This paper is devoted to the study of the constraint equations of the Lovelock gravity theories. In the case of a conformally flat, time-symmetric, and space-like manifold, we show that the Hamiltonian constraint equation becomes a generalisation of the σk-Yamabe problem. That is to say, the prescription of a linear combination of the σk-curvatures of the manifold. We search solutions in a conformal class for a compact manifold. Using the existing results on the σk-Yamabe problem, we describe some cases in which they can be extended to this new problem. This requires to study the concavity of some polynomial. We do it in two ways: regarding the concavity of a root of this polynomial, which is connected to algebraic properties of the polynomial; and seeking analytically a concavifying function. This gives several cases in which a conformal solution exists. Finally we show an implicit function theorem in the case of a manifold with negative scalar curvature, and find a conformal solution when the Lovelock theories are close to General Relativity.

1.
Bañados
,
M.
,
Teitelboim
,
C.
, and
Zanelli
,
J.
, “
Lovelock–Born–Infeld theory of gravity
,” in
J. J. Giambiagi Festschrift
(
World Scientific
,
1990
).
2.
Beltrán Jiménez
,
J.
,
Heisenberg
,
L.
,
Olmo
,
G. J.
, and
Rubiera-Garcia
,
D.
, “
Born-Infeld inspired modifications of gravity
,”
Phys. Rep.
727
,
1
129
(
2018
). Born-Infeld inspired modifications of gravity.
3.
Caffarelli
,
L.
,
Nirenberg
,
L.
, and
Spruck
,
J.
, “
The Dirichlet problem for nonlinear second order elliptic equations. III. Functions of the eigenvalues of the Hessian
,”
Acta Math.
155
,
261
301
(
1985
).
4.
Camanho
,
X. O.
,
Edelstein
,
J. D.
, and
Sánchez De Santos
,
J. M.
, “
Lovelock theory and the AdS/CFT correspondence
,”
Gen. Relativ. Gravitation
46
,
1637
(
2014
).
5.
Choquet-Bruhat
,
Y.
, “
The Cauchy problem for stringy gravity
,”
J. Math. Phys.
29
,
1891
1895
(
1988
).
6.
Choquet-Bruhat
,
Y.
,
General Relativity and the Einstein Equations
(
Oxford University Press
,
2009
).
7.
Concha
,
P. K.
,
Merino
,
N.
, and
Rodríguez
,
E. K.
, “
Lovelock gravities from Born-Infeld gravity theory
,”
Phys. Lett. B
765
,
395
401
(
2017
).
8.
Gårding
,
L.
, “
An inequality for hyperbolic polynomials
,”
J. Math. Mech.
8
,
957
965
(
1959
).
9.
Ge
,
Y.
,
Wang
,
G.
, and
Wu
,
J.
, “
The Gauss–Bonnet–Chern mass of conformally flat manifolds
,”
Int. Math. Res. Not.
2014
(
17
),
4855
4878
.
10.
Guan
,
P.
and
Wang
,
G.
, “
A fully nonlinear conformal flow on locally conformally flat manifolds
,”
J. Reine Angew. Math.
557
,
219
238
(
2003
).
11.
Guan
,
B.
, “
Conformal metrics with prescribed curvature functions on manifolds with boundary
,”
Am. J. Math.
129
(
4
),
915
942
(
2007
).
12.
Gursky
,
M. J.
and
Viaclovsky
,
J. A.
, “
Prescribing symmetric functions of the eigenvalues of the Ricci tensor
,”
Ann. Math.
166
(
2
),
475
531
(
2007
).
13.
Jana
,
S.
,
Chakravarty
,
G. K.
, and
Mohanty
,
S.
, “
Constraints on Born-Infeld gravity from the speed of gravitational waves after GW170817 and GRB 170817A
,”
Phys. Rev. D
97
,
084011
(
2018
).
14.
Labbi
,
M.-L.
,
Riemannian Curvature: Variations on Different Notions of Positivity
(
Habilitation à Diriger des Recherches, Université Montpellier II—Sciences et Techniques du Languedoc
,
2006
).
15.
Lachaume
,
X.
, “
On the concavity of a sum of elementary symmetric polynomials
,” e-print arXiv:1712.10327 (
2017
).
16.
Lachaume
,
X.
, “
n + 1 formalism of f(Lovelock) gravity
,”
Classical Quantum Gravity
35
(
11
),
115007
(
2018
).
17.
Li
,
A.
and
Li
,
Y. Y.
, “
A fully nonlinear version of the Yamabe problem and a Harnack type inequality
,”
C. R. Acad. Sci. Paris
336
(
4
),
319
324
(
2003
).
18.
Li
,
A.
and
Li
,
Y. Y.
, “
On some conformally invariant fully nonlinear equations
,”
Commun. Pure Appl. Math.
56
(
10
),
1416
1464
(
2003
).
19.
Li
,
A.
and
Li
,
Y. Y.
, “
On some conformally invariant fully nonlinear equations. II. Liouville, Harnack and Yamabe
,”
Acta Math.
195
(
1
),
117
154
(
2005
).
20.
Lovelock
,
D.
, “
Divergence-free tensorial concomitants
,”
Aequationes Math.
4
,
127
138
(
1970
).
21.
Lovelock
,
D.
, “
The einstein tensor and its Generalizations
,”
J. Math. Phys.
12
(
3
),
498
501
(
1971
).
22.
Marcus
,
M.
and
Lopes
,
L.
, “
Inequalities for symmetric functions and Hermitian matrices
,”
Can. J. Math.
9
,
305
312
(
1957
).
23.
Mardones
,
A.
and
Zanelli
,
J.
, “
Lovelock-Cartan theory of gravity
,”
Classical Quantum Gravity
8
(
8
),
1545
(
1991
).
24.
Mehdizadeh
,
M. R.
,
Zangeneh
,
M. K.
, and
Lobo
,
F. S. N.
, “
Higher-dimensional thin-shell wormholes in third-order Lovelock gravity
,”
Phys. Rev. D
92
(
4
),
044022
(
2015
).
25.
Patterson
,
E. M.
, “
A class of critical riemannian metrics
,”
J. London Math. Soc.
s2–23
(
2
),
349
358
(
1981
).
26.
Reall
,
H. S.
,
Tanahashi
,
N.
, and
Way
,
B.
, “
Causality and hyperbolicity of Lovelock theories
,”
Classical Quantum Gravity
31
,
205005
(
2014
).
27.
Reall
,
H. S.
,
Tanahashi
,
N.
, and
Way
,
B.
, “
Shock formation in Lovelock theories
,”
Phys. Rev. D
91
,
044013
(
2015
).
28.
Sheng
,
W.-M.
,
Trudinger
,
N. S.
, and
Wang
,
X.-J.
, “
The Yamabe problem for higher order curvatures
,”
J. Differ. Geom.
77
(
3
),
515
553
(
2007
).
29.
Sheng
,
W.-M.
,
Trudinger
,
N. S.
, and
Wang
,
X.-J.
, “
The k-Yamabe problem
,”
Surv. Differ. Geom.
17
,
427
458
(
2012
).
30.
Teitelboim
,
C.
and
Zanelli
,
J.
, “
Dimensionally continued topological gravitation theory in Hamiltonian form
,”
Classical Quantum Gravity
4
(
4
),
L125
(
1987
).
31.
Torii
,
T.
and
Shinkai
,
H.
, “
n + 1 formalism in Einstein-Gauss-Bonnet gravity
,”
Phys. Rev. D
78
,
084037
(
2008
).
32.
Troncoso
,
R.
and
Zanelli
,
J.
, “
Higher-dimensional gravity, propagating torsion and AdS gauge invariance
,”
Classical Quantum Gravity
17
(
21
),
4451
(
2000
).
33.
Viaclovsky
,
J. A.
, “
Conformal geometry, contact geometry, and the calculus of variations
,”
Duke Math. J.
101
(
2
),
283
316
(
2000
).
34.
Viaclovsky
,
J. A.
, “
Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds
,”
Commun. Anal. Geom.
10
(
4
),
815
846
(
2002
).
35.
Willison
,
S.
, “
Local well-posedness in Lovelock gravity
,”
Classical Quantum Gravity
32
(
2
),
022001
(
2015
).
36.
Willison
,
S.
, “
Quasilinear reformulation of Lovelock gravity
,”
Int. J. Mod. Phys. D
24
(
09
),
1542010
(
2015
).
37.
Yano
,
K.
and
Kon
,
M.
,
Structures on Manifolds
(
World Scientific
,
1984
).
38.
Yano
,
K.
,
Integral Formulas in Riemannian Geometry
(
Dekker
,
1970
).
39.
Zanelli
,
J.
, “
Chern–Simons forms in gravitation theories
,”
Classical Quantum Gravity
29
(
13
),
133001
(
2012
).
40.
Zangeneh
,
M. K.
,
Lobo
,
F. S. N.
, and
Dehghani
,
M. H.
, “
Traversable wormholes satisfying the weak energy condition in third-order Lovelock gravity
,”
Phys. Rev. D
92
,
124049
(
2015
).
41.
Zou
,
D.-C.
,
Yue
,
R.-H.
, and
Yang
,
Z.-Y.
, “
Thermodynamics of third order Lovelock anti-de sitter black holes revisited
,”
Commun. Theor. Phys.
55
(
3
),
449
(
2011
).
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