Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds

[1] W. Ambrose, A theorem of Myers, Duke Math. J. 24 (1957) 345.

[2] M. Berger, Blaschke's conjecture for spheres, Appendix D in A. L. Besse, Manifolds all of who geodesics are closed, Ergebnisse Math, und ihrer Grenzgebiete, Vol. 93, Springer, Berlin, 1978, 236-242.

[3] R. Bishop and B. Crittenden, Geometry of manifolds, Academic Press, New York, 1964.

[4] E. Bombieri, Theory of minimal surfaces, and a counter-example to the Bernstein conjecture in high dimension, Lecture Notes, Courant Inst., New York, 1970.

[5] J. Cheeger, The relation between the Laplacian and the diameter for manifolds of non-negative curvature, Arch. Math. 19 (1968) 558-560.

[6] J. Cheeger, Finiteness theorems for riemannian manifolds, Amer. J. Math. 92 (1970), 61-64.

[7] J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, North Holland, Amsterdam, 1975.

[8] J. Cheeger and M. Taylor, On the diffraction of waves by conical singularities, Comm. Pure Appl. Math., to appear.

[9] J. Cheeger and S. T. Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34 (1981) 465-480.

[10] S. Y. Cheng, P. Li and S. T. Yau, On the upper estimate of the heat kernel of a complete riemannian manifold, Amer. J. Math. 103 (1981) 1021-1063.

[11] P. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis 12 (1973) 401-414.

[12] J. Clerc and E. Stein, Lp multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974) 3911-3912.

[13] C. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Ecole Norm. Sup. 13 (1980) 419-435.

[14] B. V. Dekster and I. Kupka, symptotics of curvature in a space of positive curvature, J. Differential Geometry 15 (1980) 553-568.

[15] G. de Rham, Varieties differentiables, Publ. Inst. Math. Univ. Nancago III, Hermann, Paris, 1960.

[16] H. Donnelly, Spectral geometry for certain noncompact riemannian manifolds, Math. Z. 169 (1979) 63-76.

[17] H. Federer and W. Flemming, Normal integral currents Ann. of Math. 72 (1960) 458-520.

[18] G. Galloway, Compactness criteria for Riemannian manifolds, Proc. Amer. Math. Soc, 84 (1982) 106-110.

[19] L. Garding, Vecteurs analytiques dans les representations des groupes de Lie, Bull. Soc. Math. France 188 (1960)73-93.

[20] D. Gilbarg and M. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, 1977.

[21] M. Gromov, Almost flat manifolds, J. Differential Geometry 13 (1978) 231-241.

[22] M. Gromov, Almost flat manifolds, Curvature, diameter, and Betti numbers, Comment. Math. Helv. 56 (1981) 179-197.

[23] M. Gromov, Almost flat manifolds, Paul Levy's isoperimetric inequality, Inst. Hautes Etudes Sci. l Publ. Math., to appear.

[24] E. Heintze and H. Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. Ecole Norm. Sup. 11 (1978) 451-470.

[25] J. Kazdan, An inequality arising in geometry, Appendix E, in A. L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse Math, und ihrer Grenzgebiete, Vol. 93, Springer, Berlin, 1978, 243-246.

[26] N. N. Lebedev, Special functions and their applications, Dover, New York, 1972.

[27] J. Lions and E. Magenes, Non-homogeneous boundary value problems and appliations, Vol. 3, Springer, New York, 1972.

[28] V. G. Maz'ya, Classes of domains and imbedding theorems for function spaces, Dokl. Akad. Nauk SSSR 133 (1960) 527-530

English Translation, Soviet Math. Dokl. 1 (1960) 882-885.

[29] J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968) 1-7.

[30] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964) 101-134.

[31] R. Stanton and P. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta. Math. 140 (1978) 251-276.

[32] M. Taylor, Pseudodifferential operators, Princeton Univ. Press, Princeton, 1981.

[33] M. Taylor, Pseudodifferential operators, Fourier integral operators and harmonic analysis on compact manifolds, Proc. Sympos. Pure Math. Vol. 35, Part 2, 1979, 115-136.

[34] M. Taylor, Pseudodifferential operators, Analytic properties of elliptic and conditionally elliptic operators, Proc. Amer. Math. Soc. 28 (1971), 317-318.

[35] S. T. Yau, Some function theoretic properties of complete Riemannian manifolds, and their applications to geometry, Indiana Univ. Math. J. 25 (1976) 659-670.

[36] S. T. Yau, Survey on partial differential equations in differential geometry, preprint.

[37] Y. Kannai, Off diagonal short time asymptotics for fundamental solutions of diffusion equations, Comm. Partial Differential Equations 2 (1977) 781-830.