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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

  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-Amp\`ere 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