Romildo Pina

Conformal metrics and ricci tensors on the sphere

We consider tensors T = fg on the unit sphere S, where n > 3, g is the standard metric and f is a differentiable function on Sn. For such tensors, we consider the problems of existence of a Riemannian metric 9, conformal to g, such that Ric g = T, and the existence of such a metric that satisfies Ric g - Kg/2 = T, where K is the scalar curvature of g. We find the restrictions on the Ricci candidate for solvability, and we construct the solutions g when they exist. We show that these metrics are unique up to homothety, and we characterize those defined on the whole sphere. As a consequence of these results, we determine the tensors T that are rotationally symmetric. Moreover, we obtain the well-known result that a tensor T = αg, a > 0, has no solution g on S n if α ¬= n - 1 and only metrics homothetic to g admit (n - 1)g as a Ricci tensor. We also show that if α ¬= -(n-1)(n-2)/2, then equation Ric g - Kg/2 = αg has no solution g, conformal to g on S n , and only metrics homothetic to g are solutions to this equation when a = - (n - 1)(n - 2)/2. Infinitely many C∞ solutions, globally defined on S n , are obtained for the equation -φΔ g φ + n 2|⊇ g φ| 2 -n 2(λ+φ 2 )=0, where λ ∈ R. The geometric interpretation of these solutions is given in terms of existence of complete metrics, globally defined on R n and conformal to the Euclidean metric, for certain bounded scalar curvature functions that vanish at infinity.

On metrics satisfying equation Rij−12Kgij=Tij for constant tensors T☆

Necessary and sufficient conditions are given on a constant symmetric tensor Tij on Rn, n≥3, for which there exist metrics ḡ, conformal to a pseudo-Euclidean metric g, such that Rij−12Kgij=Tij, where Rij and K are the Ricci tensor and the scalar curvature of ḡ. All solutions ḡ are given explicitly and it is shown that there are no complete metrics ḡ conformal and nonhomothetic to g.

Invariant solutions for the static vacuum equation

We consider the static vacuum Einstein space-time when the spatial factor (or, base) is conformal to a pseudo-Euclidean space, which is invariant under the action of a translation group. We characterize all such solitons. Moreover, we give examples of static vacuum Einstein solutions for Einstein’s field equation. Applications provide an explicit example of a complete static vacuum Einstein space-time.

On the study of a class of non-linear differential equations on compact Riemannian manifolds

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds

Invariant solutions for the Einstein field equation

(M^n,g), n\geq 2

On solutions of the Ricci curvature equation and the Einstein equation

, \Delta_p u + a(x)u^{p-1} = \lambda f(u,x), (E2) where

On gradient Ricci solitons conformal to a pseudo-Euclidean space

\Delta_p

On the Ricci and Einstein equations on the pseudo-euclidean and hyperbolic spaces

is the

A class of solutions of the Ricci and Einstein equations

p-

A family of warped product semi-Riemannian Einstein metrics

laplacian, with