I am an assistant professor at the University of Miami, Department of Mathematics. Previously, I was a Postdoctoral Fellow at the University of Waterloo from 2020 to 2022, and an L. E. Dickson Instructor at the University of Chicago from 2017 to 2020. I obtained my PhD in mathematics from Stanford University in 2017 under Prof. Richard Schoen.
My research interests are partial differential equations, differential geometry and geometric measure theory.
Research
Existence of constant mean curvature disks in R^3 with capillary boundary condition. [arXiv]
(joint with Spiro Karigiannis and Jesse Madnick) A variational characterization of calibrated submanifolds. Calculus of Variations and Partial Differential Equations 62 (2023), no. 6, Paper No. 174. [Journal][arXiv]
Existence of free boundary disks with constant mean curvature in R^3. [arXiv]
(joint with Xin Zhou) Existence of curves with constant geodesic curvature in a Riemannian 2-sphere. Transactions of the American Mathematical Society 374 (2021), no.12, 9007-9028. [Journal][arXiv]
(joint with Xin Zhou) Existence of constant mean curvature 2-spheres in Riemannian 3-spheres. Communications on Pure and Applied Mathematics 76 (2023), no. 11, 3374-3436. [Journal][arXiv]
Stable solutions to the abelian Yang-Mills-Higgs equations on S^2 and T^2. The Journal of Geometric Analysis 31 (2021), no. 10, 9551-9572. [Journal][arXiv]
Instability of solutions to the Ginzburg-Landau equation on S^n and CP^n. Journal of Functional Analysis 279 (2020), No. 8, 108669. [Journal][arXiv]
(joint with Spiro Karigiannis and Jesse Madnick) Bubble tree convergence of conformally cross product preserving maps. Asian Journal of Mathematics 24 (2020), No. 6, 903-984. [Journal][arXiv]
Asymptotics for the Ginzburg-Landau equation on manifolds with boundary under homogeneous Neumann condition. Journal of Functional Analysis 278 (2020), No. 4, 108364. [Journal][arXiv]
Geometric Variational Problems: Regular and Singular behavior. PhD Thesis, Stanford University, June 2017.
A compactness result for energy-minimizing harmonic maps with rough domain metric. Communications in Analysis and Geometry 25 (2017), No.5, 927-940. [Journal][arXiv]
Teaching
at the University of Miami
Fall 2023: "Differential Geometry I"
Spring 2023: "Introduction to Linear Algebra"
Spring 2023: "Calculus II"
Fall 2022: "Introduction to ODEs"
at the University of Waterloo
Fall 2021 & Winter 2022: "Introduction to Real Analysis"
Winter 2021: "Introduction to Lie Groups and Lie Algebras"
Fall 2020: "Calculus I for the Sciences"
at the University of Chicago
Spring 2020: "Analysis in Rn"
Autumn 2019: "Basic Theory of ODEs"
Spring 2019: "Introduction to Differentiable Manifolds"
Autumn 2018: "Anaylsis in Rn (accelerated)"
Spring 2018: "Abstract Linear Algebra"
Winter 2018: "Analysis in Rn"
Fall 2017: "Mathematical Methods for Physical Sciences I"
Notes
I have collaborated with my colleagues to write notes on topics courses taught at Stanford.
Topics in minimal submanifolds (Taught by Richard Schoen, Spring 2015)
Joint with C. Li and C. Mantoulidis.
Harmonic analysis and isoperimetric inequalities (Taught by Yi Wang, Spring 2014)
Joint with O. Chodosh, N. Edelen, C. Henderson, P. Hintz, C. Mantoulidis.
Contact
E-mail: darong.cheng "at" miami.edu