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" (to appear in Proc. AMS). .DVI FILE and .PDF FILE
44. "Small volume on big n-spheres" (to appear in Proc. AMS). .DVI FILE and .PDF FILE
45. (with Florent Balacheff, and Mikhail Katz) "A Zoll counterexample to a geodesic length conjecture" (to appear in GAFA). .DVI FILE and .PDF FILE

The above papers were partially supported by various NSF grants, the latest by Grant#0202536. 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.