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Register a new userOn a Sharp Inequality Relating Yamabe Invariants on a Poincare-Einstein Manifold
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For a Poincare-Einstein manifold under certain restrictions, X. Chen, M. Lai and F. Wang proved a sharp inequality relating Yamabe invariants. We show that the inequality is true without any restriction.
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We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n , [h])$ of a Poincare-Einstein manifold $(X^{n+1} , g^+ )$ with either $n = 2$ or $n geq 3$ and $(M^n , [h])$ is locally flat - namely $(M, h)$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits also a local situation and a global one. Furthermore the latter global situation includes the case of conformal infinities of Poincare-Einstein manifolds of dimension either 2 or of dimension greater than $2$ and which are locally flat, and hence the minimizing technique of Aubin- Schoen in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau which is not known to hold. Using the algebraic topological argument of Bahri-Coron, we bypass the latter positive mass issue and show that any conformal infinity of a Poincare-Einstein manifold of dimension either $n = 2$ or of dimension $n geq 3$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.
We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some general hypothesis on the potential.
We consider the Ricci flow $frac{partial}{partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rmgeq 0, |Rm(p)|to 0, ~as ~d(o,p)to 0.$ We prove that the Ricci flow on such a manifold is nonsingular in any finite time.
We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar results are obtained for planar d-webs, d > 4, provided that additional d-4 second-order invariants vanish.
It is shown that on every closed oriented Riemannian 4-manifold $(M,g)$ with positive scalar curvature, $$int_M|W^+_g|^2dmu_{g}geq 2pi^2(2chi(M)+3tau(M))-frac{8pi^2}{|pi_1(M)|},$$ where $W^+_g$, $chi(M)$ and $tau(M)$ respectively denote the self-dual Weyl tensor of $g$, the Euler characteristic and the signature of $M$. This generalizes Gurskys inequality cite{gur} for the case of $b_1(M)>0$ in a much simpler way. We also extend all such lower bounds of the Weyl functional to 4-orbifolds including Gurskys inequalities for the case of $b_2^+(M)>0$ or $delta_gW^+_g=0$, and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature.
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