Building an online 3D LiDAR mapping system that produces a detailed surface reconstruction while remaining computationally efficient is a challenging task. In this paper, we present PlanarMesh, a novel incremental, mesh-based LiDAR reconstruction system that adaptively adjusts mesh resolution to achieve compact, detailed reconstructions in real-time. It introduces a new representation, planar-mesh, which combines plane modeling and meshing to capture both large surfaces and detailed geometry. The planar-mesh can be incrementally updated considering both local surface curvature and free-space information from sensor measurements. We employ a multi-threaded architecture with a Bounding Volume Hierarchy (BVH) for efficient data storage and fast search operations, enabling real-time performance. Experimental results show that our method achieves reconstruction accuracy on par with, or exceeding, state-of-the-art techniques-including truncated signed distance functions, occupancy mapping, and voxel-based meshing-while producing smaller output file sizes (10 times smaller than raw input and more than 5 times smaller than mesh-based methods) and maintaining real-time performance (around 2 Hz for a 64-beam sensor).