We examine the evolution of a planar Alfv¿n wave propagating along and across a pressure-balanced structure. The structure has mostly a magnetic field component perpendicular to a uniform background magnetic field B0 and gives an almost sinusoidal Alfv¿n speed gradient across B0. The Alfv¿n wave propagates both oblique to the structure and B0. A hybrid simulation code with particle protons and massless fluid electrons is used to follow evolution. The structure advects little with the wave so that the wave evolves mostly in accord with a fixed Alfv¿n speed gradient. The Alfv¿n wave undergoes resonant absorption wherein one set of resonant field lines loses wave energy to another set of resonant field lines by cross-field energy transport. We derive the timescale of the cross-field transfer and associate it with global modes. This feature of wave evolution occurs independently of ion kinetics. On resonant field lines, wave refraction generates ever smaller scales perpendicular to B0. When the perpendicular scales are of order 10 proton inertial lengths (10c/ωpi), a nonpropagating density fluctuation begins to grow to large amplitudes. The growth saturates by exciting highly oblique, compressive, and low-frequency waves which dissipate and heat protons. Protons are heated approximately by equal amounts along and across the local magnetic field. Heating continues until most of the resonantly absorbed Alfv¿n wave energy is exhausted. This process limits density fluctuations to scales across B0, which are greater than ≈ c/ωpi. This process could play a role in the solar corona where a cutoff of density fluctuations at scales near and below c/ωpi is inferred from radio observations.