In this talk, I will explain the notion of Hofer energy of
J-holomorphic curves in a noncompact symplectic manifold M. If M
comes from puncturing a closed symplectic manifold, we prove that the
Hofer energy can by bounded by a constant times the symplectic
energy. As an immediate consequence, we prove a version of Gromov's
monotonicity theorem with multiplicity for J-holomorphic curves.