On some conformally invariant fully nonlinear equations

Li, Aobing; Li, Yanyan

doi:10.1002/cpa.10099

Cited by 180 publications

(301 citation statements)

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“…In this section we consider a generalization of the classical Yamabe problem, namely, the Yamabe problem for the Gauss‐Bonnet curvature

{scriptS}^{(2 k)}

. As explained below, in the class of locally conformally flat manifolds this problem is equivalent to the so‐called

σ_{k}

‐Yamabe problem and has already been considered under a certain ellipticity assumption on the background metric (see , ). As a consequence of a formula for the linearization of the Gauss‐Bonnet curvature on

(X, g) \in {scriptH}_{n, k}

(see Proposition below) we shall prove a local version of the Yamabe problem for the Gauss‐Bonnet curvatures in a neighborhood of a subclass of

H_{n, k}

which includes all nonflat space forms, except for the round sphere.…”

Section: The Yamabe Problem For Gauss‐bonnet Curvaturesmentioning

confidence: 99%

“…The

σ_{k}

‐Yamabe problem for conformally flat manifolds (or, equivalently, the Yamabe problem for the Gauss‐Bonnet curvatures) were considered in and , assuming that the background metric satisfies a certain ellipticity condition. The next theorem solves the Yamabe problem for the Gauss‐Bonnet curvatures in a neighborhood of Riemannian manifolds in the class

H_{n, k}

, except for the round spheres, and provides many new examples of non ‐conformally flat manifolds for which this problem is affirmatively solved.…”

Section: The Yamabe Problem For Gauss‐bonnet Curvaturesmentioning

confidence: 99%

“…It is thus natural to formulate the corresponding problem for the Gauss‐Bonnet curvatures: given a metric g in X , is there

g^{'}

conformal to g so that

{scriptS}_{g^{'}}^{(2 k)}

is constant? This problem, which admits a nice variational characterization (Proposition ), has been considered so far in case g is locally conformally flat , , for it is then equivalent to the

σ_{k}

‐Yamabe problem (Proposition ). In this work we present new examples of manifolds

(X, g)

with non‐null Weyl tensor for which the problem has a positive solution.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Deformation and rigidity results for the 2k‐Ricci tensor and the 2k‐Gauss‐Bonnet curvature

Caúla¹,

Lima

Santos

2013

Mathematische Nachrichten

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Abstract. We present several deformation and rigidity results within the classes of closed Riemannian manifolds which either are 2k-Einstein (in the sense that their 2k-Ricci tensor is constant) or have constant 2k-Gauss-Bonnet curvature. The results hold for a family of manifolds containing all non-flat space forms and the main ingredients in the proofs are explicit formulae for the linearizations of the above invariants obtained by means of the formalism of double forms.

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“…In this section we consider a generalization of the classical Yamabe problem, namely, the Yamabe problem for the Gauss‐Bonnet curvature

{scriptS}^{(2 k)}

. As explained below, in the class of locally conformally flat manifolds this problem is equivalent to the so‐called

σ_{k}

(X, g) \in {scriptH}_{n, k}

(see Proposition below) we shall prove a local version of the Yamabe problem for the Gauss‐Bonnet curvatures in a neighborhood of a subclass of

H_{n, k}

which includes all nonflat space forms, except for the round sphere.…”

Section: The Yamabe Problem For Gauss‐bonnet Curvaturesmentioning

confidence: 99%

“…The

σ_{k}

H_{n, k}

, except for the round spheres, and provides many new examples of non ‐conformally flat manifolds for which this problem is affirmatively solved.…”

Section: The Yamabe Problem For Gauss‐bonnet Curvaturesmentioning

confidence: 99%

“…It is thus natural to formulate the corresponding problem for the Gauss‐Bonnet curvatures: given a metric g in X , is there

g^{'}

conformal to g so that

{scriptS}_{g^{'}}^{(2 k)}

is constant? This problem, which admits a nice variational characterization (Proposition ), has been considered so far in case g is locally conformally flat , , for it is then equivalent to the

σ_{k}

‐Yamabe problem (Proposition ). In this work we present new examples of manifolds

(X, g)

with non‐null Weyl tensor for which the problem has a positive solution.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Deformation and rigidity results for the 2k‐Ricci tensor and the 2k‐Gauss‐Bonnet curvature

Caúla¹,

Lima

Santos

2013

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…It can be viewed as a fully nonlinear version of the Yamabe problem, which was solved by Trudinger [1968], Aubin [1976] and Schoen [1984]. The solvability of the higher order k-Yamabe problem was shown for k = 2 in ] (see also [Chang et al 2002;Ge and Wang 2006]), for k = n/2 in [Trudinger and Wang 2010], for k > n/2 in [Gursky and Viaclovsky 2007], and for locally conformally flat manifolds in [Guan and Wang 2003a;Li and Li 2003;. For results concerning the modified Schouten tensor on closed manifolds, see [Gursky and Viaclovsky 2003;Li and Sheng 2005] for the case τ < 1, and [Sheng and Zhang 2007] for the case τ ≥ n − 1.…”

Section: Introductionmentioning

confidence: 99%

Some Dirichlet problems arising from conformal geometry

Li¹,

Sheng²

2011

Pacific J. Math.

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We study the problem of finding complete conformal metrics determined by some symmetric function of the modified Schouten tensor on compact manifolds with boundary; which reduces to a Dirichlet problem. We prove the existence of the solution under some suitable conditions. In particular, we prove that every smooth compact n-dimensional manifold with boundary, with n ≥ 3, admits a complete Riemannian metric g whose Ricci curvature Ric g and scalar curvature R g satisfyThis result generalizes Aviles and McOwen's in the scalar curvature case.

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“…(1.1) in this case. We refer to [2,3,8,9,11,13,16,14]. The study of geometric inequalities in [10] leads us naturally to consider the general form of Eq.…”

Section: Introductionmentioning

confidence: 99%