Glaciological processes are modeled to investigate precipitation patterns and the resulting mass flux of snow and ice across Himalayan topography. Our model tracks the accumulation and ablation of snow and ice and the transport of snow and ice across the topography by glacier motion. We investigate high elevation precipitation on the Annapurna Massif by comparing the existing ice cover with model-simulated glaciers produced by a suite of different precipitation scenarios. Our results suggest that precipitation reaches a maximum level well below the elevation of the highest peaks. Further, essentially no snow accumulates on the topography above an elevation of 6200--6300 m. Hence, the upper 1000+ m of the massif is a high elevation desert with little flux of snow and ice. Active glaciers are limited to a band of intermediate elevations where a maximum of about 60% of the landscape is covered by moving ice.