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Most real-world 3D sensors such as LiDARs perform fixed scans of the entire environment, while being decoupled from the recognition system that processes the sensor data. In this work, we propose a method for 3D object recognition using light curtains, a resource-efficient controllable sensor that measures depth at user-specified locations in the environment. Crucially, we propose using prediction uncertainty of a deep learning based 3D point cloud detector to guide active perception. Given a neural network's uncertainty, we derive an optimization objective to place light curtains using the principle of maximizing information gain. Then, we develop a novel and efficient optimization algorithm to maximize this objective by encoding the physical constraints of the device into a constraint graph and optimizing with dynamic programming. We show how a 3D detector can be trained to detect objects in a scene by sequentially placing uncertainty-guided light curtains to successively improve detection accuracy.