Research interest:

Geometric analysis and geometric PDEs. I am interested in understanding the formation of singularities and developing/improving regularity theory of geometric partial differential equations such as minimal surfaces, mean curvature flows, Allen-Cahn and Ginzburg-Landau equations, etc.

Publications and preprints:

• Some properties of closed hypersurfaces of small entropy and the topology of hypersurfaces through singularities of mean curvature flow. ETD - Doctoral Dissertations. Johns Hopkins University (2018). [Link]
• Round spheres are Hausdorff stable under small perturbation of entropy. J. Reine Angew. Math. 758, 261-280 (2020). [Link]
• On the topological rigidity of self shrinkers in R^3. (with Alexander Mramor) Int. Math. Res. Not. 2020, 1933-1941 (2020). [Link]
• The level set flow of a hypersurface in R^4 of low entropy does not disconnect. (with Jacob Bernstein) Comm. Anal. Geom. 29, 1523-1543 (2021). [Link]
• Warped tori with almost non-negative scalar curvature. (with Brian Allen, Lisandra Hernandez-Vazquez, Davide Parise, Alec Payne) Geometriae Dedicata 200, 153-171 (2019). [Link]
• Integrability of scalar curvature and normal metric on conformally flat manifolds. (with Yi Wang) J. Differential Equations 265, 1353-1370 (2018). [Link]
• Low entropy and the mean curvature flow with surgery. (with Alexander Mramor) Calc. Var. Partial Differential Equations. 60, (2021) [Link]
• Extended abstract "The level set flow of a hypersurface in R^4 of low entropy does not disconnect" in the Proceedings of the John H. Barrett Memorial Lectures at the University of Tennessee, Knoxville, May 29 - June 1, 2018 (edited by Theodora Bourni and Mat Langford).  De Gruyter Proc. Math. (2020). [Link]
• Precise asymptotics near a generic S^1 × R^3 singularity of mean curvature flow. (with Gang Zhou) To appear in Nonlinear Anal.
• Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn. (with Huy Nguyen) Int. Math. Res. Not. 2024, 6759-6789 (2024) [Link]
• Brakke regularity for the Allen-Cahn flow. (with Huy Nguyen) Preprint.
• Quantization of the energy for the inhomogeneous Allen-Cahn mean curvature. (with Huy Nguyen) To appear in Math. Ann.