Research

My research lies at the interface of geometry, analysis, and topology. Using min–max theory and fine analysis of singularities, I study the existence, geometric complexity, and regularity of minimal surfaces and their anisotropic counterparts.

Publications

Published and accepted articles.

  1. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    With Adrian C. Chu

    Cambridge Journal of MathematicsAcceptedarXiv:2409.09315

    See also an overview of questions and results in enumerative differential geometry.

  2. Strong multiplicity one theorems and homological min-max theory

    With Adrian C. Chu

    Advances in Mathematics482, Part B (2025)arXiv:2309.07741

  3. Minimal hypersurfaces for generic metrics in dimension 8

    With Zhihan Wang

    Inventiones mathematicae240, 1193–1303 (2025)arXiv:2205.01047

  4. Equivariant manifolds with positive scalar curvature

    With Tsz-Kiu Aaron Chow

    Transactions of the American Mathematical Society377, 5993–6020 (2024)arXiv:2109.12725

  5. Existence of Infinitely Many Minimal Hypersurfaces in Higher-dimensional Closed Manifolds with Generic Metrics

    Journal of Differential Geometry124 (2), 381–395 (2023)arXiv:1901.08440

  6. An Improved Morse Index Bound of Min-Max Minimal Hypersurfaces

    Calculus of Variations and Partial Differential Equations62, 179 (2023)arXiv:2007.14506

  7. On the Existence of Minimal Hypersurfaces with Arbitrarily Large Area and Morse Index

    Geometry & Topology26 (6), 2713–2729 (2022)arXiv:2001.07689

Preprints

Current manuscripts, listed in reverse chronological order.

  1. Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary

    With Gioacchino Antonelli and Paul Sweeney Jr

    arXiv:2605.11384May 2026

  2. An enumerative min-max theorem for minimal surfaces

    With Adrian C. Chu and Zhihan Wang

    arXiv:2601.01736January 2026

  3. Min-max theory and minimal surfaces with prescribed genus

    With Adrian C. Chu and Zhihan Wang

    arXiv:2507.23239July 2025

  4. Non-persistence of strongly isolated singularities, and geometric applications

    With Alessandro Carlotto and Zhihan Wang

    arXiv:2411.12677November 2024

  5. Existence and regularity of min-max anisotropic minimal hypersurfaces

    With Guido De Philippis and Antonio De Rosa

    arXiv:2409.15232September 2024

  6. Generic Regularity of Minimal Hypersurfaces in Dimension 8

    With Zhihan Wang

    arXiv:2012.05401December 2020

    Superseded by arXiv:2205.01047.

Lecture notes

Expository material from courses and minicourses.

Min-max Theory for Geodesics and Minimal Surfaces

Five-day minicourse at the School of Mathematical Sciences, Zhejiang University · June 1–5, 2026

Lecture notes: TBA

Introduction to Almgren–Pitts Min–max Theory

Notes for a five-day minicourse at Tsinghua University · July 2019

Read notes