Math Research

• Differential (Complex) Geometry and Partial Differential Equation (Complex Analysis) with details below:
   

  Differential/Complex Geometry:

  • Riemannian geometry;
    • Curvature and global shapes/topology, fundamental groups/holonomy;
    • Geodesics, minimal surfaces, and related problems;
  • Complex geometry of Kähler manifolds and holomorphic vector bundles;
    • Function theory and complex structure of noncompact complete Kähler manifolds;
    • Geometric implications of various notions of positivity in Kähler geometry and algebraic geometry;
  • Geometry of curves, surfaces and hypersurfaces;
  • Lie groups/algebras and geometries invariant under group actions;
  • Isoperimetric inequalities and curvature independent geometric inequalities;
  • Geometry of convex bodies.

  Partial Differential Equations:

  • Partial differential equations related to differential geometry;
    • Heat flows method applications in global differential geometry;
      • Moving curves, surfaces and hypersurfaces;
      • Harmonic map heat flow;
      • Deforming Riemannian metrics or connections via the Ricci flow or other parabolic equations;
      • Entropy monotonicity and Hessian (differential Harnack) estimates for parabolic flows;
    • Minimal surfaces, surfaces of prescribed curvature, and related problems;
    • Harmonic functions and harmonic maps, related variational problems;
  • Fully nonlinear elliptic and parabolic equations;
    • Regularity theory;
    • Qualitative and asymptotic behaviour;
  • Eigenvalue problems arising in geometry.

  Applications:

  • Physics, particularly quantum mechanics, general relativity, condense matters, and string theory;
  • Image processing, quantum computing/representation theory and information geometry;
  • Moving interfaces, phase changes, and related problems;
  • Reaction-diffusion systems in mathematical biology.

Seminars and Confenrences:

• Geometry Seminar at UCSD
  
• Chongqing workshop in geometric analysis, October, 2022
  
• Kansas Geometric Analysis Seminar, April, 2024
  
• 29th Southern California Geometric Analysis Seminar
  
• Workshop on geometric evolution equations
  

Current CV

List of research papers and preprints:

• A classification of locally Chern homogenous Hermitian manifolds (with F. Zheng), preprint.
• Kähler manifolds and cross quadratic bisectional curvature (with F. Zheng), J. Eur. Math. Soc.(JEMS) To appear.
• A Schur's theorem via a monotonicity and the expansion module, J. reine angew. Math. (Crelle) 805 (2023), 101–114.
• A multiplicity estimate for the Jacobi operator of a nonflat Yang-Mills field over 𝕊 ⁿ, Geometriae Dedicata. 217 (2023), no. 5, Article 92, 12 pp .
• On Hermitian manifolds whose Chern connection is Ambrose-Singer (with F. Zheng), Trans. Amer. Math. Soc. 376 (2023), no. 9, 6681–6707.
• An effective estimate of Betti numbers, J. Geom. Anal.(a special volume in honor of P. Li), 32 (2022), no. 6, Paper No. 176, 11 pp.
• Positivity and Kodaira embedding theorem (with F. Zheng), Geom. Topol. 26 (2022), no. 6, 2491–2506.
• An application of a C²-estimate for a complex Monge-Ampère equation (with C. Li and X.-H. Zhu), Internat. J. Math. (a special volume in honor of S.-S. Chern's 110th anniversary), 32 (2021), no.12, 2140007.
• The fundamental group, rational connectedness and the positivity of Kähler manifolds, J. reine angew. Math. (Crelle), 774 (2021), 267–299.

• Manifolds with positive orthogonal Ricci curvature (with Q. Wang and F. Zheng), Amer. J. Math., 143 (2021), no. 3, 833–857.
• Liouville theorems and a Schwarz lemma for holomorphic mappings between Kähler manifolds (Top Cited Article of the Journal, 2020-2021), Comm. Pure Appl. Math., 74 (2021), 1100–1126.
• An alternate induction argument in Simons' proof of holonomy theorem , Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs (Nankai, 2019, A. Grigor’yan, Y. Sun, Eds.) Advances in Analysis and Geometry, 3 (2021), 443–458.
• Kähler-Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature (with X. Li), J. Math. Pures Appl. (Liouville), 138 (2020), 28–45.
• Gap theorem on Kähler manifolds with nonnegative orthogonal bisectional curvature (with Y.-Y. Niu), J. reine angew. Math. (Crelle), 763 (2020), 111–127.
• On orthogonal Ricci curvature (with F. Zheng), Advances in Complex Geometry, Contemp. Math. , 735 (2019), 203–215.
• Schwarz lemmata and their applications, Internat. J. Math. (a special volume in honor of L.-F. Tam), 30 (2019), no.13, 1940007.
• Comparison and vanishing theorems for Kähler manifolds (with F. Zheng), Calc. Var. Partial Differential Equations, 57 (2018), no. 6, Art. 151, 31 pp.
• Four-dimensional gradient shrinking solitons with positive isotropic curvature (with X. Li and K. Wang), Int. Math. Res. Not. (IMRN), 2018 (2018), 949–959.
• Entropy and a convergence theorem for Gauss curvature flow in high dimensions (with P. Guan), J. Eur. Math. Soc.(JEMS), 19 (2017), 3735–3761.
• Flow by the power of the Gauss curvature (with B. Andrews and P. Guan), Adv. Math., 299 (2016), 174–201.
• Isoperimetric comparison via viscosity (with K. Wang), J. Geom. Anal., 26 (2016), no. 4, 2831–2841.
• Parabolic frequency monotonicity and a theorem of Hardy-Pólya-Szegö , Contemp. Math., 644 (2015), 203–210.
• Erratum to: An optimal gap theorem [MR2957306], Invent. Math., 196 (2014), no. 2, 511–514.

• Ricci flow and manifolds with positive curvature , Symmetry: representation theory and its applications, 491–504, Progr. Math., 257, Birkhδuser/Springer, New York, 2014.
• A Perron type theorem on the principal eigenvalue of nonsymmetric elliptic operators, Amer. Math. Monthly, 121 (2014), no. 10, 903–908.
• Poincaré-Lelong equation via the Hodge-Laplace heat equation (with L.-F. Tam), Compos. Math., 149 (2013), no. 11, 1856–1870.
• Estimates on the modulus of expansion for vector fields solving nonlinear equations, J. Math. Pures Appl. (Liouville), 99 (2013), 1–16.
• Eigenvalue comparison on Bakry-Emery manifolds (with B. Andrews), Comm. Partial Differential Equations, 37 (2012), 2081–2092.
• An optimal gap theorem, Invent. Math., 189 (2012), 737–761. Erratum.
• Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations (with I. Bakas and S.-L. Kong), J. reine angew. Math. (Crelle), 663 (2012), 209–248.
• Sharp differential estimates of Li-Yau-Hamilton type for positive (p,p)-forms on Kähler manifolds (with Y.-Y. Niu), Comm. Pure Appl. Math., 64 (2011), 920–974.
• Manifolds with 1/4-pinched flag curvature (with B. Wilking), Geom. Funct. Anal. (GAFA), 20 (2010), 571–591.
• Monotonicity and holomorphic functions, Geometry and Analysis I, Adv. Lect. Math. (special volume in honor of S.-T. Yau), 17 (2010), 447–457.
• The large time asymptotics of the entropy , Complex Analysis. Trends Math., 2010, 301–306.
• Sharp logarithmic Sobolev inequalities on gradient solitons and applications (with J. Carrillo), Comm. Anal. Geom., 17 (2009), 721–753.
• Gradient estimate for p-harmonic functions, 1/H flow and an entropy formula (with B. Kotschwar), Ann. Sci. Éc. Norm. Supér., 42 (2009), no. 1, 1–36.
• Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds (with P. Lu, J.-L. Vázquez and C. Villani), J. Math. Pures Appl. (Liouville), 91 (2009), no. 1, 1–19.
• Closed type-I ancient solutions to Ricci flow, Recent advances in geometric analysis, Adv. Lect. Math., 11 (2009), 147–150.
• On 4-dimensional gradient shrinking solitons (with N. Wallach), Int. Math. Res. Not.(IMRN), 2008 (2008), 152, 13 pages.
• On a classification of the gradient shrinking solitons (with N. Wallach), Math. Res. Lett., 15 (2008), no. 5, 941–955.
• Monotonicity and Li-Yau-Hamilton inequality, Surv. Differ. Geom., 12 (2008), 251–302.
• Local monotonicity and mean value formulas for evolving Riemanniann manifolds (with K. Ecker, D. Knopf and P. Topping), J. reine angew. Math. (Crelle), 618 (2008), 89–130.
• Complete manifolds with nonnegative curvature operator (with B. Wu), Proc. Amer. Math. Soc., 135 (2007), 3021–3028.
• Mean value theorem on manifolds, Asian J. Math. (special volume in honor of S. Baouendi), 11 (2007), 277–304.
• A matrix Li-Yau-Hamilton inequality for Kähler-Ricci flow, J. Differential Geom., 75 (2007), 303–358.
• A note on Perelman's LYH type inequality, Comm. Anal. Geom. , 14 (2006), 883–905.
• Ancient solutions to Kähler-Ricci flow, Math. Res. Lett, 12 (2005), 633–654.
• Entropy and reduced distance for Ricci expanders (with M. Feldman and T. Ilmanen), J. Geom. Anal., 15 (2005), 49–62.
• Matrix Li-Yau-Hamilton estimates on Kähler manifolds (with H.-D. Cao), Math. Ann., 331 (2005), 795–807.
• Kähler-Ricci flow on complete manifolds, Written version of lectures given at 2005, Clay-MSRI workshop.

• A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature, J. Amer. Math. Soc.(JAMS), 17 (2004), 909–946.
• Addenda to "The entropy formula for linear heat equation", J. Geom. Anal., 14 (2004), 369–374. Appendix
• The entropy formula for linear heat equation, J. Geom. Anal., 14 (2004), 87–100;
• Ricci flow and nonnegativity of sectional curvature, Math. Res. Lett., 11 (2004), 883–904.
• Kähler-Ricci flow and the Poincaré-Lelong equation (with L.-F. Tam), Comm. Anal. Geom., 12 (2004), 111–141.
• Plurisubharmonic functions and structure of complete Kähler manifolds with nonnegative curvature (with L.-F. Tam), J. Differential Geom., 64 (2003), 457–524.
• The Lefschetz theorems for CR submanifolds and the nonexistence of the real analytic Levi flat submanifolds (with J. Wolfson), Comm. Anal. Geom. , 11 (2003), 551–562.
• Plurisubharmonic functions and Kähler-Ricci flow (with L.-F. Tam), Amer. J. Math., 125 (2003), 623–645.
• Liouville properties of plurisubharmonic functions (with L.-F. Tam), unpublished manuscript (results sharpened in J. Differential Geom. 2003 paper).
• Ricci Flatness of ALE manifolds (with Y.-G. Shi and L.-F. Tam), Trans. Amer. Math. Soc., 355 (2003), 1933–1959.
• Monotonicity and Kähler-Ricci flow , Geometric Evolution Equations (National Tsing Hua Univ. 2002), Contemp. Math., 367 (2005), 149–165.
• A Bernstein type theorem for minimal volume preserving maps, Proc. Amer. Math. Soc. , 130 (2002), 1207–1210.
• Poisson equation and Hermitian-Einstein metrics on holomorphic vector bundles over complete noncompact Kähler manifolds , Indiana Univ. Math. J., 51 (2002). no 3, 679–704.
• Poisson equation, Poincaré-Lelong equation and curvature decay on complete Kähler manifolds (with Y.-G. Shi and L.-F. Tam), J. Differential Geom., 57 (2001), 339–388.
• Gap Theorems for minimal submanifolds in ℝⁿ, Comm. Anal. Geom. , 9 (2001), no 3, 641–656.
• Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds (with H. Ren), Trans. Amer. Math. Soc., 353 (2000), 441–456.
• Holomorphicity of proper harmonic maps between unit balls with Bergamn metrics (with S.-Y. Li), Math. Ann. , 316 (2000), 333–354.
• Hermitian harmonic maps from complete Hermitian manifolds to complete Riemannian manifolds , Math. Z., 232 (1999), 331–335.
• Vanishing theorems on complete Kähler manifolds and their applications , J. Differential Geom. , 50 (1998), 89–122.
• Non-existence of some quasi-conformal diffeomorphisms , Michigan Math. J. , 45 (1998), 489–495.
• Notes on transnormal functions and its updated version, unpublished manuscript.

• Book Review on Extrinsic geometric flows , by Ben Andrews, Bennett Chow, Christine Guenther, and Mat Langford. Bulletin of AMS, (2021)
• The Ricci Flow: Techniques and Applications, Part IV: Long-Time Solutions and Related Topics. (With B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu and F. Luo) Mathematical Surveys and Monographs. 206. American Mathematical Society, Providence, RI, 2015.
• The Ricci Flow: Techniques and Applications, Part III: Geometric-Analytic Aspects. (With B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu and F. Luo) Mathematical Surveys and Monographs. 163. American Mathematical Society, Providence, RI, 2010.
• The Ricci Flow: Techniques and Applications, Part II: Analytic Aspects. (With B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu and F. Luo) Mathematical Surveys and Monographs. 144. American Mathematical Society, Providence, RI, 2008.
• The Ricci Flow: Techniques and Applications, Part I: Geometric Aspects. (With B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu and F. Luo) Mathematical Surveys and Monographs. 135. American Mathematical Society, Providence, RI, 2007.
• Hamilton's Ricci Flow. (With B. Chow and P. Lu.) Graduate Studies in Mathematics. 77. American Mathematical Society, Providence, RI, 2006.

Links

Mathematics Genealogy Project
MathSciNet
Chern Number: 2
Erdös Number: 3
Einstein Number: 4
Riemann Number: 5
Gauss Number: 6