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

- Equality in the spacetime positive mass theorem II (with D. Lee) arXiv:2302.06040 [math.DG]
- Scalar curvature deformation and mass rigidity for ALH manifolds with boundary (with H. C. Jang),
**Transactions of the AMS**, 375 (2022), 8151-8191 - Equality in the spacetime positive mass theorem (with D. Lee),
**Communications in Mathematical Physics**376 (2020), no. 3, 2379-2407 - Mass rigidity for hyperbolic manifolds (with H.C. Jang and D. Martin),
**Communications in Mathematical Physics**376 (2020), no. 3, 2329-2349 (pdf) - 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) - 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

- Local structure theory of Einstein manifolds with boundary (with Z. An), arXiv:2405.17577 [math.DG]
- 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) - 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 - 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 - Static potentials and area minimizing hypersurfaces (with D. Martin and P. Miao),
**Proceedings of the AMS**146 (2018), no. 6, 2647-2661 (pdf) - Geometric inequalities and rigidity theorems on equatorial spheres (with D. Wu),
**Communications in Analysis and Geometry**25 (2017), no. 1, 185-206 (pdf) - The equality case of the Penrose inequality for asymptotically flat graphs (with D. Wu),
**Transactions of the AMS**367 (2015), 31-47 (pdf) - Hypersurfaces with nonnegative scalar curvature (with D. Wu),
**Journal of Differential Geometry**95 (2013) 249-278 (pdf) - 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

- 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 - 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) - 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) - 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) - 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) - Solutions of special asymptotics to the Einstein constraint equations,
**Classical and Quantum Gravity**27 (2010), no. 24, 245002 (10pp) (pdf) - 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) - On the center of mass of isolated systems with general asymptotics,
**Classical and Quantum Gravity**26 (2009) 015012 (25pp) (pdf)

### (Intrinsic) Flat Convergence

- 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) - 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

- 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) - 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

- Trapped surfaces, topology of black holes, and positive mass theorem (with D. Lee),
**Notices of the AMS,**69 (2022), no. 4, 536–545 (link) - Book Review of "Geometric Relativity'' by Dan A. Lee,
**Bulletin of the AMS**58, Number 3, July 2021, Pages 461–466 (link) - 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

- 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) - 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)