Math Research

• Differential (Complex) Geometry and Partial Differential Equation (Complex Analysis)

Summary of Recent Researches:

• Summary-2008
  

• Geometry Seminar at UCSD
  
• UCI-UCSD joint Geometry Seminar
  
• 16th Southern California Geometric Analysis Seminar
  
• Workshop on geometric evolution equations
  

List of research papers and preprints:

• Sharp logarithmic Sobolev inequalities on gradient solitons and applications, (joint with Jose Carrillo), submitted preprint.
• Local Aronson-Benilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds, (joint with Peng Lu, Juan-Luis Vazquez and Cedric Villani), Jour. Math. Pures Appl., to appear.
• Closed type-I ancient solutions to Ricci flow, Contemp. Math., to appear.
• Local gradient estimates of $p$-harmonic functions, $1/H$-flow, and an entropy formula, (joint with Brett Kotschwar), Ann. Sci. Ecole Norm. Sup. to appear.
• On $4$-dimensional gradient shrinking solitons, (joint with Nolan Wallach), IMRN, Vol 2008 (2008), rnm152, 13 pages.
• On a classification of the gradient shrinking solitons, (joint with Nolan Wallach), Math. Res. Lett, to appear.
• Monotonicity and Li-Yau-Hamilton inequality, Surveys in Differential Geometry, vol 12, pages 251--302.
• Complete manifolds with nonnegative curvature operator (joint with B. Wu), Proceedings of AMS. 135(2007), 3021--3028.
• Mean value theorem on manifolds, Asian J. Math. 11(2007), 277--304.
• A note on Perelman's LYH type inequality, Comm. Anal. Geom. 14(2006), 883--905.
• Local monotonicity and mean value formulas for evolving Riemanniann manifolds (joint with K. Ecker, D. Knopf and P. Topping), J. reine angew. Math. (Crelle), 618(2008), 89--130.
• Kaehler-Ricci flow on complete manifolds, Written version of lectures given at 2005, Clay-MSRI workshop.

• A matrix Li-Yau-Hamilton inequality for Kaehler-Ricci flow, J. Differential Geom. 75(2007), 303--358.
• Ancient solutions to Kaehler-Ricci flow, Math. Res. Lett, 12 (2005), 633--654.
• Entropy and reduced distance for Ricci expanders (joint with M. Feldman and T. Ilmanen), J. Geom. Anal. 15 (2005), 49--62.
• A monotonicity formula on complete Kaehler manifolds with nonnegative bisectional curvature, J. Amer. Math. Soc., 17 (2004), 909--946.
• The entropy formula for linear heat equation, J. Geom. Anal., 14(2004), 87-100; Addenda to "The entropy formula for linear heat equation" , J. Geom. Anal., 14 (2004), 369-374. Appendix
• Ricci flow and nonnegativity of sectional curvature, Math. Res. Lett, 11 (2004), 883-904.
• Plurisubharmonic functions and structure of complete Kaehler manifolds with nonnegative curvature, (with L.-F. Tam), J. Differential Geom. 64(2003), 457-524.
• Monotonicity and Kaehler-Ricci flow , Geometric Evolution Equations (National Tsing Hua Univ. 2002), Contemp. Math., 367, pages 149--165, Amer. Math. Soc., Providence, RI, 2005.
• 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.
• Matrix Li-Yau-Hamilton estimates on Kaehler manifolds (with H.-D. Cao), Math. Ann. 331(2005), 795-807.
• Liouville properties of plurisubharmonic functions (with L.-F. Tam), unpublished manuscript (results sharpen in JDG 2003 paper).
• Plurisubharmonic functions and Kaehler-Ricci flow (with L.-F. Tam), Amer. J. Math. 125 (2003), 623-645.
• Kaehler-Ricci flow and the Poincare-Lelong equation (with L.-F. Tam), Comm. Anal. Geom. 12(2004), 111-141.
• Ricci Flatness of ALE manifolds (with Y.-G. Shi and L.-F. Tam), Transactions of AMS. 355(2003), 1933-1959.
• Poisson equation and Hermitian-Einstein metrics on holomorphic vector bundles over complete noncompact Kahler manifolds , Indiana Univ. Math. J. 51(2002). no 3, 679-704.
• Poisson equation, Poincare-Lelong equation and curvature decay on complete Kaehler manifolds (with Y.-G. Shi and L.-F. Tam), J. Differential Geom. 57(2001), 339-388.
• A Bernstein type theorem for minimal volume preserving maps, Proceedings of AMS. 130(2002), 1207-1210.
• Gap Theorems for Minimal Submanifolds in $R^n$ , Comm. Anal. Geom. vol 9(2001), no 3, 641-656.
• Hermitian-Einstein metrics for vector bundles on complete Kaehler manifolds (with H. Ren), Transactions of AMS. 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.
• Vanishing theorems on complete Kaehler manifolds and their applications , J. Differential Geom. 50(1998), 89-122.
• Hermitian harmonic maps from complete Hermitian manifolds to complete Riemannian manifolds , Math. Z. 232(1999), 331-335.
• Non-existence of some quasi-conformal diffeomorphisms , Michigan Math. J. 45(1998), 489-495.
• Notes on transnormal functions , unpublished manuscript.

• The Ricci Flow: Techniques and Applications, Part III: Geometric-Analytic Aspects. (With Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, Jim Isenberg, Tom Ivey, Dan Knopf, Peng Lu and Feng Luo.) Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI. To appear.
• The Ricci Flow: Techniques and Applications, Part II: Analytic Aspects. (With Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, Jim Isenberg, Tom Ivey, Dan Knopf, Peng Lu and Feng Luo.) Mathematical Surveys and Monographs, Vol. 144. American Mathematical Society, Providence, RI, 2008.
• The Ricci Flow: Techniques and Applications, Part I: Geometric Aspects. (With Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu and Feng Luo.) Mathematical Surveys and Monographs, Vol. 135. American Mathematical Society, Providence, RI, 2007.
• Hamilton's Ricci Flow. (With Bennet Chow and Peng Lu.) Graduate Studies in Mathematics, Vol 77. American Mathematical Society, Providence, RI, 2006.