On Harnack inequalities and singularities of admissible metrics in the Yamabe problem

Trudinger, Neil S.; Xu-Jia Wang
July 2009
Calculus of Variations & Partial Differential Equations;Jul2009, Vol. 35 Issue 3, p317
Academic Journal
In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds ( M, g0) of dimension n = 3. For n/2 < k < n, we prove a sharp Harnack inequality for admissible metrics when ( M, g0) is not conformally equivalent to the unit sphere S n and that the set of all such metrics is compact. When ( M, g0) is the unit sphere we prove there is a unique admissible metric with singularity. As a consequence we prove an existence theorem for equations of Yamabe type, thereby recovering as a special case, a recent result of Gursky and Viaclovsky on the solvability of the k-Yamabe problem for k > n/2.


Related Articles

  • Multiple closed geodesics on Riemannian 3-spheres. Yiming Long; Wei Wang // Calculus of Variations & Partial Differential Equations;Oct2007, Vol. 30 Issue 2, p183 

    In this paper, we prove that for every Riemannian Q-homological 3-sphere ( M, g) with injectivity radius $$inj(M)\ge \pi$$ and the sectional curvature K satisfying $${\frac{1}{16} < K \le 1}$$ there exist at least two geometrically distinct closed geodesics.

  • Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary. Qinian Jin; Aobing Li; YanYan Li // Calculus of Variations & Partial Differential Equations;Apr2007, Vol. 28 Issue 4, p509 

    In this paper we consider a fully nonlinear version of the Yamabe problem on compact Riemannian manifold with boundary. Under various conditions we derive local estimates for solutions and establish some existence results.

  • Kahler-Einstein structures of general natural lifted type on the cotangent bundles. Simona-Luiza Druţă // Balkan Journal of Geometry & Its Applications;2009, Vol. 14 Issue 1, p30 

    We study the conditions under which the cotangent bundle T*M of a Riemannian manifold (M; g), endowed with a Kählerian structure (G; J) of general natural lift type (see [4]), is Einstein. We first obtain a general natural Kähler-Einstein structure on the cotangent bundle T*M. In this...

  • Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds. Li Chen; Wenyi Chen // Annals of Global Analysis & Geometry;Jun2009, Vol. 35 Issue 4, p397 

    In this paper, we derive a local gradient estimate for the positive solution to the following parabolic equation , where a, b are real constants, M is a complete noncompact Riemannian manifold. As a corollary, we give a local gradient estimate for the corresponding elliptic equation: , which...

  • A Condition for Warped Product Semi-Invariant Submanifolds to be Riemannian Product Semi-Invariant Submanifolds in Locally Riemannian Product Manifolds. At�eken, Mehmet // Turkish Journal of Mathematics;2008, Vol. 32 Issue 3, p349 

    In this article, we give a necessary and sufficient condition for warped product semi-invariant submanifolds to be Riemannian product semi-invariant submanifolds in a locally Riemannian product manifold whose factor manifolds are real space form.

  • Minimal lightlike hypersurfaces in R42 with integrable screen distribution. Sakaki, Makoto // Balkan Journal of Geometry & Its Applications;2009, Vol. 14 Issue 1, p84 

    We give the necessary and sufficient condition for a lightlike hypersurface in R42 with integrable screen distribution to be minimal. Using the condition we can get many minimal lightlike hypersurfaces in R42 which are not totally geodesic.

  • On constant isotropic submanifold by generalized null cubic. Onat, Leyla // Differential Geometry--Dynamical Systems;2008, p249 

    In this paper we shall be concerned with curves in an Lorentzian submanifold M1, and give a characterization of each constant isotropic immersion by generalized null cubic with constant curvature on the Lorentzian submanifold.

  • On submanifolds of L xfF satisfying Chen's basic equality. Funabashi, S.; Kim, Y.-M.; Pak, J. S. // Acta Mathematica Hungarica;2003, Vol. 99 Issue 3, p189 

    It is well known that the warped product L xfF of a line L and a Kaehler manifold F is an almost contact Riemannian manifold which is characterized by some tensor equations appeared in (1.7) and (1.8). In this paper we determine submanifolds of L xfF which are tangent to the structure vector...

  • Partial Regularity for Weak Heat Flows into a General Compact Riemannian Manifold. Liu, Xian-Gao // Archive for Rational Mechanics & Analysis;Jun2003, Vol. 168 Issue 2, p131 

    Discusses the partial regularity for weak heat flows into a general compact riemannian manifold. Information on monotonicity inequality and energy inequality; Discussion of hodge decomposition and parabolic hardy space; Construction of a tangent frame and rewriting of the equation.


Read the Article


Sign out of this library

Other Topics