7 papers
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…
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 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…