  We prove the upper-tail Large Deviation Principle (LDP) for the parabolic
Airy process and characterize the limit shape of the directed landscape under
the upper-tail conditioning. The LDP result answers Conjecture 10.1 in Das,
Dauvergne, and Vir\'{a}g (2024). The starting point of our proof is the
metric-level LDP for the directed landscape from Das, Dauvergne, and Vir\'{a}g
(2024) that reduces our work to solving a variational problem. Our proof is
PDE-based and uses geometric arguments, connecting the variational problem to
the weak solutions of Burgers' equation. Further, our method may generalize to
the setting of the upper-tail LDP for the KPZ fixed point under the multi-wedge
initial data, and we prove a decomposition result in this direction.
