My research is in the fields interconnecting analysis, differential geometry, topology, and partial differential equations. My main research is in general relativity and differential geometry, but I have a strong passion to physics and general scientific fields. For example, I have collaborated with researchers in mechanical engineering under NSF Grant CMMI-2130668.

Positive Mass Theorem

  1. Equality in the spacetime positive mass theorem II (with D. Lee) arXiv:2302.06040 [math.DG]
  2. Scalar curvature deformation and mass rigidity for ALH manifolds with boundary (with H. C. Jang), Transactions of the AMS, 375 (2022), 8151-8191
  3. Equality in the spacetime positive mass theorem (with D. Lee), Communications in Mathematical Physics 376 (2020), no. 3, 2379-2407 (pdf)
  4. Mass rigidity for hyperbolic manifolds (with H.C. Jang and D. Martin), Communications in Mathematical Physics 376 (2020), no. 3, 2329-2349 (pdf)
  5. Spacetime positive mass theorem in dimensions less than eight (with M. Eichmair, D. Lee, and R. Scheon), Journal of the European Mathematical Society, 18 (2016), no. 1, 83-121 (pdf)
  6. Stability of the positive mass theorem for graphical hypersurfaces of Euclidean space (with D. Lee), Communications in Mathematical Physics 337 (2015), no. 1, 151-169 (pdf)

Ricci and Scalar Curvatures

  1. Local structure theory of Einstein manifolds with boundary (with Z. An), arXiv:2405.17577 [math.DG]
  2. Static vacuum extensions with prescribed Bartnik boundary data near a general static vacuum metric (with Z. An), Annals of PDE Vol. 10, No. 6 (2024)
  3. Existence of static vacuum extensions with prescribed Bartnik boundary data (with Z. An), Cambridge Journal of Mathematics, Vol. 10, No. 1 (2022), pp. 1-68
  4. New asymptotically flat static vacuum metrics with near Euclidean boundary data (with Z. An), Journal of Mathematical Physics, Proceedings of ICMP XX, 63 (2022), no. 5
  5. Static potentials and area minimizing hypersurfaces (with D. Martin and P. Miao), Proceedings of the AMS 146 (2018), no. 6, 2647-2661 (pdf)
  6. Geometric inequalities and rigidity theorems on equatorial spheres (with D. Wu), Communications in Analysis and Geometry 25 (2017), no. 1, 185-206 (pdf)
  7. The equality case of the Penrose inequality for asymptotically flat graphs (with D. Wu), Transactions of the AMS 367 (2015), 31-47 (pdf)
  8. Hypersurfaces with nonnegative scalar curvature (with D. Wu), Journal of Differential Geometry 95 (2013) 249-278 (pdf)
  9. Rigidity theorems on hemispheres in non-positive space forms (with D. Wu), Communications in Analysis and Geometry 18 (2010), no. 2, 339-363 (pdf)

Mathematical Relativity

  1. Bartnik mass minimizing initial data sets and improvability of the dominant energy scalar (with D. Lee), Journal of Differential Geometry, 126 (2) 741 - 800, 2024
  2. Localized deformation for initial data sets with the dominant energy condition (with J. Corvino), Calculus Variations and Partial Differential Equations (2020), no. 1, No. 42 (pdf)
  3. On the validity of the definition of angular momentum in general relativity (with P.-N. Chen, M.-T. Wang, and S.-T. Yau), Annales Henri Poincaré, 17 (2016), no. 2, 253-270 (pdf)
  4. On the center of mass in general relativity. Fifth International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 51, pt. 1, vol. 2, Amer. Math. Soc., Providence, RI (2012) pp. 575-591 (pdf)
  5. Specifying angular momentum and center of mass for vacuum initial data sets (with R. Schoen and M.-T. Wang), Communications in Mathematical Physics 306 (2011), no. 3, 785-803 (pdf)
  6. Solutions of special asymptotics to the Einstein constraint equations, Classical and Quantum Gravity 27 (2010), no. 24, 245002 (10pp) (pdf)
  7. Foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics, Communications in Mathematical Physics 300 (2010), no. 2, 331-373 (pdf)
  8. On the center of mass of isolated systems with general asymptotics, Classical and Quantum Gravity 26 (2009) 015012 (25pp) (pdf)

(Intrinsic) Flat Convergence

  1. Intrinsic flat convergence of points and applications to stability of the positive mass theorem (with D. Lee and R. Perales), Annales Henri Poincaré, 23, pages 2523–2543 (2022)
  2. Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space (with D. Lee and C. Sormani), Journal für die reine und angewandte Mathematik (Crelle's Journal), 727 (2017) (pdf)
    (Corrigendum J. Reine Angew. Math.785 (2022), 273–274)

Harmonic Maps

  1. Existence of harmonic maps into CAT(1) spaces (with C. Breiner, A. Fraser, C. Mese, P. Sargent, Y. Zhang), Communications in Analysis and Geometry 28 (2020), no. 4, 781-835 (special issue in honor of Uhlenbeck's 75th birthday) (pdf)
  2. Regularity of harmonic maps from Polyhedra to CAT(1) Spaces (with C. Breiner, A. Fraser, C. Mese, P. Sargent, Y. Zhang), Calculus Variations and Partial Differential Equations 57 (2018), no. 1, Art. 12, 35 pp. (pdf)

Expository Articles

  1. Trapped surfaces, topology of black holes, and positive mass theorem (with D. Lee), Notices of the AMS, 69 (2022), no. 4, 536–545 (link)
  2. Book Review of "Geometric Relativity'' by Dan A. Lee,  Bulletin of the AMS 58, Number 3, July 2021, Pages 461–466 (link)
  3. The Mathematics of Richard Schoen (with Bray, Minicozzi, Eichmair, Yau, Uhlenbeck, Kusner, Marques, Mese, and Fraser), Notices of the AMS 65 (2018), no. 11, 1349-1376 (link)

Lecture Notes

  1. The equality case in positive mass theorems (Lecture notes for EWM-EMS summer school: The Cauchy Problem in General Relativity at Institut Mittag-Leffler in June 13-17, 2022)
  2. On the center of mass and constant mean curvature surfaces of an initial data set, Lecture Notes,  2013 MSRI Summer School in General Relativity (to be published by the Cambridge University Press as a chapter)

Selected Talk Recordings

  • Spacetime Positive Mass Theorem (positivity)
    Variational Methods in Geometry Seminar in Institute for Advanced Study on Feb 5, 2019
    (based on the paper "Spacetime positive mass theorem in dimensions less than eight" joint with M. Eichmair, D. Lee, and R. Schoen)

 

  • Equality in spacetime positive mass theorem
    Workshop on Geometric Functionals: Analysis and Applications in Institute for Advanced Study on March 5, 2019.
    (based on the paper "Equality in the spacetime positive mass theorem" joint with D. Lee)
  • Existence of static vacuum extension
    Joint Online Mathematical Relativity Colloquium on May 6, 2021
    (based on "Existence of static vacuum extensions with  prescribed Bartnik boundary data" joint with Z. An)
    Video Link
  • Bartnik's quasi-local mass conjectures
    Geometric Analysis Colloquium at Fields Institute on Feb 8, 2021
    (based on the paper "Bartnik mass minimizing initial data sets and improvability of the dominant energy scalar" joint with D. Lee)