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Research

I do research in Geometric Analysis and Complex Differential Geometry. Recently, I am mainly interested in the regularity theory of nonlinear partial differential equations on complex manifolds, and their applications to the geometry of Kähler metrics.

Publications (in peer-reviewed journals)

  1. Uniform entropy and energy bounds for fully non-linear equations, with D. Phong,
    Comm. Anal. Geom. Volume 32, Number 8, (2024) 2305 – 2325. arXiv:2207.08983
  2. Diameter estimates in Kähler geometry II: removing the small degeneracy assumption, with D. Phong, J. Song and J. Sturm,
    Math. Z. 308, 43 (2024)., journal; arXiv:2405.18280
  3. CscK metrics near the canonical class, with W. Jian, Y. Shi and J. Song,
    accepted by J. Reine Angew. Math. (Crelle’s Journal), (2024), arXiv:2401.02655
  4. Green’s functions and complex Monge-Ampère equations, with D. Phong and J. Sturm,
    J. Differential Geom. Vol. 127, No. 3 (2024), pp. 1083-1119. journal; arXiv:2202.04715
  5. On L∞ estimates for fully nonlinear partial differential equations, with D. Phong,
    Ann. of Math. 200 (2024), no. 1, 365-398 journal; arXiv:2204.12549
  6. Uniform L∞ estimates: subsolutions to fully nonlinear partial differential equations, with D. Phong,
    accepted by the AMS Contemporary Mathematics volume in memory of Steve Zelditch (2024),  arXiv:2401.11572
  7. Schauder estimates for equations with cone metrics, II,  with J. Song,
    Anal. PDE, Vol. 17 (2024), No. 3, 757 – 830. journal; arXiv:1809.03116
  8. Diameter estimates in Kähler geometry, with D. Phong, J. Song and J. Sturm,
    Comm. Pure Appl. Math. Volume 77, Issue 8 (2024), 3520-3556, journal; arXiv:2209.09428
  9. Auxiliary Monge-Ampère equations in geometric analysis, with D. Phong,
    Notices of the ICCM, Volume 11 (2023) Number 1, 98 – 135. journal; arXiv:2210.13308
  10. Ann. of Math. (2) 198 (2023), no.1, 393-418. journal; arXiv:2106.02224
  11. Local noncollapsing for complex Monge-Ampère equations, with J. Song,
    J. Reine Angew. Math. (Crelle’s Journal) 793 (2022), 225-238. journal; arXiv:2201.02930
  12. On L∞ estimates for Monge-Ampère and Hessian equations on nef classes, with D. Phong, F. Tong and C. Wang,
    Anal. PDE, Vol. 17 (2024), No. 2, 749-756. journal; arXiv:2111.14186
  13. On the Hessian-cscK equations, with K. Smith and F. Tong,
    Math. Z. 303, 33 (2023). journal; arXiv:2110.11169
  14. Kähler-Einstein Metrics and Eigenvalue Gaps, with D. Phong and J. Sturm,
    Comm. Anal. Geom. Vol. 31, No. 9 (2023), pp. 2255-2275. journal; arXiv:2001.05794
  15. Stability estimates for the complex Monge-Ampère and Hessian equations, with D. Phong and F. Tong,
    Calc. Var. Partial Differ. Equ. 62, 7 (2023). journal; arXiv:2106.03913
  16. A new gradient estimate for the complex Monge-Ampère equation, with D. Phong and F. Tong,
    Math. Ann. Vol. 388, 1045 – 1051, (2024).  journal; arXiv:2106.03308
  17. On the degeneration of asymptotically conical Calabi-Yau metrics, with T. Collins and F. Tong,
    Math. Ann. Vol. 383, 867 – 919 (2022) journal; arXiv:2006.15752
  18. Compactness of Kähler-Ricci solitons on Fano manifolds, with D. Phong, J. Song and J. Sturm,
    Pure Appl. Math. Q. 18 (2022), no. 1, 305 – 316. journal; arXiv:1805.03084
  19. On the Kähler-Ricci flow on Fano manifolds, with D. Phong and J. Sturm,
    Pure Appl. Math. Q. 18 (2022), no. 2, 573 – 581. journal; arXiv:2001.06329
  20. Positivity of Weil-Petersson currents on canonical models, with J. Song,
    Pure Appl. Math. Q. 17 (2021), no. 3, 1045-1059. journal;  pdf
  21. Schauder estimates for equations with cone metrics, I, with J. Song,
    Indiana Univ. Math. J. 70 (2021), no. 5, 1639 – 1676. journal; arXiv:1612.00075
  22. Parabolic Dimensional Reductions of 11D Supergravity, with T. Fei and D. Phong,
    Anal. PDE 14 (2021), no. 5, 1333 -1361. journal; arXiv:1806.00583
  23. Geometric estimates for complex Monge-Ampère equations, with X. Fu and J. Song,
    J. Reine Angew. Math. (Crelle’s Journal) 765 (2020), 69-99. journal; arXiv:1706.01527
  24. On convergence criteria for the coupled flow of Li-Yuan-Zhang, with T. Fei and D. Phong,
    Math. Z., 292, 473–497(2019). journal; arXiv:1808.06968
  25. Kähler-Ricci flow on blowups along submanifolds,
    Math. Ann., 375,  1147–1167(2019). journal; arXiv:1809.03654
  26. A Geometric Construction of Solutions to 11D Supergravity, with T. Fei and D. Phong,
    Comm. Math. Phys. 369, 811-836(2019). journal; arXiv:1805.07506
  27. Pseudo-locality for a coupled Ricci flow, with Z. Huang and D. Phong,
    Comm. Anal. Geom., Vol. 26, No. 3 (2018), pp. 585-626. journal; arXiv:1510.04332
  28. Connecting toric manifolds by conical Kähler-Einstein metrics, with V. Datar, J. Song and X. Wang,
    Adv. Math. 323 (2018), 38-83. journal; arXiv:1308.6781
  29. On the Kähler Ricci flow on projective manifolds of general type,
     Int. Math. Res. Not. IMRN 2017, no. 7, 2139-2171. journal; arXiv:1501.04239
  30. On Feldman-Ilmanen-Knopf conjecture for the behavior of the Kähler Ricci flow, with J. Song,
    Math. Res. Lett. 23 (2016), no. 6, 1681-1719. journal; arXiv:1505.04869
  31. Geometric convergence of the Kähler-Ricci flow on complex surfaces of general type, with J. Song and B. Weinkove,
    Int. Math. Res. Not. IMRN 2016, no. 18, 5652-5669. journal; arXiv:1505.00705
  32. Some variational problems in conformal geometry, with H. Li,
    5th ICCM. Part 1, 2, 221–231, AMS/IP Stud. Adv. Math., 51, pt. 1, 2, Amer. Math. Soc., Providence, RI, 2012. pdf
  33. The second variational formula for the functional  ∫v (6)(g)dVg, with H. Li,
    Proc. Amer. Math. Soc. 139 (2011), no. 8, 2911-2925. journal; arXiv:1006.0156
  34. Two Kazdan-Warner-type identities for the renormalized volume coefficients and the Gauss-Bonnet curvatures of a Riemannian metric, with Z. Han and H. Li,
    Pacific J. Math. 251 (2011), no. 2, 257-268. journal; arXiv:0911.4649

Preprints

  1. Sup-slopes and sub-solutions for fully nonlinear elliptic equations, with J. Song,
    preprint, (2024), arXiv:2405.03074
  2. Sobolev inequalities on Kähler spaces, with D. H. Phong, J. Song and J. Sturm,
    preprint, (2023), arXiv:2311.00221
  3. On the modulus of continuity of solutions to complex Monge-Ampère equations, with D.H. Phong, F. Tong and C. Wang,
    preprint, (2021), arXiv:2112.02354

Thesis and Notes

  1. Auxiliary Complex Monge-Ampère Equations in Nonlinear PDE Estimates: A Survey, a survey article written for ICBS, 2024.
  2. Lecture notes (I, II, III) in a summer school at Tsinghua University, July 2023
  3. Diameter estimates in Kähler geometry, Oberwolfach report, Differentialgeometrie im Grossen, July 2 – 7, 2023
  4. Some parabolic and elliptic problems in complex Riemannian geometry, PhD Thesis, Rutgers University – New Brunswick, 2015