Paper · Apr 22, 2026
A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary
As on arXiv
Abstract
In this paper, we show that starting from a geodesic ball in , for , with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class on the boundary, the solution to the normalized Ricci flow which is continuous up to the boundary, exists for all and converges locally uniformly in to a complete hyperbolic metric as (see Theorem 1.2 for details). Moreover, the sectional curvature of maintains less than for . For dimension , to achieve such a convergence result, we need the additional assumption that the mean curvature on the boundary increases in a certain speed to infinity as .
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 10A.12Curve Shortening and Minimal Discs
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.
- 11A.1The Equation and Its First Solutions
A heat equation for the metric: spheres, cylinders and scaling.