Every continuous monotone family of simply connected domains in C containing 0 can be encoded into a family of driving measures on the unit circle via the Loewner-Kufarev equation. Characterizing the geometric property of the domains in terms of the measure is a challenging problem in geometric function theory. Inspired by results in random conformal geometry (without using them), we show that if the driving measure has a finite Loewner-Kufarev energy, then each domain in the monotone family is bounded by a Weil-Petersson quasicircle, and they form a foliation of the Riemann sphere. The space of WP quasicircles can be identified with the Weil-Petersson universal Teichmuller space, and a functional called the universal Liouville action (or Loewner energy) is shown to be a Kahler potential by Takhtajan-Teo. We show further that there is a duality between the Loewner-Kufarev energy of the foliation with the Loewner energy of each leaf which implies any WP quasicircle can be embedded into a foliation with finite Loewner-Kufarev energy. This is joint work with Fredrik Viklund (KTH).