11 citations · 23 across the 16 of their papers we have counts for
18 papers
Modulus of Concavity and Fundamental Gap Estimates on Surfaces
Gabriel Khan, Malik Tuerkoen, Guofang Wei
The fundamental gap of a domain is the difference between the first two eigenvalues of the Laplace operator. In a series of recent and celebrated works, it was shown that for conve…
Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow
Gabriel Khan
We study curve-shortening flow for twisted curves in (i.e., curves with nowhere vanishing curvature and torsion ) and define a notion of torsion-curvature ent…
Negative curvature constricts the fundamental gap of convex domains
Gabriel Khan, Xuan Hien Nguyen
We consider the Laplace-Beltrami operator with Dirichlet boundary conditions on convex domains in a Riemannian manifold , and prove that the product of the fundamental gap…
Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature
Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen +1
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 a…
When Optimal Transport Meets Information Geometry
Gabriel Khan, Jun Zhang
Information geometry and optimal transport are two distinct geometric frameworks for modeling families of probability measures. During the recent years, there has been a surge of r…
An Illustrated Introduction to the Ricci Flow
Gabriel 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…
A Hall of Statistical Mirrors
Gabriel Khan, Jun Zhang
The primary objects of study in information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher-Rao metric and a pa…
Kähler-Ricci Flow preserves negative anti-bisectional curvature
Gabriel Khan, Fangyang Zheng
In recent work (Pure Appl. Anal. 2 (2020), 397-426), the first named author and J. Zhang found a connection between the regularity theory of optimal transport and the curvature of…
Positively curved Kähler metrics on tube domains and their applications to optimal transport
Gabriel Khan, Jun Zhang, Fangyang Zheng
In this article, we study a class of Kähler manifolds defined on tube domains in , and in particular those which have symmetry. For these,…
A Conjectural Inequality for Visible Points in Lattice Parallelograms
Gabriel Khan, Mizan R. Khan, Joydip Saha +1
Let , with and . Let denote the lattice parallelogram spanned by and , that is, $$P_{a,n} = \left\{ t_1(1,0)+ t_2(…
On The Expected Total Curvature of Confined Equilateral Quadrilaterals
Gabriel Khan
In this paper, we prove that the total expected curvature for random spatial equilateral quadrilaterals with diameter at most decreases as increases. To do so, we prove sev…
Eigenvalue estimates without Bakry-Emery-Ricci bounds
Gabriel Khan
We establish a lower bound for the real eigenvalues of a Laplace-Beltrami operator with an -drift term. We make no assumptions that the operator is self-adjoint or that t…
On the Hermitian Geometry of -Gauduchon Orthogonal Complex Structures
Gabriel Khan
The purpose of this note is to study the complex structures orthogonal to a given Riemannian metric. For another paper on this topic, we highly recommend the work of Salamon. His w…
On the Kähler Geometry of Certain Optimal Transport Problems
Gabriel Khan, Jun Zhang
Let and be domains of equipped with respective probability measures and . We consider the problem of optimal transport from to with respect t…
Hall's Conjecture on Extremal Sets for Random Triangles
Gabriel Khan
In this paper we partially resolve Hall's conjecture about the distribution of random triangles. We consider the probability that three points chosen uniformly at random, in a boun…
The set of all orthogonal complex structures on the flat -tori
Gabriel Khan, Bo Yang, Fangyang Zheng
In \cite{BSV}, Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori for any . We will call these exampl…
The Drift Laplacian and Hermitian Geometry
Gabriel Khan
Let be a compact Hermitian manifold. Suppose is the lowest eigenvalue of the complex Laplacian on . We prove that where depends only on the dimensio…
A Condition Ensuring Spatial Curves Develop Type-II Singularities Under Curve Shortening Flow
Gabriel Khan
We provide a condition for spatial curves which rules out the development of a type I singularity. The condition is that after the last time for which an inflection point develops,…