https://neckpinch.com/authors/gabriel-khan/New arXiv papers by Gabriel Khan, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2305.07171arxiv:2305.07171Thu, 11 May 2023 00:00:00 GMTGabriel Khan — We study curve-shortening flow for twisted curves in ℝ³ (i.e., curves with nowhere vanishing curvature κ and torsion τ) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type II singularity which admits sequences along which fracτκ² → ∞. This contrasts strongly with Altschuler's planarity theorem [J. Differential Geom. (1991)], which shows that along any essential blow-up sequence, fracτκ → 0.Curve shorteningSingularities https://arxiv.org/abs/2211.06403arxiv:2211.06403Fri, 11 Nov 2022 00:00:00 GMTGabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen, Guofang Wei — We study the log-concavity of the first Dirichlet eigenfunction of the Laplacian for convex domains. For positively curved surfaces satisfying a condition involving the curvature and its second derivatives, we show that the first eigenfunction is strongly log-concave. Previously, for general convex domains, the log-concavity of the first eigenfunctions were only known when lying in ℝⁿ and Sⁿ. Using this estimate, we establish lower bounds on the fundamental gap of such regions. Furthermore, we study the behavior of these estimates under Ricci flow and other deformations of the metric.Ricci flowSurfacesEigenvalues & spectrum https://arxiv.org/abs/2201.04923arxiv:2201.04923Thu, 13 Jan 2022 00:00:00 GMTGabriel Khan — The Ricci flow is one of the most important topics in differential geometry, and a central focus of modern geometric analysis. In this paper, we give an illustrated introduction to the subject which is intended for a general audience. The goal is to provide a working definition of the Ricci flow as well as some intuition for its behavior without assuming any prerequisite knowledge of differential geometry or topology.Ricci flow