Quantitative Volume Space Form Rigidity Under Lower Ricci Curvature Bound

Let $M$ be a compact $n$-manifold of $\operatorname{Ric}_M\ge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $\rho>0$ such that for any $x\in M$, the open $\rho$-ball at $x^*$ in the (local) Riemannian universal covering space, $(U^*_\rho,x^*)\to (B_\rho(x),x)$, has the maximal volume i.e., the volume of a $\rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H=-1$, the volume entropy of $M$ is maximal i.e. $n-1$ ([LW1]). The main results of this paper are quantitative space form rigidity i.e., statements that $M$ is diffeomorphic and close in the Gromov-Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H=1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].Comment: The only change from the early version is an improvement on Theorem A: we replace the non-collapsing condition on $M$ by on $\tilde M$ (the Riemannian universal cover), and the corresponding modification is adding "subsection c" in Section

Multi-task 3D building understanding with multi-modal pretraining

This paper explores various learning strategies for 3D building type classification and part segmentation on the BuildingNet dataset. ULIP with PointNeXt and PointNeXt segmentation are extended for the classification and segmentation task on BuildingNet dataset. The best multi-task PointNeXt-s model with multi-modal pretraining achieves 59.36 overall accuracy for 3D building type classification, and 31.68 PartIoU for 3D building part segmentation on validation split. The final PointNeXt XL model achieves 31.33 PartIoU and 22.78 ShapeIoU on test split for BuildingNet-Points segmentation, which significantly improved over PointNet++ model reported from BuildingNet paper, and it won the 1st place in the BuildingNet challenge at CVPR23 StruCo3D workshop.Comment: 8 pages, 9 figures, 9 table