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Research

I do research in Geometric Analysis and Complex Differential Geometry.

 

Publications

  • Schauder estimates for equations with cone metrics, II,  with J. Song, Anal. PDE, Vol. 17 (2024), No. 3, 757 – 830. journal; arXiv
  • Auxiliary Monge-Ampère equations in geometric analysis, with D. H. Phong, Notices of the ICCM, Volume 11 (2023) Number 1, 98 – 135. journal; arXiv
  • Uniform entropy and energy bounds for fully non-linear equations, with D. H. Phong, accepted by Comm. Anal. Geom. arXiv
  • On L∞ estimates for complex Monge-Ampere equations, .pdf, with D. H. Phong and F. Tong, Ann. of Math. (2) 198 (2023), no.1, 393-418. journal; arXiv
  • Diameter estimates in Kähler geometry, with D. H. Phong, J. Song and J. Sturm, to appear in Comm. Pure Appl. Math. journal; arXiv
  • On L∞ estimates for fully nonlinear partial differential equations, with D. H. Phong, to appear in Ann. of Math. journal; arXiv
  • Green’s functions and complex Monge-Ampère equations, with D. H. Phong and J. Sturm, accepted by J. Differential Geom. arXiv
  • Local noncollapsing for complex Monge-Ampère equations, with J. Song, J. Reine Angew. Math. (Crelle’s Journal) 793 (2022), 225-238. journal; arXiv
  • On L∞ estimates for Monge-Ampère and Hessian equations on nef classes, with D.H. Phong, F. Tong and C. Wang, Anal. PDE, Vol. 17 (2024), No. 2, 749-756. journal; arXiv
  • On the Hessian-cscK equations, with K. Smith and F. Tong, Math. Z. 303, 33 (2023). journal; arXiv
  • Stability estimates for the complex Monge-Ampère and Hessian equations, with D.H. Phong and F. Tong, Calc. Var. Partial Differ. Equ. 62, 7 (2023). journal; arXiv
  • A new gradient estimate for the complex Monge-Ampère equation, with D.H. Phong and F. Tong, Math. Ann. Vol. 388, 1045 – 1051, (2024).  journal; arxiv
  • On the degeneration of asymptotically conical Calabi-Yau metrics, with T. Collins and F. Tong, Math. Ann. Vol. 383, 867 – 919 (2022) journal; arXiv
  • Compactness of Kähler-Ricci solitons on Fano manifolds, with D.H. Phong, J. Song and J. Sturm, Pure Appl. Math. Q. 18 (2022), no. 1, 305 – 316. journal; arXiv
  • On the Kähler-Ricci flow on Fano manifolds, with D. H. Phong and J. Sturm, Pure Appl. Math. Q. 18 (2022), no. 2, 573 – 581. journal; arXiv
  • Kähler-Einstein Metrics and Eigenvalue Gaps, with D. H. Phong and J. Sturm, accepted by Comm. Anal. Geom. arXiv
  • Positivity of Weil-Petersson currents on canonical models, with J. Song, Pure Appl. Math. Q. 17 (2021), no. 3, 1045-1059. journal;  pdf
  • Schauder estimates for equations with cone metrics, I, with J. Song, Indiana Univ. Math. J. 70 (2021), no. 5, 1639 – 1676. journal; arXiv
  • Parabolic Dimensional Reductions of 11D Supergravity, with T. Fei and D.H. Phong, Anal. PDE 14 (2021), no. 5, 1333 -1361. journal; arXiv
  • 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
  • On convergence criteria for the coupled flow of Li-Yuan-Zhang, with T. Fei and D.H. Phong, Math. Z., 292, 473–497(2019). journal; arXiv
  • Kähler-Ricci flow on blowups along submanifolds, Math. Ann., 375,  1147–1167(2019). journal; arXiv
  • A Geometric Construction of Solutions to 11D Supergravity, with T. Fei and D.H. Phong, Comm. Math. Phys. 369, 811-836(2019). journal; arXiv
  • Pseudo-locality for a coupled Ricci flow, with Z. Huang and D. H. Phong, Comm. Anal. Geom., Vol. 26, No. 3 (2018), pp. 585-626. journal; arXiv
  • Connecting toric manifolds by conical Kähler-Einstein metrics, with V. Datar, J. Song and X. Wang, Adv. Math. 323 (2018), 38-83. journal; arXiv
  • On the Kähler Ricci flow on projective manifolds of general type, Int. Math. Res. Not. IMRN 2017, no. 7, 2139-2171. journal; arXiv
  • 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
  • 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
  • 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
  • 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
  • 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

 

Preprints

  • Uniform L∞ estimates: subsolutions to fully nonlinear partial differential equations, with D. H. Phong, preprint arXiv
  • CscK metrics near the canonical class, with W. Jian, Y. Shi and J. Song, preprint arXiv
  • Sobolev inequalities on Kähler spaces, with D. H. Phong, J. Song and J. Sturm, preprint arXiv
  • On the modulus of continuity of solutions to complex Monge-Ampère equations, with D.H. Phong, F. Tong and C. Wang, preprint arXiv

 

Thesis and Notes