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c-Biharmonic maps: a higher order conformally invariant generalization of harmonic maps

Cezar Oniciuc, Iasi University, Romania

10/12/2024 ore 17:00, Aula B, Palazzo delle Scienze

Seminario del ciclo Seminari di Geometria Differenziale e Topologia.

Abstract: Biharmonic maps between Riemannian manifolds are critical points of the bienergy functional E?, and represent a natural generalization of harmonic maps. These maps are characterized by a fourth-order non-linear elliptic equation and exhibit intriguing properties, especially in the case of biharmonic immersions in spheres. However, a limitation of biharmonic maps is their lack of invariance under conformal changes of the domain metric (M?,g) for any dimension n. In contrast, harmonic maps are conformally invariant when n=2. This suggests the need for a new generalization of harmonic maps that preserves conformal invariance for some n.

In this talk, we introduce the concept of c-biharmonic maps. By modifying the bienergy functional with additional lower-order terms, we define a new functional Ec?, called the c-biharmonic functional, whose critical points are known as c-biharmonic maps. The c-biharmonic equation is also a fourth-order non-linear elliptic equation, but c-biharmonic maps are conformally invariant when n=4. The lower-order modification of the bienergy functional does not change the analytic properties of c-biharmonic maps comparing with the biharmonic ones. On the other hand, from geometric point of view, and especially for isometric immersions, the behavior of c-biharmonic immersions is very different from that of biharmonic ones.
This is a joint work with Volker Branding and Simona Nistor.