Nonnegative Ricci curvature and escape rate gap

Let M be an open n-manifold of nonnegative Ricci curvature and let p ∈ M {p\in M} . We show that if ( M , p ) {(M,p)} has escape rate less than some positive constant ϵ ⁢ ( n ) {\epsilon(n)} , that is, minimal representing geodesic loops of π 1 ⁢ ( M , p ) {\pi_{1}(M,p)} escape from any bounded balls at a small linear rate with respect to their lengths, then π 1 ⁢ ( M , p ) {\pi_{1}(M,p)} is virtually abelian. This generalizes the author’s previous work [J. Pan, On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature, Geom. Topol. 25 2021, 2, 1059–1085], where the zero escape rate is considered.

Efficiently Computing Geodesic loop for Interactive Segmentation of 3D Mesh

Introduction: Shape segmentation is a fundamental problem of computer graphics and geometric modeling. Although the existence segmentation algorithms of shapes have been widely studied in mathematics community, little progress has been made on how to compute them on polygonal surfaces interactively using geodesic loops. Method: We compute the geodesic distance fields with improved Fast March Method (FMM) proposed by Xin and Wang. We propose a new algorithm to compute geodesic loops over a triangulate surface and a new interactive shape segmentation manner on triangulate surface. Result: The average computation time on 50K vertices model is less than 0.08s. Discussion: In the future, we will use an accurate geodesic algorithm and parallel computing techniques to improve our algorithm to obtain better smooth geodesic loop. Conclusion: A large number of experimental results show that the algorithm proposed in this paper can effectively achieve high precision geodesic loop paths, and our method can also be used to interactive shape segmentation in real time.