Research Interests

• differential geometry, geometric analysis and PDE

Papers

• A simpler proof of Frank and Lieb's sharp inequality on the Heisenberg group (joint with Fengbo Hang), preprint.

• A Sharp Inequality Relating Yamabe Invariants on Asymptotically Poincare-Einstein Manifolds with a Ricci Curvature Lower Bound (joint with Zhixin Wang), preprint.

• On a Sharp Inequality Relating Yamabe Invariants on a Poincare-Einstein Manifold (joint with Zhixin Wang), Proc. AMS. 150 (2022), no. 11, 4923-4929.

• Uniqueness Results on a geometric PDE in Riemannian and CR Geometry Revisited , Math.Z. 301 (2022), no. 2, 1299-1314.

• Remarks on an Inequality for Closed Hypersurfaces in Complete Manifolds with Nonnegative Ricci Curvature , to appear in Annales de la Faculte des Sciences de Toulouse.

• Improved Sobolev inequality with constraints (joint with Fengbo Hang), IMRN 2022, no. 14, 10822-10857.

• Liouville type theorems on manifolds with nonnegative curvature and strictly convex boundary (joint with Qianqiao Guo and Fengbo Hang), Math. Res. Lett. 28 (2021), no. 5, 1419-1439.

• Uniqueness results for positive harmonic functions on Bⁿ satisfying a nonlinear boundary condition (joint with Qianqiao Guo), Calculus of Variations and PDE 59 (2020), no. 5, Paper No. 146, 8 pp.

• On compact Riemannian manifolds with convex boundary and Ricci curvature bounded from below , J. Geom. Anal. 31 (2021), no. 4, 3988-4003.

• Remarks on a mean field equation on S² , (joint work with Changfeng Gui, Fengbo Hang and Amir Moradifam). Journal of Mathematical Study, 54 (2021), no. 1, 81-88. (special issue for 70th birthdays of Professor Alice Chang and Paul Yang)

• Uniqueness results on surfaces with boundary , Calculus of Variations and PDE 56 (2017), no. 3, Paper No. 87, 11 pp.

• An integral formula in Kahler geometry with applications , Comm. Contemporary Math. 19 (2017), no. 5, 1650063, 12 pp.

• Brendle's inequality on static manifolds (joint with Ye-Kai Wang), Journal of Geom. Anal. 28 (2018), no. 1, 152-169.

• Characterization by symmetry of solutions of a nonlinear subelliptic equation on the Heisenberg group (joint with Yazhou Han and Meijun Zhu), J. Math. Study 50 (2017), no. 1, 17-27.

• Boundary effect of Ricci curvature (joint work with Pengzi Miao), J. Diff. Geom. 103 (2016), 59-82.

• Some recent results in CR geometry , Tsinghua lectures in mathematics, 469-484, Adv. Lect. Math. (ALM), 45, Int. Press, Somerville, MA.

• On a remarkable formula of Jerison and Lee in CR geometry , Math. Res. Letters. 22 (2015), no.1, 279-299.

• The isoperimetric constant of symmetric spaces of noncompact type , Proc. AMS.143 (2015), no.11, 4885-4891.

• A new characterization of the CR sphere and the sharp eigenvalue estimate for the Kohn Laplacian ( joint with Song-Ying Li, Duong Ngoc Son), Advances in Math. 281 (2015), 1285-1305.

• An Obata-type theorem in CR geometry ( joint with Song-Ying Li), J. Diff. Geom. 95 (2013), 483-502.

• Pinching theorems for the volume entropy ( joint with Francois Ledrappier), preprint, 2011.

• Bottom of the spectrum of Kahler manifolds with strongly pseudo-convex boundary ( joint with Song-Ying Li), IMRN 19 (2012), 4351-4371.

• Compactifications of complete Riemannian manifolds , Surveys in Geometric Analysis and relativity ALM 20, pp. 517-529.

• Local gradient estimate for p-harmonic functions on Riemannian manifolds (joint with Lei Zhang), to appear in Comm. Anal. Geom. 19 (2011), no. 4, 759-771.

• An integral formula for the volume entropy with applications to rigidity (joint with Francois Ledrappier), Journal Diff. Geom. 85 (2010), 461-478.

• Extension of a theorem of Shi and Tam (joint with Michael Eichmair and Pengzi Miao), Calculus of Variations and PDE. 43 (2012), no. 1-2, 45-56.

• Rigidity Theorems for Compact Manifolds with Boundary and Positive Ricci Curvature (joint with Fengbo Hang), J. Geom. Anal 19 (2009), 628-642.

• A rigidity theorem for the hemishpere (joint with Fengbo Hang), preprint. (now subsumed in the paper above)

• Harmonic functions, entropy, and a characterization of the hyperbolic space, Journal Geom. Anal. 18 (2008), 272-284.

• An integral equation in conformal geometry (joint with Fengbo Hang and Xiaodong Yan), Annales de l'Institut Henri Poincare (C) 26 (2009), 1-21.

• A note on a conjecture of Schroeder and Strake (joint with Fengbo Hang), Pacific Math. Journal 232 (2007), 283-287.

• Remark on Zhong-Yang's eigenvalue estimate (joint with Fengbo Hang), Int. Math. Res. Notices 2007.

• Sharp integral inequalities for harmonic functions (joint with Fengbo Hang and Xiaodong Yan), Comm. Pure Appl. Math. 61 (2008), 54-95.

• On the stability of Kahler-Einstein manifolds (joint with Xianzhe Dai and Guofang Wei), Comm. Anal. Geom 15 (2007), 669-693.

• On asymptotically complex hyperbolic Kahler manifolds , Proc. AMS 135 (2007), no.9, 2949-2960.

• Some rigidity and nonrigidity results on the sphere (joint with Fengbo Hang), Comm. Anal. Geom. 14 (2006) no.1, 91-106.

• A new approach to some nonlinear geometric equations in dimension two (joint with Fengbo Hang), Cal. of Var. PDE. 26 (2006) no. 1, 119-135.

• On the stability of Riemannian manifolds with a parallel spinor (joint with Xianzhe Dai and Guofang Wei), Invent. Math. 161 (2005), 151-176.

• Uniqueness of the AdS spacetime in any dimensions , Acta Math. Sinica 21 (2005) no. 4, 917-922.

• On the L² cohomology of a convex cocompact hyperbolic manifold, Duke Math. J. 115 (2002) no.2, 311-327.

• On the geometry of conformally compact Einstein manifolds, Math. Research Letters 8 (2001) no. 5&6 (2001) 671--688.

• A new proof of Lee's theorem on the spectrum of conformally compact Einstein manifolds, Comm. Anal. Geom. 10 (2002), no.3, 647-651.

• The mass of asymptotically hyperbolic manifolds, Journal Diff. Geom. 57 (2001), no.2, 273-299.

Notes

• Notes on Perelman's paper on the entropy formula for Ricci flow and its geometric applications (joint with Natasa Sesum and Gang Tian), 2003.