arXiv subjects

Jiewon Park

Publications and source records attributed to Jiewon Park.

10 recordsLinked to original sources

arXiv2026

Rigidity and volume pinching for the sharp gradient estimate in positive Ricci curvature

Colding established a sharp gradient estimate for the Green function on manifolds with nonnegative Ricci curvature, which was extended to positive Ricci curvature by Manea recently. We prove almost rigidity of such gradient estimates for positive Ricci curvature. We show that the average of the gradient deficit, namely $\fint (1-|\nabla b|^2)\,dV$, controls the volume deficit in a quantitative manner; here $b$ is the distance-like function defined using the Green function. This implies a quantitative almost rigidity theorem for manifolds with $\mathrm{Ric}\ge(n-1)g$ and a gap theorem for Einstein manifolds. We also prove a rigidity theorem for closed 4-dimensional Einstein manifolds by finding a new monotonicity formula.

math.DG↗

arXiv2025

Rigidity of the gradient estimate for Einstein manifolds

We study the rigidity of Ricci-flat manifolds with quadratic curvature decay under conditions on the Green function. We show that if the gradient of the Green function is uniformly bounded from below, then the manifold is flat. Furthermore, we prove that for a Ricci-flat manifold with quadratic curvature decay and Euclidean volume growth, the curvature is in $L^p$ for any $p \ge 2$. Combining with Cheeger-Tian \cite{CT} and Kröncke-Szabó \cite{KS}, we obtain that the manifold must be ALE of optimal order.

math.DG↗

arXiv2025

Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane

We prove three related quantitative results for the relative isoperimetric problem outside a convex body $Ω$ in the plane: (1) Łojasiewicz estimates and quantitative rigidity for critical points, (2) rates of convergence for the gradient flow, and (3) quantitative stability for minimizers. These results come with explicit constants and optimal exponents/rates, and hold whenever a simple two-dimensional auxiliary variational problem for circular arcs outside of $Ω$ is nondegenerate. The proofs are inter-related, and in particular, for the first time in the context of isoperimetric problems, a flow approach is used to prove quantitative stability for minimizers.

math.AP↗

arXiv2025

Quantitative Rigidity Using Colding's Monotonicity Formulas for Ricci Curvature

In \cite{Colding}, Colding proved monotonicity formulas for the Green function on manifolds with nonnegative Ricci curvature. Inspired by the sharp estimates relating the pinching of monotone quantities to the splitting function in \cite{cjn}, in this paper we investigate quantitative control obtained from pinching of Colding's monotone functionals. From the Green functions with poles at $(k+1)$-many independent points, $k$-splitting functions are constructed with regularity quantitatively controlled by the pinching. Moreover, the pinching at these independent points controls the distance to the nearest cone of the form $\mathbb{R}^k \times C(X)$.

math.DG↗

arXiv2025

Geometric medians on product manifolds

Product manifolds arise when heterogeneous geometric variables are jointly observed. While the Fr\'{e}chet mean on Riemannian manifolds separates cleanly across factors, the canonical geometric median couples them, and its behavior has remained largely unexplored. In this paper, we give the first systematic treatment of this problem. After formulating the coupled objective, we establish general existence and uniqueness results that the median is unique on any Hadamard product, and remains locally unique under sharp conditions on curvature and injectivity radius even when one or more factors have positive curvature. We then prove that the estimator enjoys Lipschitz stability to perturbations and the optimal breakdown point, extending classical robustness guarantees to the product-manifold setting. Two practical solvers are proposed, including a Riemannian subgradient method with global sublinear convergence and a product-aware Weiszfeld algorithm that achieves local linear convergence. Both algorithms update the factors independently while respecting the latent coupling term, enabling implementation with standard manifold primitives. Simulations on parameter spaces of univariate and multivariate Gaussian distributions endowed with the Bures-Wasserstein geometry show that the median is more resilient to contamination than the Fr\'{e}chet mean.

stat.ME↗

arXiv2024

Monotonicity formulas and Hessian of the Green function

Based on an assumption on the Hessian of the Green function, we derive some monotonicity formulas on nonparabolic manifolds. This assumption is satisfied on manifolds that meet certain conditions including bounds on the sectional curvature and covariant derivative of the Ricci curvature, as shown in the author's previous work \cite{P}. We also give explicit examples of warped product manifolds on which this assumption holds.

math.DG↗

arXiv2022

A Matrix Li-Yau-Hamilton estimate for the Green function on Kähler manifolds

In this paper we prove a matrix Li-Yau-Hamilton inequality for the Green function on complete Kähler manifolds with nonnegative holomorphic bisectional curvature. This estimate can be seen as an elliptic analogue of the matrix estimate of Cao and Ni for the heat equation on Kähler manifolds, or the complex analogue of the estimate for Riemannian manifolds obtained previously by the author.

math.DG↗

arXiv2019

Canonical identification at infinity for Ricci-flat manifolds

We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.

math.DG↗

arXiv2018

A Compactness Theorem for Rotationally Symmetric Riemannian Manifolds with Positive Scalar Curvature

Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit spaces have $H^1$ warping function that has nonnegative scalar curvature in a weak sense, and have Euclidean tangent cones almost everywhere.

math.DG↗

arXiv2017

Matrix Inequality for the Laplace Equation

Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci flow, Kähler-Ricci flow, and mean curvature flow, to name a few. As an elliptic analogue, Colding proved a sharp gradient estimate for the Green function on a manifold with nonnegative Ricci curvature. In this paper we prove a related matrix inequality on manifolds with suitable curvature and volume growth assumptions.

math.DG↗