  We consider the model of the Brownian plane, which is a pointed non-compact
random metric space with the topology of the complex plane. The Brownian plane
can be obtained as the scaling limit in distribution of the uniform infinite
planar triangulation or the uniform infinite planar quadrangulation and is
conjectured to be the universal scaling limit of many others random planar
lattices. We establish sharp bounds on the probability of having a short cycle
separating the ball of radius $r$ centered at the distinguished point from
infinity. Then we prove a strong version of the spatial Markov property of the
Brownian plane. Combining our study of short cycles with this strong spatial
Markov property we obtain sharp isoperimetric bounds for the Brownian plane.
