Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds

• Autores: María del Mar González Nogueras, Mariel Saez Trumper
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 24, Nº 4, 2023, págs. 2047-2081
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • In this paper we study bounds for the first eigenvalue of the Paneitz op- erator P and its associated third-order boundary operator B3 on four-manifolds.

    We restrict to orientable, simply connected, locally conformally flat manifolds that have at most two umbilic boundary components. The proof is based on showing that under the hypotheses of the main theorems, the considered mani- folds are confomally equivalent to canonical models. This equivalence is proved by showing the injectivity of suitable developing maps. Then the bounds on the eigenvalues are obtained through explicit computations on the canonical models and its connections with the classes of manifolds that we are considering. In particular, we give an explicit bound for a 4-dimensional annulus with a radially symmetric metric. The fact that P and B3 are conformal in four dimensions is key in the proof.