Venerdì 20 Dicembre, alle ore 11:00 in Aula 3014 (3° Piano) - Edificio U5, Bruno Premoselli (ULB) terrà il seguente seminario
Titolo: Non-existence of extremals for the second conformal eigenvalue in dimensions 3 to 10
Abstract: Let $(M,g)$ be a closed manifold of dimension $n \ge 3$. We consider in this talk the conformal Laplacian of $g$, defined as $L_g = \Delta_g + c_n S_g$, where $S_g$ is the scalar curvature of $(M,g)$ and $c_n$ is an explicit numerical constant. We define the second conformal eigenvalue of $(M, [g])$ as the infimum, over all unit-volume metrics $h$ in the conformal class of $[g]$, of the second eigenvalue of $L_h$. In dimensions larger than 11 it was proven by Ammann and Humbert that the second conformal eigenvalue is attained provided $(M,g)$ is not locally conformally flat. In this talk we investigate the lower-dimensional case $3 \le n \le 10$. We prove that there is an open neighbourhood of the round metric on the sphere in which the second conformal eigenvalue is never attained. This is the first non-trivial non-existence result for conformal eigenvalues in all settings and in particular implies that, in low dimensions, local modifications of $g$ do not ensure that the second conformal eigenvalue is attained. This is a joint work with J. Vétois (Mc Gill) based on arXiv:2408.07823.
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