Critical metrics of the total scalar curvature functional on 4-manifolds

Curvature and Volume Renormalization of Ahe Metrics on 4-manifolds

This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V , as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M . In addition we show that the differential or variation dV of V , or equivalently the variation of the L norm of the Weyl curvature, is intrinsically determined by the conformal inf...

On the Scalar Curvature of Einstein Manifolds

We show that there are high-dimensional smooth compact manifolds which admit pairs of Einstein metrics for which the scalar curvatures have opposite signs. These are counter-examples to a conjecture considered by Besse [6, p. 19]. The proof hinges on showing that the Barlow surface has small deformations with ample canonical line bundle.

Constant T -curvature conformal metrics on 4-manifolds with boundary

In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M, g) with smooth boundary there exists a metric conformal to g with constant T -curvature, zero Q-curvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to solving a fourth order nonlinear elliptic boundary value problem (BVP) with boundary conditions gi...

Prescribed Scalar Curvature problem on Complete manifolds

Conditions on the geometric structure of a complete Riemannian manifold are given to solve the prescribed scalar curvature problem in the positive case. The conformal metric obtained is complete. A minimizing sequence is constructed which converges strongly to a solution. In a second part, the prescribed scalar curvature problem of zero value is solved which is equivalent to find a solution to ...

Constant scalar curvature metrics on toric surfaces