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Picture of Jianguo Cao

Address:

Jianguo Cao

Room 279 Hurley

Department of Mathematics

University of Notre Dame

Notre Dame, Indiana 46556-4618

Phone:

(574) 631-8847

(574) 631-6579 (Fax)

Email: mailto:cao.7@nd.edu

Teaching Math 20550 and Math 20580, Fall 2008

(I) 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 Thursday's tutorial session, October 17: Stewart's book Chapters 16.1, 16.2 and 16.3

(II) For Math 20580 students:

  • (1) For course information of Math 20580, click   Math20580-info.pdf.
  • (2) For a tentative schedule of Math 20580, click   Math20580-schedule.pdf.
  • (3) For homework assignments of Math 20580 of entire semester, click   Math20580-assignments.pdf.
  • (4) Homework due Thursday's tutorial session, October 17: Lay's book Chapter 4.6, 4.7, 5.1 and 5.2
  • (5a) Exam 2 will take place on October 14, Tuesday, 8am - 9:15 am in Room 127 Hayes.
  • (5b) Our Exam 2 will cover up to Chapter 4.7.
  • (5c) On Monday (Oct 13) evening 7pm-9pm, we have extra Review session in HAYE 125.
  • (5d) For a copy of old Exam 2, click   PracExam2.pdf.

Research Interests and Vita:

  • Differential Geometry
  • For an updated vitae, click Sept2008.pdf.

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

The book was co-authored with Professor Youde Wang, Institute of Math., Academy Sinica. The book was published by Science Press in China, February 2006. Hard copies of the book are available at some bookstores in mainland, China. Price per copy: 28 Chinese Yuans ($4 US dolloars). The following PDF file was updated on Jan. 16, 2006

   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), publication  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 semi-rigidity of abelian structures on generalized graph-manifolds of nonpositive sectional curvature" (with J. Cheeger and X. Rong), preprint 2005, to be posted soon.
  • 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, 2006.
  • 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) preprint Feb 2008, to be posted soon.
  • 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. ``The center of mass flow and applications to the topological rigidity of injective F-structure", (with Jeff Cheeger and Xiaochun Rong ), in preparation, Feb 2008.
  • 31. ``Certain 4-manifolds with non-negative curvature". Preprint 1993. This paper was written in 1993. From time to time, I often receive requests for this preprint. Thus, it might be helpful to post a PDF-file of this paper here, see 4-manifolds.pdf. ( to appear in ``International Journal of Geometry and Topology").
  • 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. ``Microlocalization and nonexistence of $C^2$ Levi-flat hypersurfaces in $\Bbb CP^2$" (with Mei-Chi Shaw), Preprint, Feb 2008, submitted.
  • 36. ``Positive harmonic functions on singular spaces with non-negative Ricci curvature" (with Bobo Hua), Preprint, September 2008.
  • 37. ``Minimal volume and filling invariants of visibility manifolds" (with Xaioyang Chen), in preparation.

 

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 2008-2009, click 2008-2009.pdf.

Notre Dame Mathematics Department Homepage

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