes the two leading-order equations of Section 6.2 linearly independent and keeps the gauge-fixed linearization elliptic; Theorems 1.1 and 6.1 are stated without this hypothesis, but Section 6.2 invokes it to start the induction, so if a Lovelock combination had $A_1=0$ the argument as written would not go through.

Editorial extensions

If this is right

  • Every asymptotically hyperbolic Lovelock metric with smooth conformal infinity has a genuine polyhomogeneous expansion; the formal Fefferman–Graham expansion computed by earlier deformation arguments is therefore the actual expansion of the metric.
  • In even boundary dimension, the coefficient of $x^{n-1}\log x$ is controlled by the obstruction tensor $\mathcal{O}=\frac{A_1(\alpha)}{c_n}\Delta^{n/2-2}(P^k_{ij,k}-P^k_{k,ij})$, so smoothness of the expansion is governed to leading order by the vanishing of that tensor built from the boundary Schouten tensor.
  • When $\tilde B_{1,2}(\beta,\kappa)\neq 0$, the singular Yamabe-$(2q)$ problem has a formal solution $u=x+u_2x^2+\cdots+u_{n+1}x^{n+1}+L^{(2q)}x^{n+2}\log x$, making the linear combination of scalar-$(2q)$ curvatures vanish up to $O(x^{n+2}\log x)$.
  • For a compact spin manifold of dimension $4k$ with a boundary metric of positive Yamabe invariant, a nonvanishing Dirac index $I(X,h)$ forbids a conformally compact Lovelock filling with $\operatorname{scal}_g\geq -n(n+1)$.
  • Odd-dimensional conformal infinities admit smooth expansions, while even-dimensional ones carry the logarithmic obstruction; this dichotomy matches the Einstein setting.

Reading between the lines

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

  • Beyond the paper, the same Green-operator induction should apply to any divergence-free curvature-tensor equation whose gauge-fixed linearization is a sum of Laplace-type operators with computable indicial roots; the Lovelock case is a testbed, not the only possible one.
  • Beyond the paper, the obstruction-tensor formula implies a family of conformal boundary invariants indexed by $\alpha$: when $A_1(\alpha)$ changes sign or vanishes, the locus of smoothly extendable conformal classes can move, so different higher-curvature theories may disagree on which conformal infinities are smoothly fillable.
  • Beyond the paper, the filling obstruction likely extends to other spinorial index invariants or nonzero eta invariants, and the scalar-curvature bound $\operatorname{scal}_g\ge -n(n+1)$ is probably not optimal; this is an extension, not a claim of the paper.
  • Beyond the paper, the singular Yamabe-$(2q)$ obstruction $L^{(2q)}$, which transforms with conformal weight $-(n+1)$, is a natural candidate for a Lovelock analogue of $Q$-curvature; computing it explicitly in low dimensions would be a concrete test of that analogy.
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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

3 major / 5 minor

Summary. The paper studies conformally compact (asymptotically hyperbolic) metrics satisfying the Lovelock equations, a higher-curvature generalization of the Einstein equation. Its main claim is that every asymptotically hyperbolic Lovelock metric is polyhomogeneous near the conformal boundary, thereby realizing formally the Fefferman--Graham expansion obtained by Albin. It also computes the leading-order ambient obstruction tensor in even dimensions, constructs formal solutions of the singular Yamabe-(2q) problem, and proves an index-theoretic obstruction to conformally compact Lovelock fillings under a scalar-curvature lower bound. The proof strategy adapts Biquard--Herzlich's elliptic-regularity approach from the Einstein case, using DeTurck gauge fixing, indicial analysis, and Green's integral operators on a hyperbolic semi-ball.

Significance. If the main regularity theorem holds, it is a substantial extension of the polyhomogeneity theory of conformally compact Einstein metrics to a large class of Lovelock theories, and it gives a rigorous justification for treating Albin's formal expansions as actual asymptotic expansions. The obstruction-tensor computation in Section 7 and the index obstruction in Section 9 are concrete and potentially useful for constructing examples. The paper is not merely programmatic: the algebraic computations of the Lovelock linearization and the obstruction tensor are explicit, and the proof follows the cited framework of BH14 in detail. However, the central theorem as stated is stronger than the proof supports because a nondegeneracy hypothesis is silently assumed.

major comments (3)
  1. [§6.2, Theorem 6.1 (and Theorem 1.1)] The statement of Theorem 6.1 omits the hypothesis 1 ∈ LimSec(α) (equivalently A1(α) ≠ 0) that the proof requires. In the proof of Theorem 6.6, the text states 'The assumption on LimSec(α) implies that A1(α) is not equal to 0' and uses this to make equations (6.6) and (6.7) linearly independent, allowing the inductive determination of the expansion coefficients. The same nonvanishing is needed for the linearization formula in Lemma 6.2 and Lemma 6.5, where the displayed operator is proportional to A1(α); if A1(α) = 0, the linearized modified Lovelock tensor degenerates at the hyperbolic metric and the elliptic-regularity engine of Section 6 collapses. The excluded case is nonempty: for n = 6, the choice α1 = 72, α2 = 1 gives A1(α) = 0, while the hyperbolic metric still satisfies the Lovelock equation by the normalization of λ(2q). The theorem should either be restricted to nondegenerate couplings A1(α) ≠ 0, or the degenerate case must be analyzed separately.
  2. [§7, Theorem 7.2; compare §4 and §6.2] There is an inconsistency in the definition of A1(α) between the statement of Theorem 7.2 (and Theorem 1.2) and the definition used in Section 4 and Section 6.2. In §4 and §6.2, A1(α) is defined as ∑_q α_q (−1/2)^{q−1} ((n−2)!/2) (2q)!/(n−2q)!, while Theorem 7.2 states A1(α) = ∑_q α_q (−1/2)^{q−1} ((n−2)!/2) (2q−1)!/(n−2q)!. These differ by a factor of 2q. Since the leading obstruction term in formula (7.1) is A1(α)/c_n times a generically nonzero curvature expression, the two versions yield different invariants and cannot both be correct. Please reconcile the definitions and verify the displayed formula with the correct constant.
  3. [§8, proof of Theorem 8.1] The proof of the singular Yamabe-(2q) result is substantially sketched. In particular, the sentence introducing 'A key observation based on the commutativity of Cg and ∂x' is not a proof; the extraction of ∂x^{s+1}u from the term C_g^{2q−1}(T^{q−1}·η) is the main technical step, and it is asserted rather than demonstrated. Since Theorem 8.1 is one of the stated results of the paper, the derivation of the recurrence (8.6) and the subsequent handling of the log term should be written out in sufficient detail to be checked.
minor comments (5)
  1. [§3.2 (Kulkarni-Nomizu product)] In the definition of the Kulkarni-Nomizu product for simple double forms, the displayed formula 'ω.η = (α1 ∧ β1) ⊗ (α1 ∧ β2)' appears to contain a typo; it should presumably be '(α1 ∧ β1) ⊗ (α2 ∧ β2)'.
  2. [§6.2, proof of Lemma 6.2] The sentence 'As discussed in Example 5.6, the Laplace term of Lϕ0 is an isomorphism' suppresses the coefficient A1(α) in front of both Laplace-type factors; the isomorphism statement is only valid when A1(α) ≠ 0, so the role of the nondegeneracy hypothesis should be made explicit in the displayed argument.
  3. [§7, proof of Theorem 7.2] The transition from the displayed derivative formula to 'This leads to the same result for the highest-order term in the derivatives of h' is terse: the reader is asked to compare with [GH05, Theorem 2.1] without a full explanation of why the 'terms involving ∂x^k h with k < s' do not contribute to the trace-free leading-order term. A short justification or a more detailed computation would improve readability.
  4. [§8, notation] In the final paragraph of Section 8, the notation alternates between 'β_q' and 'βg' in the expression for eFβ(g); please use a single consistent notation for the components of β.
  5. [§9.2, Lemma 9.7] The proof of Lemma 9.7 says that the argument of [CQY04, Lemma 2.1] 'only uses the Fefferman-Graham expansion and thus holds' for Lovelock metrics. This is plausible, but since the Lovelock expansion can contain logarithmic terms in even boundary dimensions, the statement that the eigenfunction has the displayed form without a log term should be checked explicitly or referenced to the relevant part of the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the polyhomogeneity proof upgrades Albin's independent formal expansion via external elliptic regularity, and the unstated A1(alpha) != 0 hypothesis is a correctness gap, not a circular reduction.

full rationale

The paper's central derivation, Theorem 6.1, is an upgrade step: starting from the formal Fefferman-Graham expansion constructed in [Alb20], it uses the Biquard-Herzlich elliptic-regularity machinery (Green's operators, indicial roots, Proposition 5.11, Theorem 6.4) to show that an asymptotically hyperbolic Lovelock metric agrees with a polyhomogeneous expansion. None of these inputs includes the conclusion that the metric is polyhomogeneous; [Alb20] is a published, independent formal-expansion result, and [BH14], [Lee06], [GL91], and [GH05] are external analytic and geometric tools. The only load-bearing dependence on the author's advisor's work is via [Alb20], but this is substantive external support rather than a self-referential definition. The proof in Section 6.2 requires the nondegeneracy condition 1 in LimSec(alpha), which by definition entails A1(alpha) != 0, and uses this to make equations (6.6) and (6.7) linearly independent; Theorems 1.1 and 6.1 state polyhomogeneity without this hypothesis. That is a genuine gap for couplings with A1(alpha) = 0, and it should be flagged as a correctness risk, but it is not circular: the theorem does not define polyhomogeneity in terms of A1(alpha), and no fitted parameter is later relabeled as a prediction. The obstruction tensor and singular Yamabe-(2q) coefficients are computed from the formal expansion rather than fitted to it. Accordingly, there is no specific equation that reduces to its own input, and the circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central proof is largely imported technology from prior elliptic and conformal geometry literature; the paper's own contribution is adapting it to Lovelock equations. No genuinely new physical entities are introduced.

free parameters (2)
  • Lovelock coupling constants alpha_q = unspecified
    The family F_alpha depends on arbitrary constants alpha_q; the theorems hold only when LimSec(alpha) is nonempty and A1(alpha) is nonzero, a nondegeneracy condition used in Eqs. (6.6)-(6.7).
  • asymptotic sectional curvature kappa = 1 by rescaling
    The metric is rescaled so the sectional curvature tends to -1, as stated in Section 3 and used throughout; other values in LimSec(alpha) could occur but are normalized away.
assumptions (6)
  • standard math Elliptic regularity and Fredholm mapping properties for Laplace-type operators on weighted Holder spaces
    Invoked from Lee [Lee06] and Biquard-Herzlich [BH14] in Sections 5 and 6.
  • domain assumption Linearization formula for the Lovelock tensor and contraction formula for double forms
    Taken from Albin [Alb20, Lemma 1.1, Lemmas 3.2-3.4]; this is the main external input and the central proof fails if these formulas are incorrect.
  • domain assumption Nondegeneracy LimSec(alpha) nonempty and A1(alpha) nonzero
    Needed for ellipticity and for iteratively solving the expansion in Section 6.2; not stated in Theorem 6.1.
  • standard math DeTurck gauge and Ebin slice theorem provide a local diffeomorphism gauge
    Used in Lemma 6.2 and Remark 6.3 to impose the gauge condition.
  • standard math Atiyah-Patodi-Singer index theory and Lichnerowicz vanishing for the Dirac operator
    Used in Section 9 to transfer the GHS21 index obstruction to Lovelock fillings.
  • domain assumption Conformal transformation formulas for scalar-(2q) curvature and scalar curvature
    Used in Section 8, Equation (8.3), and Section 9.2; these are standard curvature transformation formulas.

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Cite this review

Pith. "Pith review of Conformally compact metrics and the Lovelock tensors." pith.science (2026). https://pith.science/paper/4QBDV4DV

@misc{pith2026250524188,
  author       = {Pith},
  title        = {Pith review of: Conformally compact metrics and the Lovelock tensors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4QBDV4DV}},
  note         = {Machine review of arXiv:2505.24188}
}
read the original abstract

We study conformally compact metrics satisfying the Lovelock equations, which generalize the Einstein equation. We show that these metrics admit polyhomogeneous expansions, thereby naturally realizing the Fefferman-Graham expansion, which is an important tool in conformal geometry and the AdS/CFT correspondence. In even dimensions, we identify a boundary obstruction to smoothness near the boundary that generalizes the ambient obstruction tensor in the Einstein setting. Under appropriate regularity and curvature conditions, we also construct a formal solution to the singular Yamabe-(2q) problem and provide an index obstruction for the conformally compact Lovelock filling problem of spin manifolds.

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Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Poincaré-Lovelock metrics on conformally compact manifolds

    Pierre Albin. Poincaré-Lovelock metrics on conformally compact manifolds. Advances in Mathematics , 367:107108, 2020

    work page 2020

  2. [2]

    Anderson

    Michael T. Anderson. A survey of Einstein metrics on 4-manifolds. In Handbook of geometric analysis, N o. 3 , volume 14 of Adv. Lect. Math. (ALM) , pages 1--39. Int. Press, Somerville, MA, 2010

    work page 2010

  3. [3]

    Arthur L. Besse. Einstein manifolds , volume 32. Springer-Verlag, Berlin, Heidelberg, New York, 1987

    work page 1987

  4. [4]

    Analyse sur un demi-espace hyperbolique et poly- homog\'en\'eit\'e locale

    Olivier Biquard and Marc Herzlich. Analyse sur un demi-espace hyperbolique et poly- homog\'en\'eit\'e locale. Calculus of Variations and Partial Differential Equations , 51(3):813--848, Nov 2014

    work page 2014

  5. [5]

    Chruściel, Erwann Delay, John M

    Piotr T. Chruściel, Erwann Delay, John M. Lee, and Dale N. Skinner. Boundary regularity of conformally compact Einstein metrics. Journal of Differential Geometry , 69(1):111 -- 136, 2005

    work page 2005

  6. [6]

    Chang and Yuxin Ge

    Sun-Yung A. Chang and Yuxin Ge. On conformally compact Einstein manifolds. Revista de la Union Matematica Argentina , 64, 2022

    work page 2022

  7. [7]

    Chang, Jie Qing, and Paul Yang

    Sun-Yung A. Chang, Jie Qing, and Paul Yang. On the topology of conformally compact Einstein 4-manifolds. In Noncompact problems at the intersection of geometry, analysis, and topology , Contemp. Math., 350, 2004

    work page 2004

  8. [8]

    Dennis M. DeTurck . Existence of metrics with prescribed Ricci curvature: Local theory. Inventiones mathematicae , 65(2):179--207, 1981

    work page 1981

Show all 31 references
  1. [9]

    M. H. Dehghani and S. H. Hendi. Taub-NUT /bolt black holes in Gauss-Bonnet-Maxwell gravity. Physical Review D , 73(8), April 2006

    2006

  2. [10]

    David G. Ebin. The manifold of Riemannian metrics. Proceedings of Symposia in Pure Mathematics , 15(2):11--40, 1970

    1970

  3. [11]

    Robin Graham

    Charles Fefferman and C. Robin Graham. Conformal invariants. Élie Cartan et les mathématiques d'aujourd'hui - Lyon , S131:95--116, 1985

    1985

  4. [12]

    Robin Graham

    Charles Fefferman and C. Robin Graham. The Ambient Metric . Annals of Mathematics Studies. Princeton University Press, 2012

    2012

  5. [13]

    Robin Graham and Kengo Hirachi

    C. Robin Graham and Kengo Hirachi. The ambient obstruction tensor and Q -curvature. AdS/CFT correspondence\: Einstein metrics and their conformal boundaries , 8:59--71, 2005

    2005

  6. [14]

    Gursky, Q

    M.J. Gursky, Q. Han, and S. Stolz. An invariant related to the existence of conformally compact Einstein fillings. Transactions of the American Mathematical Society , 374(6):4185--4205 -- 4205, 2021

    2021

  7. [15]

    Einstein metrics with prescribed conformal infinity on the ball

    C.Robin Graham and John M Lee. Einstein metrics with prescribed conformal infinity on the ball. Advances in Mathematics , 87(2):186--225, 1991

    1991

  8. [16]

    Robin Graham

    C. Robin Graham. Volume and area renormalizations for conformally compact Einstein metrics . Rend. Circ. Mat. Palermo S , 63:31--42, 2000

    2000

  9. [17]

    C. Graham. Volume renormalization for singular Yamabe metrics . Proc. Am. Math. Soc. , 145(4):1781--1792, 2017

    2017

  10. [18]

    Hendi and M.H

    S.H. Hendi and M.H. Dehghani. Taub–NUT black holes in third order Lovelock gravity. Physics Letters B , 666(2):116–120, August 2008

    2008

  11. [19]

    Kulkarni

    Ravindra S. Kulkarni. On the Bianchi identities. Mathematische Annalen , 199:175--204, 1972

    1972

  12. [20]

    M.-L. Labbi. Double forms, curvature structures and the (p,q) -curvatures. Transactions of the American Mathematical Society , 357(10):3971--3992, 2005

    2005

  13. [21]

    On Gauss-Bonnet curvatures

    Mohammed-Larbi Labbi. On Gauss-Bonnet curvatures. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only] , 3:Paper 118, 11 p., electronic only--Paper 118, 11 p., electronic only, 2007

    2007

  14. [22]

    Variational properties of the Gauss-Bonnet curvatures

    Mohammed Labbi. Variational properties of the Gauss-Bonnet curvatures. Calculus of Variations and Partial Differential Equations , 32:175--–189, 07 2008

    2008

  15. [23]

    On generalized Einstein metrics, 2008

    Mohammed Larbi Labbi. On generalized Einstein metrics, 2008

    2008

  16. [24]

    John M. Lee. Fredholm operators and Einstein metrics on conformally compact manifolds . Memoirs of the American Mathematical Society, no. 864. American Mathematical Society, Providence, R.I, 2006

    2006

  17. [25]

    Partial differential equations invariant under conformal or projective transformations

    Charles Loewner and Louis Nirenberg. Partial differential equations invariant under conformal or projective transformations. In Lars V. Ahlfors, Irwin Kra, Bernard Maskit, and Louis Nirenberg, editors, Contributions to Analysis , pages 245--272. Academic Press, 1974

    1974

  18. [26]

    The Einstein tensor and its generalizations

    David Lovelock. The Einstein tensor and its generalizations. Journal of Mathematical Physics , 12(3):498--501, 10 1971

    1971

  19. [27]

    The large-n limit of superconformal field theories and supergravity

    Juan Maldacena. The large-n limit of superconformal field theories and supergravity. International Journal of Theoretical Physics , 38(4):1113--1133, Apr 1999

    1999

  20. [28]

    The Hodge cohomology of a conformally compact metric

    Rafe Mazzeo. The Hodge cohomology of a conformally compact metric . Journal of Differential Geometry , 28(2):309 -- 339, 1988

    1988

  21. [29]

    Elliptic theory of differential edge operators I

    Rafe Mazzeo. Elliptic theory of differential edge operators I . Communications in Partial Differential Equations , 16(10):1615--1664, 1991

    1991

  22. [30]

    Noncollapsed degeneration of Einstein 4–manifolds, I

    Tristan Ozuch. Noncollapsed degeneration of Einstein 4–manifolds, I . Geometry and Topology , 26(4):1483 – 1528, 2022. Cited by: 3; All Open Access, Green Open Access

    2022

  23. [31]

    Anti-de Sitter space and holography

    Edward Witten. Anti-de Sitter space and holography. Advances in Theoretical and Mathematical Physics , 2(2):253--291, 1998

    1998

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