I do research in geometric analysis and complex geometry.

Publications:

• Diameter estimates in Kahler geometry II: removing the small degeneracy assumption, with B. Guo, D.H. Phong and J. Sturm, arXiv

• Sup-slopes and sub-solutions for fully nonlinear elliptic equations, with B. Guo, arXiv

• CscK metrics near the canonical class,  with B. Guo, W. Jian and Y. Shi, arXiv

• Convergence of the Kahler-Ricci flow on minimal models of general type, with W. Jian

• Minimal slopes and bubbling for complex Hessian equations, with V. Datar and R. Mete, arXiv

• Sobolev inequalities on Kahler spaces, with B. Guo, D.H. Phong and J. Sturm, arXiv

• Geometric regularity of blow-up limits of the Kahler-Ricci flow, with M. Hallgren, W. Jian and G. Tian, arXiv

• Finite time singularities of the Kahler-Ricci flow, with W. Jian and G. Tian, arXiv

• A new proof of Perelman's scalar curvature and diameter estimates for the Kahler-Ricci flow on Fano manifolds, with W. Jian and G. Tian, arXiv

• Diameter estimates in Kähler geometry, with B, Guo, D.H. Phong and J. Sturm, arXiv, accepted at CPAM

• Local noncollapsing for complex Monge-Ampère equations, with B. Guo, J. Reine Angew. Math. 793 (2022), 225–238, arXiv

• Kahler-Einstein metric near an isolated log canonical singularity, with V. Datar and X. Fu, J. Reine Angew. Math. 797 (2023), 79–116, arXiv

• Diameter estimates for long-time solutions of the Kahler-Ricci flow, with W. Jian, Geom. Funct. Anal. 32 (2022), no. 6, 1335–1356, arXiv

• Nakai-Moishezon criterions for complex Hessian equations, arXiv

• Metric rigidity of Kahler manifolds with lower Ricci lower bounds and almost maximal volume, with V. Datar and H. Seshadri, Proc. Amer. Math. Soc. 149 (2021), no. 8, 3569–3574arXiv

• An Analytic Proof of the Stable Reduction Theorem, with J. Sturm and X. Wang, arXiv

• Continuity of the Weil-Petersson potential, with J. Sturm and X. Wang, arXiv

• Riemannian geometry of Kahler-Einstein currents III: compactness of Kahler-Einstein manifolds of negative scalar curvature, with J. Sturm and X. Wang, arXiv

• Positivity of Weil-Petersson currents on canonical models, with B. Guo, Pure Appl. Math. Q. 17 (2021), no. 3, 1045–1059 

• Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow, with G. Tian and Z. Zhang, arXiv

• Schauder estimates for equations with cone metrics, II,  with B. Guo, to appear in Anal. PDE, arXiv

• A remark on constant scalar curvature Kahler metrics on minimal models, with W. Jian and Y. Shi, Proc. Amer. Math. Soc. 147 (2019), no. 8, 3507–3513, arXiv

• Compactness of Kahler-Ricci solitons on Fano manifolds, B. Guo, D.H. Phong and J. Sturm, arXiv

• Degeneration of Kahler-Einstein manifolds of negative scalar curvature, arXiv 

• Geometric estimates for complex Monge-Ampere equations, with X. Fu and B. Guo, to appear in Crelle's journal, arXiv 

• Connecting toric conical Kahler-Einstein manifolds, with V. Datar, B. Guo and X. Wang, Adv. Math. 323 (2018), 38-83, arXiv 

• The Kahler-Ricci flow through singularities, with G. Tian, Invent. Math. 207 (2017), no. 2, 519-595, arXiv

• On Feldman-Ilmanen-Knopf conjecture for the blow-up behavior of the Kahler-Ricci flow, with B. Guo, Math. Res. Lett. 23 (2016), no. 6, 1681-1719, arXiv

• Geometric convergence of the Kahler-Ricci flow on complex surfaces of general type, with B. Guo and B. Weinkove, I.M.R.N. 2016, no. 18, 5652-5669, arXiv

• Schauder estimates for equations with cone metrics, I, with B. Guo, Indiana Univ. Math. J. 70 (2021), no. 5, 1639–1676, arXiv

• Riemannian geometry of Kahler-Einstein currents II: an analytic proof of Kawamata’s base point free theorem, arXiv

• The Ricci flow on the sphere with marked points, with D.H. Phong, J. Sturm and X. Wang, J. Differential Geom. 114 (2020), no. 1, 117–170, arXiv

• Riemannian geometry of Kahler-Einstein currents, arXiv

• A remark on Kahler metrics with conical singularities along a simple normal crossing divisor, with V. Datar, Bull. of Lond. Math. Soc., 47 (2015), no. 6, 1010-1013, arXiv

• The degenerate J-flow and the Mabuchi energy on minimal surfaces of general type, with B. Weinkove, Universitatis Iagellonicae Acta Mathematica, no. 50, (2013), 89-106, arXiv

• Ricci flow and birational surgery, arXiv

• Degeneration of Kahler-Ricci solitons on Fano manifolds, with D.H. Phong and J. Sturm, Universitatis Iagellonicae Acta Mathematica, No. 52 (2015), 29-43, arXiv

• The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality, with X. Wang, Geom. Topol., 20 (2016), no. 1, 49-102, arXiv

• The J-flow on Kahler surfaces: a boundary case, with H. Fang, M. Lai and B. Weinkove, Anal. PDE 7 (2014), no. 215-226, arXiv

• Some Type I solutions of Ricci flow with rotational symmetry, I.M.R.N., no. 16, 7365-7381, arXiv

• On a conjecture of Candelas and de la Ossa, Comm. Math. Phys. 334 (2015), no. 2, 697--717, arXiv

• Bounding scalar curvature for global solutions of the Kahler-Ricci flow, with G. Tian, American Journal of Mathematics, vol. 138, no. 3, 2016, arXiv

• The Kahler-Ricci flow on projective bundles, with B. Szekelyhidi and B. Weinkove, Int. Math. Res. Not. IMRN 2013, no. 2, 243-257, arXiv

• Contracting exceptional divisors by the Kahler-Ricci flow II, with B. Weinkove, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1529-1561, arXiv

• Metric flips with Calabi ansatz, with Y. Yuan, Geom. Func. Anal. 22 (2012), no. 1, 240-265, arXiv

• Contracting exceptional divisors by the Kahler-Ricci flow, with B. Weinkove, Duke Math. J. 162 (2013), no.2, 367-415, arXiv

• Convergence of the Kahler-Ricci flow on singular Calabi-Yau varieties, with Y. Yuan, ALM 21, Advances in Geometric Analysis (volume dedicated to Professor Shing-Tung Yau), 119-138, 2012 

• Finite time extinction of the Kahler-Ricci flow, Math. Res. Lett., 21 (2014), no. 6, 1435-1449, arXiv

• The Kahler-Ricci flow on Hirzebruch surfaces, with B. Weinkove, J. Reine Angew. Math. 659 (2011), 141-168, arXiv

• The modified Kahler-Ricci flow and solitons, with D.H. Phong, J. Sturm and B. Weinkove, Comment. Math. Helv. 86 (2011), no. 1, 91-112, arXiv

• Canonical measures and Kahler-Ricci flow, with G. Tian, J. Amer. Math. Soc. 25 (2012), no. 2, 303-353, arXiv

• Test configurations, large deviations and geodesic rays on toric varieties, with S. Zelditch, Adv. Math. 229 (2012), no. 4 2338-2378, arXiv

• Bergman metrics and geodesics in the space of Kahler metrics on toric varieties, with S. Zelditch, Anal. PDE 3 (2010), no. 3, 295-358, arXiv

• The Kahler-Ricci flow with positive bisectional curvature, with D.H. Phong, J. Sturm and B. Weinkove, Invent. Math. 173 (2008), no. 3, 651-665, arXiv

• The Kahler-Ricci flow and the $\bar\partial$ operator on vector fields, with D.H. Phong, J. Sturm and B. Weinkove, J. Differential Geom. 81 (2009), no. 3, 631-647, arXiv

• Constructions of Kahler-Einstein metrics with negative scalar curvature, with B. Weinkove, Math. Ann. 347 (2010), no. 1, 59-79, arXiv

• Convergence of Bergman geodesics on CP^1, with S. Zelditch, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 7, 2209-2237, arXiv

• On Donaldson's flow of surfaces in a hyperkahler four-manifold, with B. Weinkove, Math. Z. 256 (2007), no. 4, 769-787, arXiv

• The Moser-Trudinger inequality on Kahler-Einstein manifolds, with D.H. Phong, J. Sturm and B. Weinkove, Amer. J. Math. 130 (2008), no. 3, 651-665, arXiv

• The Kahler-Ricci flow on surfaces of positive Kodaira dimension, with G. Tian, Invent. Math. 170 (2007), no. 3, 609-653, arXiv

• Energy functionals and canonical Kahler metrics, with B. Weinkove, Duke Math. J. 137 (2007), no. 1, 159-184, arXiv

• The Kahler-Ricci flow on Kahler surfaces, Proceedings of Gokova Geometry-Topology Conference 2006, 123-135

• On the convergence and singularities of the J-flow with applications to the Mabuchi energy, with B. Weinkove, Comm. Pure Appl. Math. 61 (2008), no. 2, 210-229, arXiv

• The α-invariant on Toric Fano Manifolds, Amer. J. Math. 127 (2005), no. 6, 1247-1259, arXiv

• The α-invariant on CP^2 blown up at two points, Trans. Amer. Math. Soc. 357 (2005), no. 1, 45-57, arXiv

Notes:

• Lecture notes on the Kahler-Ricci flow, with B. Weinkove, ‘An introduction to the Kahler-Ricci flow’, 89-188, Lecture Notes in Math., 2086, Springer, Cham, 2013, arXiv

• Complex Monge-Ampere equations, with D.H. Phong and J. Sturm, Surveys in Differential Geometry, vol. 17, 327-411 (2012), arXiv

• The Szego kernel on an orbifold circle bundle, thesis at Columbia University, 2004