Research Papers of Christopher B. Croke

• 1. Some New Riemannian Invariants", J. Diff. Geom., 15 (1980), 443-466.

• 2. "Riemannian Manifolds with Large Invariants", J. Diff. Geom., 15 (1980), 467-491.

• 3. "Some Isoperimetric Inequalities and Eigenvalue Estimates", Ann. Scient. Ec. Norm. Sup., 4e serie, t. 13 (1980), 419-435.

• 4. "A `Maximal Torus' Type Theorem for Complete Riemannian Manifolds", Pacific J. of Math., Vol. 97, no. 1 (1981), 65-73.

• 5. "The First Eigenvalue of the Laplacian for Plane Domains", Proc. Amer. Math. Soc., Vol. 81, no. 2 (1981), 304-305.

• 6. (with Alan Weinstein) "Closed Curves on Convex Hypersurfaces and Periods of Nonlinear Oscillations", Inventiones Mathematicae, Vol. 64, Fase. 2 (1981), 199-202.

• 7. "An Eigenvalue Pinching Theorem", Inventiones Mathematicae, Vol. 68 (1982), 253-256.

• 8. "Poincare's Problem and the Length of the Shortest Closed Geodesic on a Convex Hypersurface", J. Diff. Geom., 17 (1982), 595-634.

• 9. "On the Volume of Metric Balls", Proc. Amer. Math. Soc., Vol. 88, no. 4 (1983), 660-664.

• 10. "A Sharp Four Dimensional Isoperimetric Inequality", Comment. Math. Helvetici, 59 (1984), 187-192.

• 11. "Curvature Free Volume Estimates", Inventiones Mathematicae 76 (1984), 515-521.

• 12. "Area and the Length of the Shortest Closed Geodesic", J. Diff. Geom. 27 (1988), pp. 1-21.

• 13. (with V. Schroeder) "The Fundamental Group of Compact Manifolds Without Conjugate Points", Comm. Math. Helv. 61 (1986), 161-175.

• 14. (with A. Derdzinski) "A Lower Bound for 1 on Manifolds with Boundary", Comm. Math. Helv. 62 (1987), 106-121.

• 15. "Lower Bounds on the Energy of Maps", Duke J. Math., vol. 55, No. 4 (1987), pp. 901-908.

• 16. (with Hermann Karcher) "Volume of Small Balls on Open Manifolds: Lower Bounds and Examples", Trans. Amer. Math. Soc., vol. 309, no. 2 (1988), pp. 753-762.

• 17. "An Isoembolic Pinching Theorem" Invent. Math., 92 (1988), pp. 385-387.

• 18. "Rigidity and the Distance Between Boundary Points" J. Diff. Geom. 33(1991),445-464.

• 19. "Rigidity for Surfaces of Non-Positive Curvature" Comm. Math. Helv. 65 (1990), pp. 150-169.

• 20. (with Albert Fathi), "An Inequality Between Engergy and Intersection" Bull. London Math. Soc. 22(1990), 489-494.DVI FILE

• 21. "Rigidity for Simply Connected Manifolds with No Conjugate Points which are Flat Outside a Compact Set" Proceedings of AMS. 111 no. 1 (1991), 297-298.

• 22. (with Albert Fathi and Jack Feldman), "The Marked Length- Spectrum of a Surface of Non-Positive Curvature" Topology 31, no. 4 (1992) 847-855.DVI FILE

• 23. (with Bruce Kleiner) "A Warped Product Splitting Theorem" Duke Journal 67, no. 3 (1992) 571-574.DVI FILE

• 24. "Volumes of Balls in Manifolds without Conjugate Points" International Journal of Mathematics 3, no. 4 (1992) 455- 467.DVI FILE

• 25. (with Bruce Kleiner) "Conjugacy and Rigidity for Manifolds with a Parallel Vector Field" J. Diff. Geom., 39 (1994), 659-680.DVI FILE

• 26. (with Bruce Kleiner) On Tori Without Conjugate Points Invent. Math., 120 (1995),pp. 241-257.DVI FILE

• 27 (with Patrick Eberlein and Bruce Kleiner) "Conjugacy and Rigidity for Nonpositively Curved Manifolds of Higher Rank" Topology, 35(1996), 273-286.

• 28. (with Bruce Kleiner) "A Rigidity Theorem for Simply connected Manifolds without Conjugate Points" Ergodic Theory and Dynamical Systems, 18 pt.4(1998), 807-812 .DVI FILE

• 29. (with Bruce Kleiner) "A Rigidity Theorem for Manifolds without Conjugate Points" Ergodic Theory and Dynamical Systems, 18 pt.4(1998), 813-829 .DVI FILE

• 30. (with Vladimir Sharafutdinov) "Spectral rigidity of a compact negatively curved manifold" Topology, 37(1998), 1265-1273.DVI FILE

• 31. (with N. Dairbekov and Vladimir Sharafutdinov) "Local boundary rigidity of a Riemannian manifold with curvature bounded from above" (Russian) Dokl. Akad. Nauk 365 (1999), no. 3, 309--310.

• 32. (with N. Dairbekov and Vladimir Sharafutdinov) "Local boundary rigidity of a compact Riemannian manifold with curvature bounded above" Trans. Amer. Math. Soc. 352 (2000), no. 9, 3937--3956.DVI FILE

• 33. (with Bruce Kleiner) "Spaces with nonpositive curvature and their ideal boundaries" Topology, 39 (2000), no. 3, 549--556.DVI FILE

• 34. (with Bruce Kleiner) "The geodesic flow of a nonpositively curved graph manifold" GAFA, 12 (2002) 479-545.DVI FILE

• 35. "The volume and lengths on a three sphere" Comm. Anal. Geom. 10 (2002), no. 3, 467--474.DVI FILE and PDF FILE

• 36. "Conjugacy rigidity for nonpositively curved graph manifolds" Ergodic Theory Dynam. Systems 24 (2004), no. 3, 723--733.DVI FILE

• 37. "Rigidity theorems in Riemannian geometry", Chapter in Goemetric Methods in Inverse Problems and PDE Control, C. Croke, I. Lasiecka, G. Uhlmann, and M. Vogelius eds., Springer 2004.PDF FILE

• 38. (with Nurlan Dairbekov) "Lengths and volumes in Riemannian manifolds" Duke Math. J. 125 (2004), no. 1, 1--14.DVI FILE and PDF FILE

• 39. (with Mikhail Katz) "Universal volume bounds in Riemannian manifolds" Chapter in Surveys in Differential Geometry S.-T. Yau ed., Intrernational Press 2003.DVI FILE and PDF FILE

• 40. (with Victor Bangert, Sergei Ivanov, and Mikhail Katz) "Filling area conjecture and real ovalless hyperelliptic surfaces", Geom. Funct. Anal. 15 (2005), no. 3, 577--597. .PS FILE

• 41. "Boundary and lens rigidity of finite quotients", Proc. Amer. Math. Soc. 133 (2005), no. 12, 3663--3668. .PDF FILE and DVI FILE

• 42. (with Victor Bangert, Sergei Ivanov, and Mikhail Katz) "Boundary case of equality in optimal Loewner-type inequalities" , Trans. Amer. Math. Soc. 359 (2007), no. 1, 1--17. .PS FILE and .PDF FILE

• 43. "A synthetic characterization of the hemisphere", Proc. Amer. Math. Soc. 136 (2008), no. 3, 1083--1086. .DVI FILE and .PDF FILE

• 44. "Small volume on big n-spheres", Proc. Amer. Math. Soc. 136 (2008),no. 2, 715--717. .DVI FILE and .PDF FILE

• 45. (with Florent Balacheff, and Mikhail Katz) "A Zoll counterexample to a geodesic length conjecture" , Geom. Funct. Anal. 19 (2009), no. 1, 1--10. .DVI FILE and .PDF FILE

• 46. "Area of small disks" Bull. of London Math. Soc. (2009) 41(4), 701--708. .DVI FILE and .PDF FILE

• 47. "Scattering rigidity with trapped geodesics", Ergodic Theory and Dynamical Systems, 34 (2014), pp 826-836. doi:10.1017/etds.2012.164. . .PDF FILE

• 48. (with Pilar Herreros) "Lens rigidity with trapped geodesics in two dimensions" Asian Journal of Mathematics, Vol. 20, No. 1 (2016), pp. 47-58.. .PDF FILE

• 49. (with Haomin Wen) "Scattering data versus lens data on surfaces" (preprint). .PDF FILE

• 50. (with Gregory Chambers, Yevgeny Liokumovich, and Haomin Wen) "Area of convex disks" (Preprint). .PDF FILE

The above papers were partially supported by various NSF grants, the latest by Grant DMS 10-03679. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.