Research

My research has partly been supported by the NSF (Grant DMS-1905165/2050120/2147521, and DMS-2403728) and an AMS/Simons travel grant.

Main publications and preprints

Mean curvature flow with generic low-entropy initial data II. With O. Chodosh, F. Schulze. Submitted.
Improved generic regularity of codimension-1 minimizing integral currents. With O. Chodosh, F. Schulze. Submitted.
Generic regularity for minimizing hypersurfaces in dimensions 9 and 10. With O. Chodosh, F. Schulze. Submitted.
Double-well phase transitions are more rigid than minimal hypersurfaces. Proc. Am. Math. Soc., vol. 152 (2024), 1301-1308.
Decomposing 4-manifolds with positive scalar curvature. With R. H. Bamler, C. Li. Adv. Math., vol. 430 (2023), 109231.
The p-widths of surfaces. With O. Chodosh. Publ. Math. IHÉS, vol. 137 (2023), 245-342.
Metrics with λ1(-Δ+kR) >= 0 and flexibility in the Riemannian Penrose Inequality. With C. Li. Commun. Math. Phys., vol. 401 (2023), 1831-1877.
Mean curvature flow with generic low-entropy initial data. With O. Chodosh, K. Choi, F. Schulze. Duke Math. J., to appear.
Variational aspects of phase transitions with prescribed mean curvature. Calc. Var. Partial Differ. Equ., vol. 61 (2022), no. 2, article 43.
Mean curvature flow with generic initial data. With O. Chodosh, K. Choi, F. Schulze. Invent. Math., to appear.
Ancient gradient flows of elliptic functionals and Morse index. With K. Choi. Am. J. Math., vol. 144 (2022), no. 2, 541-573.
Minimal hypersurfaces with arbitrarily large area. With O. Chodosh. Int. Math. Res. Not., vol. 2021 (2021), no. 14, 10841-10847.
Capacity, quasi-local mass, singular fill-ins. With P. Miao, L.-F. Tam. J. Reine Angew. Math., vol. 2020 (2020), no. 768, 55-92.
Minimal surfaces and the Allen-Cahn equation on 3-manifolds: index, multiplicity, and curvature estimates. With O. Chodosh. Ann. Math., vol. 191 (2020), no. 1, 213-328.
Positive scalar curvature with skeleton singularities. With C. Li. Math. Ann., vol. 374 (2019), no. 1, 99-131. [Known Misprints]
Allen-Cahn min-max on surfaces. J. Differ. Geom., vol. 117 (2021), no. 1, 93-135.
Total mean curvature, scalar curvature, and a variational analog of Brown-York mass. With P. Miao. Commun. Math. Phys., vol. 352 (2017), no. 2, 703-718.
On the Bartnik mass of apparent horizons. With R. Schoen. Class. Quantum Gravity, vol. 32 (2015), no. 20, 205002. Appeared on IOPselect 2015, a special edition of journal articles, on the basis of substantial advances, a high degree of novelty, and/or impact on future research.

Reports

Improved generic regularity of minimizing hypersurfaces. MFO Reports, No. 29 (2023), to appear.
Decomposing 4-manifolds with positive scalar curvature. MFO Reports, No. 28 (2022), 1593-1595.
The p-widths of a surface. MFO Reports, No. 35 (2021), 1900-1902.
Metrics with λ1(-Δ+kR) >= 0 and flexibility in the Riemannian Penrose Inequality. MFO Reports, No. 30 (2021), 1637-1640.
Ancient gradient flows of elliptic functionals and Morse index. MFO Reports, No. 34 (2019), 2055-2058.
Minimal surfaces and the Allen--Cahn equation on 3-manifolds. MFO Reports, No. 27 (2018), 1658-1661; also, No. 35 (2018), 2113-2116.
Mean curvature deficit and a quasi-local mass. With P. Miao. Harvard University Center of Mathematical Sciences and Applications (CMSA) Series in Mathematics, vol. 1., Nonlinear analysis in geometry and applied mathematics, 99-107, Int. Press, Somerville, MA, 2017.
The curvature on a black hole boundary. CQG+ Insight Piece.

Notes

T. H. Colding and W. P. Minicozzi's blowup uniqueness and Łojasiewicz inequalities.
Richard Schoen's lectures on minimal submanifolds. With D. Cheng, C. Li.
Yi Wang's lectures on harmonic analysis and isoperimetric inequalities. With D. Cheng, O. Chodosh, N. Edelen, C. Henderson, P. Hintz.
Richard Bamler's lectures on Ricci flow. With O. Chodosh.
Brian White's lectures on minimal surfaces. With O. Chodosh.

Theses

Geometric variational problems in mathematical physics. Advisor: Richard Schoen. Doctoral thesis. [Known Misprints]
Regularity of hypersurfaces in Rn+1 moving by mean curvature flow. Advisor: Leon Simon. Undergraduate honors thesis.

The exposition follows Klaus Ecker's Regularity Theory for Mean Curvature Flow.