This work is about conformally invariant equations from a geometric point of view. In other words, given a solution to an elliptic conformally invariant equation in a subdomain of the sphere, following ideas of Espinar-Gálvez-Mira, we construct an elliptic hypersurface in the Hyperbolic space. We can related analytic conditions of the solution to the conformally invariant elliptic equation and the geometry of the hypersurface.
In this work we assume that the dimension is greater than 2. We show a non-existence theorem for degenerate elliptic problems for conformal metrics on the closed northern hemisphere with minimal boundary. On a compact annulus on the sphere, we prove a uniqueness result for degenerate problem with minimal boundary. We prove a non-existence theorem for degenerate problems on a compact annulus on the sphere, under the hypothesis that there is a solution on the punctured closed northern hemisphere that satisfies certain property. We show that a solution for elliptic problems of conformal metrics on the punctured compact northern hemisphere with minimal boundary is rotationally invariant. For the non-degenerate case on such domain, the puncutured compact northern hemisphere, we have that solutions with minimal boundary are rotationally invariant.