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Conformal geometry and fully nonlinear equations
, 2006
"... This article is a survey of results involving conformal deformation of Riemannian metrics and fully nonlinear equations. ..."
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Cited by 20 (1 self)
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This article is a survey of results involving conformal deformation of Riemannian metrics and fully nonlinear equations.
Regularity of a fourth order nonlinear PDE with critical exponent
, 1999
"... Abstract. In this paper we demonstrate the regularity of minimizers for a variational problem, a special case of which arises in spectral theory and conformal geometry. The associated EulerLagrange equation is fourth order semilinear; the leading term is the bilaplacian, and lower order terms appea ..."
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Abstract. In this paper we demonstrate the regularity of minimizers for a variational problem, a special case of which arises in spectral theory and conformal geometry. The associated EulerLagrange equation is fourth order semilinear; the leading term is the bilaplacian, and lower order terms appear at critical powers. 0. Introduction. Our aim in this paper is to study the regularity of minimizers for a certain variational problem defined either on a closed compact fourdimensional Riemannian manifold or in a smooth bounded domain in R 4. The associated EulerLagrange equation is fourth order and nonlinear; moreover, the nonlinearity is “critical ” in a sense which we will soon describe.
Constant Qcurvature metrics in arbitrary dimension
 SISSA, 27/2006/M
"... Abstract: Working in a given conformal class, we prove existence of constant Qcurvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a nthorder nonlinear elliptic differential (or integral) equation with variational structure, ..."
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Abstract: Working in a given conformal class, we prove existence of constant Qcurvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a nthorder nonlinear elliptic differential (or integral) equation with variational structure, where n is the dimension of the manifold. Since the corresponding Euler functional is in general unbounded from above and below, we use critical point theory, jointly with a compactness result for the above equation.
Topological methods for an elliptic equations with exponential nonlinearities
"... abstract. We consider a class of variational equations with exponential nonlinearities on compact surfaces. From considerations involving the MoserTrudinger inequality, we characterize some sublevels of the EulerLagrange functional in terms of the topology of the surface and of the data of the equ ..."
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Cited by 16 (7 self)
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abstract. We consider a class of variational equations with exponential nonlinearities on compact surfaces. From considerations involving the MoserTrudinger inequality, we characterize some sublevels of the EulerLagrange functional in terms of the topology of the surface and of the data of the equation. This is used together with a minmax argument to obtain existence results.
Concentrationcompactness phenomena in the higher order Liouville’s equation
"... We investigate different concentrationcompactness phenomena related to the Qcurvature in arbitrary even dimension. We first treat the case of an open domain in R 2m, then that of a closed manifold and, finally, the particular case of the sphere S 2m. In all cases we allow the sign of the Qcurvatu ..."
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Cited by 15 (10 self)
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We investigate different concentrationcompactness phenomena related to the Qcurvature in arbitrary even dimension. We first treat the case of an open domain in R 2m, then that of a closed manifold and, finally, the particular case of the sphere S 2m. In all cases we allow the sign of the Qcurvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in R 2m, concentration phenomena can occur only at points of positive Qcurvature. As a consequence, on a locally conformally flat manifold of nonpositive Euler characteristic we always have compactness. 1 Introduction and statement of the main results Before stating our results, we recall a few facts concerning the Paneitz operator P 2m g and the Qcurvature Q2m g on a 2mdimensional smooth Riemannian manifold (M, g). Introduced in [BO], [Pan], [Bra] and [GJMS], the Paneitz operator
QCURVATURE AND POINCARÉ METRICS
 MATHEMATICAL RESEARCH LETTERS 9, 139–151 (2002)
, 2002
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Variational status of a class of fully nonlinear curvature prescription problems
"... Abstract. Prescribing, by conformal transformation, the k thelementary symmetric polynomial of the Schouten tensor σk(P) to be constant is a generalisation of the Yamabe problem. On compact Riemannian nmanifolds we show that, for 3 ≤ k ≤ n, this prescription equation is an EulerLagrange equation ..."
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Abstract. Prescribing, by conformal transformation, the k thelementary symmetric polynomial of the Schouten tensor σk(P) to be constant is a generalisation of the Yamabe problem. On compact Riemannian nmanifolds we show that, for 3 ≤ k ≤ n, this prescription equation is an EulerLagrange equation of some action if and only if the structure is locally conformally flat. 1.
Classification of solutions to the higher order Liouville’s equation
 in R 2m
, 2009
"... We classify the solutions to the equation (−∆) m u = (2m − 1)!e 2mu on R 2m giving rise to a metric g = e 2u g R 2m with finite total Qcurvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of ∆u at infinity. As a c ..."
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We classify the solutions to the equation (−∆) m u = (2m − 1)!e 2mu on R 2m giving rise to a metric g = e 2u g R 2m with finite total Qcurvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of ∆u at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric e 2u g R 2m at infinity, and we observe that the pullback of this metric to S 2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round. 1 Introduction and statement of the main theorems The study of the Paneitz operators has moved into the center of conformal geometry in the last decades, in part with regard to the problem of prescribing the Qcurvature. Given a 4dimensional Riemannian manifold (M, g), the Qcurvature Q4 g and the Paneitz operator P 4 g have been introduced by Branson
Conformal Metrics with Constant QCurvature
, 2007
"... We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant Qcurvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal ba ..."
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Cited by 11 (1 self)
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We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant Qcurvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.