How curved boundaries shape quantum field behavior near surfaces

Starting from a heat-kernel approach, the authors derive how the nonlocal effective action of a massless scalar field depends on the extrinsic curvature of a flat space's boundary, carried to quadratic order in that curvature. The framework is fully covariant, so it applies to surfaces that cannot be described by a single height function — a limitation of earlier Monge-patch calculations. As a concrete test, they compute the rate at which quantum particles are created by an oscillating deformed ring in 2+1 dimensions and a deformed sphere in 3+1 dimensions, connecting the geometric formalism to a measurable physical effect.