For Math 20550 students:

• (1) For course information of Math 20550, click   Math20550-info.pdf.
• (2) For a tentative schedule of Math 20550, click   Math20550-schedule.pdf.
• (3) For homework assignments of Math 20550 of entire semester, click   Math20550-assignments.pdf.
• (4) Homework due Friday's class, December 4: Stewart's book Chapters 17.5, 17.6a, 17.6b and 17.7a. Please show your work when you are doing your homework problems.

Research Interests and Vita:

• Differential Geometry
• For an updated vitae, click CV-Sept2009.pdf.

Information for the textbook ``A Rapid Course in Riemannian Geometry" in Chinese

   For the PDF file of whole book, see the whole book

Publications and Preprints in PDF files:

• 1.``Existence of an extremal joining two arbitrary points on a manifold". (Chinese) Nanjing Daxue Xuebao Ziran Kexue Ban 20 (1984), no. 4, 627--640. (Math. Review: 87c:58026);
• 2. ``Some homotopically equivalent spaces of paths, and Fréchet numbers". (Chinese) Acta Math. Sinica 28 (1985), no. 6, 817--824. (Math. Review: 87d:58036);
• 3. ``The existence of generalized isothermal coordinates for higher-dimensional Riemannian manifolds." Trans. Amer. Math. Soc. 324 (1991), no. 2, 901--920. (Math Review: 91h:53030);  see  Yamabe.pdf.
• 4. ``Prescribing Ricci curvature on open surfaces." Hokkaido Math. J. 20 (1991), no. 2, 265--278. (with D. DeTurck, Math Review: 92k:53068);
• 5. ``Simple closed geodesics on convex surfaces." J. Differential Geom. 36 (1992), no. 3, 517--549.   see  Calabi-Cao.pdf. (with E. Calabi; Math. Review: 93h:53039);
• 6. ``The Bers-Nielsen kernels and souls of open surfaces with negative curvature". Michigan Math. J. 41 (1994), no. 1, 13--30. (Math Review: 95d:30085)
• 7. ``The Ricci curvature equation with rotational symmetry". Amer. J. Math. 116 (1994), no. 2, 219--241. (with D. DeTurck; Math. Review: 94m:53052); see Ricci.pdf.
• 8. ``Rigidity for non-compact surfaces of finite area and certain Kähler manifolds". Ergodic Theory Dynam. Systems 15 (1995), no. 3, 475--516. (Math Review, 96e:53046);
• 9. ``Examples of simply-connected Liouville manifolds with positive spectrum." Differential Geom. Appl. 6 (1996), no. 1, 31--50. (with I. Benjamini, Math Review 97b:53036); see Example.pdf
• 10. ``A new isoperimetric comparison theorem for surfaces of variable curvature." Duke Math. J. 85 (1996), no. 2, 359--396. (with I. Benjamini; Math. Review, 97m:58046);
• 11. ``An isoperimetric comparison theorem for PL-manifolds of non-positive curvature". Proceedings of the Third Summer School on Differential Geometry, Partial Differential Equations and Numerical Analysis (Spanish) (Bogota, 1995), 1--7, Colec. Mem., 7, Acad. Colombiana Cienc. Exact. F. Natur., Bogota, 1996. (Math Review: 98a:53061)
• 12. ``Topics in geometric analysis via Gromov's methods". Adv. in Math. (China) 26 (1997), no. 6, 481--506. (Math Review: 99d:53038)
• 13. ``Kahler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature", (with F. Xavier), in Mathematische Annalen, Volume 319 Issue 3 (2001) pp 483-491. (Math Review: 2001m:53060). See Euler.pdf.
• 14. ``Splittings and Cr-structures for manifolds with nonpositive sectional curvature'', (with J. Cheeger and X. Rong) in Inventiones mathematicae, Volume 144 Issue 1 (2001) pp 139-167. (Math Review: 2002a:53049). See splittings.pdf.
• 15. ``Cheeger isoperimetric constants of Gromov-hyperbolic spaces with quasi-poles", ``Communication in Contemporary Math.", Vol. 2, No. 4 (2000) 511-533, (Math Review: 2001m:53068). See Quasi.ps or Martin-bounday.pdf
• 16. ``A New 3-dimensional Curvature Integral Formula for PL-manifolds of Non-positive Curvature'', (with J.  F. Escobar), Comm. Analysis and Geometry, Vol. 11, (2003), 489-551. See Cag2.ps or see Cag2.pdf.
• 17.  ``The smoothness of Riemannian submersions with nonnegative sectional curvature " (with M. Shaw),   in ``Communication in Contemporary Math. Vol 7 (2005) p137-144. ", see Pulished.pdf .
• 18. ``Finsler Geometry of Projectivized Vector Bundles " (with P. Wong), J. Math. Kyoto Univ. volume 43, No.2 (2003) 383-424, see Cao-Wong.pdf or Cao-Wong.ps.
• 19. ``Local splitting structures on nonpositively curved manifolds and semirigidity in dimension 3" (with J. Cheeger and X. Rong),  In the special volume in honor of Peter Li,  Comm. Anal. Geom., volume 12, No. 1, (2004), 391-417.  See ccr2.pdf or 2CCR.ps.
• 20. ``A closing lemma for flap trips in surfaces of non-positive curvature and applications" (with F. Xavier), Preprint, July 2001, see Close.ps or Close.pdf.
• 21. ``Estimates for the d-bar-Neumann problem and nonexistence of C^2 Levi-flat hypersurfaces in  CP^n” (with M. Shaw and L. Wang), Mathematische Zeitschrift, vol 248, No.1. (2004) p183-221, see  online.pdf.  There is an error in Proposition 4.5 of the paper. This error has been corrected (Prop. 4.5 has been by-passed). Erratum: Math. Z. 248 (2004), no. 1, 223--225. For detail, see item #26 below.
• 22. ` A new proof of the Takeuchi theorem " (with M. Shaw), Lecture Notes of Seminario Interdisplinare di Mathematica, vol. 4(2005) pp.65-72. For the PDF file of the paper, see New-proof.pdf.
• 23. ``The center of mass flow and applications to the topological rigidity of injective F-structure", (with Jeff Cheeger and Xiaochun Rong ), in preparation.
• 24. ``Martin points on open manifolds of non-positive curvature" (with HuiJun Fan and F. Ledrappier), Transaction of Amer. Math. Soc., vol 359 (2007) page 5697-5723, see Martin.pdf.
• 25. ``Rigidity for the marked length-spectrum of higher dimensional nonpositively curved manifolds with F-structure,” (with C. Croke), in preparation.
• 26. ``The d-bar-Cauchy problem and nonexistence of Lipschitz Levi-flat hypersurfaces in CP^n with n >= 3" (with M. Shaw), Math. Zeit, vol 256 (2007), No. 1, page 175-192. See Lipschitz-p.pdf.
• 27. ``K\"ahler version of Cheeger-Gromoll-Perelman soul theory with positive bi-sectional curvature" (with Jiaping Wang) in preparation.
• 28. ``An intrinsic proof of Gromoll-Grove diameter rigidity theorem" (with Hongyan Tang), vol 19 no.3 (2007) 401-419 ``Communication in Contemporary Math." , see Cao-Tang.pdf.
• 29. ``Pseudo-Einstein and Q-flat metrics with eigenvalue estimates on CR-hypersurfaces", (with Shu-Cheng CHANG), ``Indiana Univ. Math. Journal", vol 56, No. 6 (2007), pages 2839-2858, see Published.pdf.
• 30. ``Positive harmonic functions on singular spaces with non-negative Ricci curvature" (with Bobo Hua), in preparation.
• 31. ``Certain 4-manifolds with non-negative curvature". Front. Math. 2008, volume 3, No. 4 : pages 475-494. See 4-manifolds.pdf.
• 32. ``An extension of Perelman's soul theorem for singular spaces" (with B. Dai and J. Mei), updated on December 20, 2007, see soul-extension.pdf.
• 33. ``An optimal extension of Perelman's comparison theorem for quadrangles and its applications" (with B. Dai and J. Mei), see quadrangles.pdf.
• 34. ``The modified Calabi-Yau problems for CR-manifolds and applications" (with Shu-Cheng Chang),in the book ``TOPOLOGY AND PHYSICS Proceedings" of the Nankai International Conference in Memory of Xiao-Song Lin Tianjin, China 27 - 31 July 2007, (edited by Kevin Lin, ZhenghanWang and Weiping Zhang), ISBN 978-981-281-910-9 981-281-910-X, Nankai Tracts in Mathematics - Vol. 12 (2008) pages 3-17. see Calabi-Yau-CR.pdf.
• 35. ``Minimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds" (with Xaioyang Chen), see MinVol.pdf.
• 36. ``A quadrangle comparison theorem and its application to soul theory for Alexandrov spaces" (with Bo Dai and Jiaqiang Mei), see Quadrangle.pdf.
• 37. ``A proof of Perelman's collapsing theorem for $3$-manifolds" (with Jian Ge), updated on October 28, 2009 (added two figures) see Perelman-collapsing.pdf.

Links to various web pages:

Notre Dame E-mail Web-site

Differential Geometry and Global Analysis (MSRI)

MathSciNet, Mathematical Reviews on the Web ,

World-wide Mathematics Department Web Servers ,

USA Mathematics Department Web Servers,

• For future colloquium schedule of 2009-2010, click 2009-2010.pdf.

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