Cálculo variacional de aplicações harmônicas e bi-harmônicas

Abstract

In this work, we will explore some basic results of the theory of harmonic and biharmonic maps. A smooth application f : (M, g) ! (N, h) between two Riemannian manifolds with M compact will be called harmonic when it is a critical point of the energy functional, and biharmonic when it is a critical point of the bi-energy functional. Consequently, we will derive the formula for the first variation of the energy functional and prove that an application is harmonic if and only if its tension field vansh. Similarly, we will calculate the formula for the first variation of the bi-energy functional and demonstrate that a map is biharmonic if and only if its bi-tension field vanish. Finally, we will calculate the formula for the second variation for harmonic and biharmonic maps, and we will study conditions for stability.