arXiv subjects

Dongwoo Gang

Publications and source records attributed to Dongwoo Gang.

3 recordsLinked to original sources

arXiv2026

Second-Order Departure of the Gigli--Mantegazza Flow from Ricci Flow

For a closed connected Riemannian manifold $(M,g)$, the Gigli--Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding $x\mapsto p_t(x,\cdot)\,d\operatorname{vol}_g$. The resulting family $\widetilde{g}_t$ agrees with Ricci flow to first order in $t$, but in general not to second order. We prove that $$ \widetilde{g}_t =g-2t\operatorname{Ric}_g +t^2\left(-\Delta\operatorname{Ric}_g +2\operatorname{Ric}_g^2-\frac{2}{3}\mathcal{Q}_g\right) +O_{C^0}(t^3), $$ where $\mathcal{Q}_g$ is quadratic in the full curvature tensor. The term $-\Delta\operatorname{Ric}_g$ also occurs in the second-order expansion of Ricci flow, so the discrepancy depends pointwise and quadratically on the curvature. In particular, at a Ricci-flat metric that is not flat, Ricci flow is stationary while $\widetilde{g}_t$ is not. The Gromov--Hausdorff distance between the Gigli--Mantegazza and Ricci-flow metrics is $O(t^2)$, and round spheres show that this estimate is sharp.

math.DG↗

arXiv2025

Oriented Grassmannian Bundle, Normal Curvature Reduction, and Persistent Homology

We consider a smooth closed orientable submanifold $M \subset \mathbb{R}^D$ with narrow cycles. We embed $M$ into a scaled oriented Grassmannian bundle via the Gauss map in order to enlarge the scale of these cycles. Under mild assumptions, we show that this embedding reduces the normal curvature of the embedded submanifold in directions where the original normal curvature is large. For smooth closed hypersurfaces, we further show that this construction increases the distance between antipodal points of narrow cycles for fixed volume. We then obtain an explicit range of radii for which the ambient Čech complex on this Grassmannian bundle is homotopy equivalent to the embedded manifold, yielding lower bounds on the scales at which the Čech filtration recovers the homology of $M$. Since the distance induced by the embedding depends on both positions and oriented tangent spaces, we work with Whitney $C^1$ convergence of embeddings and prove that the associated Čech persistent homology is stable with respect to the interleaving distance. Finally, we describe a procedure for computing a distance matrix for a finite subset with respect to this embedding and illustrate the construction on several examples, including an approximate quasi-halo orbit in the Saturn--Enceladus system.

math.DG↗

arXiv2025

Persistent Stiefel-Whitney Classes of Tangent Bundles

Stiefel-Whitney classes are topological invariants of vector bundles, and those of the tangent bundle capture essential features of a manifold, such as whether it is orientable and how it can be embedded in Euclidean space. We present an algorithm that computes these classes for the tangent bundle directly from a finite sample of points. Starting from the point cloud, we build a filtration of simplicial complexes and compute its persistent cohomology, and then apply the Wu formula, which recovers the Stiefel-Whitney classes from the cup product and the Steenrod squares alone, without estimating tangent spaces or a smooth structure. The key step, finding the Wu classes, reduces to solving a system of linear equations, so the computation runs in polynomial time in the number of simplices. We prove that whenever the sample recovers the shape of a closed manifold, the computed classes agree with the true Stiefel-Whitney classes of its tangent bundle, and that this remains true even when the data carry spurious topological features on which the Steenrod squares vanish, so the classes can be identified over a wide range of scales rather than only where the sample matches the manifold exactly. We illustrate the method on triangulated four-dimensional manifolds and on point clouds coming from image patches and from a molecular conformation space.

math.AT↗