On long time behavior of the focusing energy-critical NLS on Rd×T via semivirial-vanishing geometry

We study the focusing energy-critical NLS(NLS)$i{\partial }_{t}u+{\Delta }_{x,y}u=-|u{|}^{\frac{4}{d-1}}u$ on the waveguide manifold $R_x^d\times T_y$ with $d\ge 2$. We reveal the somewhat counterintuitive phenomenon that despite the energy-criticality of the nonlinear potential, the long time dynamics of (NLS) are purely determined by the semivirial-vanishing geometry which possesses an energy-subcritical characteristic. As a starting point, we consider a minimization problem $m_c$ defined on the semivirial-vanishing manifold with prescribed mass c. We prove that for all sufficiently large mass the variational problem $m_c$ has a unique optimizer $u_c$ satisfying ${\partial }_y{u}_c=0$, while for all sufficiently small mass, any optimizer of $m_c$ must have non-trivial y-dependence. Afterwards, we prove that $m_c$ characterizes a sharp threshold for the bifurcation of finite time blow-up ($d=2,3$) and globally scattering ($d=3$) solutions of (NLS) in dependence of the sign of the semivirial. To the author's knowledge, the paper also gives the first large data scattering result for focusing NLS on product spaces in the energy-critical setting.