Predicting ambulance demand accurately in fine resolutions in space and time is critical for ambulance fleet management and dynamic deployment. Typical challenges include data sparsity at high resolutions and the need to respect complex urban spatial domains. To provide spatial density predictions for ambulance demand in Melbourne, Australia, as it varies over hourly intervals, we propose a predictive spatio-temporal kernel warping method. To predict for each hour, we build a kernel density estimator on a sparse set of the most similar data from relevant past time periods (labeled data), but warp these kernels to a larger set of past data irregardless of time periods (point cloud). The point cloud represents the spatial structure and geographical characteristics of Melbourne, including complex boundaries, road networks and neighborhoods. Borrowing from manifold learning, kernel warping is performed through a graph Laplacian of the point cloud and can be interpreted as a regularization toward, and a prior imposed for, spatial features. Kernel bandwidth and degree of warping are efficiently estimated via cross-validation, and can be made time- and/or location-specific. Our proposed model gives significantly more accurate predictions compared to a current industry practice, an unwarped kernel density estimation and a time-varying Gaussian mixture model.