Existence Of Conformal Metrics Research Articles

A singular Yamabe problem on manifolds with solid cones

Abstract We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 {d>\frac{n-2}{2}} . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.

The holomorphic d-scalar curvature on almost Hermitian manifolds

In this paper, we study the existence of conformal metrics with constant holomorphic d-scalar curvature and the prescribed holomorphic d-scalar curvature problem on closed, connected almost Hermitian manifolds of dimension n ⩾ 6. In addition, we obtain an application and a variational formula for the associated conformal invariant.

Existence of conformal metrics with prescribed Q-curvature on manifolds

We consider the problem of existence of conformal metrics with prescribed Q-curvature on riemannian n-dimensional manifolds (Mn,g0),5≤n≤7, not conformally diffeomorphic to the standard sphere Sn. Under the assumptions that Qg0 is semi-positive, Rg0 is non-negative and the prescribed function is flat near its critical points, we study the loss of compactness of the problem and we prove existence results through Euler-Hopf type criteria.

Existence of Conformal Metrics with Prescribed Q-Curvature on Manifolds

Existence of Conformal Metrics with Prescribed Q-Curvature on Manifolds

The scalar curvature problem on four-dimensional manifolds

We study the problem of existence of conformal metrics with prescribed scalar curvatures on a closed Riemannian \begin{document}$ 4 $\end{document} -manifold not conformally diffeomorphic to the standard sphere \begin{document}$ S^{4} $\end{document} . Using the critical points at infinity theory of A.Bahri [ 6 ] and the positive mass theorem of R.Schoen and S.T.Yau [ 32 ], we prove compactness and existence results under the assumption that the prescribed function is flat near its critical points. These are the first results on the prescribed scalar curvature problem where no upper-bound condition on the flatness order is assumed.

Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

Topological methods for the resonant [formula omitted]-curvature problem in arbitrary even dimension

This paper is devoted to the problem of existence of conformal metrics with prescribed Q-curvature on closed Riemannian manifolds of even dimension n≥4, when the kernel of the associated GJMS operator is trivial and the total integral of the corresponding Q-curvature is a positive integer multiple of the one of the n-dimensional round sphere. Indeed, exploiting the variational structure of the problem, we develop a full Morse theory and algebraic topological arguments for existence for this non-compact geometric variational problem of high order, extending the works Ahmedou and Ndiaye (2019) and Ndiaye (2015) to arbitrary even dimensions.

Algebraic topological methods for the supercritical Q-curvature problem

We study the problem of existence of conformal metrics with prescribed Q-curvature on closed four-dimensional Riemannian manifolds. This problem has a variational structure, and in the case of interest here, it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated Euler–Lagrange functional, the so-called true critical points at infinity of the associated variational problem, occur. Using the characterization of the critical points at infinity of the associated variational problem which is established in [42], combined with some arguments from Morse theory, some algebraic topological methods, and some tools from dynamical system originating from Conley's isolated invariant sets and isolated blocks theory, we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold, stable and unstable manifolds of some of the critical points at infinity of the associated Euler–Lagrange functional.

Topological tools for the prescribed scalar curvature problem on S n

AbstractIn this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].

Existence of Conformal Metrics with Prescribed Q-Curvature

We consider the problem of existence of conformal metrics with prescribed Q-curvature on standard sphere <svg style="vertical-align:-1.59705pt;width:57.700001px;" id="M1" height="15.6" version="1.1" viewBox="0 0 57.700001 15.6" width="57.700001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,13.55)"><path id="x1D446" d="M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68&#xA;l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" /></g> <g transform="matrix(.012,-0,0,-.012,8.225,5.388)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g> <g transform="matrix(.017,-0,0,-.017,14.813,13.55)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,21.51,13.55)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.017,-0,0,-.017,34.769,13.55)"><path id="x2265" d="M531 285l-474 -214v56l416 183l-416 184v56l474 -215v-50zM531 -40h-474v50h474v-50z" /></g><g transform="matrix(.017,-0,0,-.017,49.473,13.55)"><path id="x35" d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8&#xA;l-26 -65h-233z" /></g> </svg>. 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On the prescribed scalar curvature problem on [formula omitted]: The degree zero case

In this Note, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere Sn, n⩾3. We give new existence and multiplicity results based on a new Euler–Hopf formula type. Our argument also has the advantage of extending the well known results due to Y. Li (1995) [10].

Existence of conformal metrics with prescribed scalar curvature on the four dimensional half sphere

In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the four dimensional half sphere. Using dynamical and topological methods involving the study of the critical points at infinity of the associated variational structure, we prove some existence results like Bahri-Coron theorem. Furthermore, we consider the approximate subcritical problem and we construct some solutions which blow up at two different points, one of them lay on the boundary and the other one is an interior point.

Existence results for the prescribed Scalar curvature on S^{3}

This paper is devoted to the existence of conformal metrics on S 3 with prescribed scalar curvature. We extend well known existence criteria due to Bahri-Coron.

Integral pinching results for manifolds with boundary

We prove that some Riemannian manifolds with boundary satisfying an explicit integral pinching condition are spherical space forms. More precisely, we show that three-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the L-norm of the scalar curvature and the L-norm of the Ricci tensor are spherical space forms with totally geodesic boundary. Moreover, we prove also that four-dimensional Riemannian manifolds with umbilic boundary, positive Yamabe invariant and an explicit integral pinching between the total integral of the (Q,T )-curvature and the L-norm of the Weyl curvature are spherical space forms with totally geodesic boundary. As a consequence, we show that a certain conformally invariant operator, which plays an important role in Conformal Geometry, is non-negative and has trivial kernel if the Yamabe invariant is positive and verifies a pinching condition together with the total integral of the (Q,T )-curvature. As an application of the latter spectral analysis, we show the existence of conformal metrics with constant Q-curvature, constant T -curvature, and zero mean curvature under the latter assumptions.

Existence of conformal metrics with constant Q-curvature

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with a compactness result by the second author.

Prescribed [formula omitted]-curvature on manifolds of even dimension

Given a compact even dimensional manifold and a prescribed function f , under suitable hypotheses, we prove the existence of conformal metrics with Q -curvature f . We use a variational method which allows the corresponding Graham–Jenne–Mason–Sparling operator to have negative eigenvalues.

Existence of Conformal Metrics on Spheres with Prescribed Paneitz Curvature

In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the n-sphere, with n ≥ 5. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given positive function under which we prove the existence of a metric, conformally equivalent to the standard metric, with prescribed Paneitz curvature.

Existence of conformal metrics on $S^n$ with prescribed fourth-order invariant

In this paper we prescribe a fourth-order conformal invariant on the standard $n$-sphere, with $ n\geq5 $, and study the related fourth-order elliptic equation. We first find some existence results in the perturbative case. After some blow-up analysis we build a homotopy to pass from the perturbative case to the nonperturbative one under some flatness condition. Finally we state some existence results under the assumption of symmetry.

A note on the problem of prescribing Gaussian curvature on surfaces

The problem of existence of conformal metrics with Gaussian curvature equal to a given function K on a compact Riemannian 2-manifold M of negative Euler characteristic is studied. Let Ko be any nonconstant function on M with max KO = 0, and let KA = KO + A . It is proved that there exists a A* > 0 such that the problem has a solution for K = KA iff A E (-oo, A*] Moreover, if A E (0, A*), then the problem has at least 2 solutions. Let M be a closed 2-dimensional smooth manifold and g be a Riemannian metric on M. Let k denote the Gaussian curvature of g. If g' = e2ug is another Riemannian metric conformal to g, and has Gaussian curvature k', then it is well known that k' = e-2u(k Au), where A is the Laplacian of g. Given a function K E C??(M), the problem of prescribing Gaussian curvature asks whether one can find u E C? (M) such that the metric g' = e2ug has the given K as its Gaussian curvature. Obviously, this is equivalent to the problem of solvability of the following elliptic equation (1) Au-k+Ke2u=O, onM. If u is a solution of (1), then we have by integrating (1)

The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature

The existence of conformal metrics with constant scalar curvature and constant boundary mean curvature