References
Aronson, D. G., Uniqueness of positive weak solutions of second order parabolic equations.Ann. Polon. Math., 16 (1965), 285–303.
Azencott, R., Behavior of diffusion semi-groups at infinity.Bull. Soc. Math. France, 102 (1974), 193–240.
Cheeger, J. & Ebin, D.,Comparison theorems in Riemannian geometry. North-Holland Math. Library (1975).
Cheeger, J. &Gromoll, D., The splitting theorem for manifolds of nonnegative Ricci curvature.J. Differential Geom., 6 (1971), 119–128.
Cheeger, J., Gromov, M. &Taylor, M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds.J. Differential Geom., 17 (1983), 15–33.
Cheeger, J. &Yau, S. T., A lower bound for the heat kernel.Comm. Pure Appl. Math., 34 (1981), 465–480.
Cheng, S. Y., Li, P. &Yau, S. T., On the upper estimate of the heat kernel of a complete Riemannian manifold.Amer. J. Math., 103 (1981), 1021–1063.
Cheng, S. Y. &Yau, S. T., Differential equations on Riemannian manifolds and their geometric applications.Comm. Pure Appl. Math., 28 (1975), 333–354.
Donnelly, H. &Li, P., Lower bounds for the eigenvalues of Riemannian manifolds.Michigan Math. J., 29 (1982), 149–161.
Fischer-Colbrie, D. &Schoen, R., The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature.Comm. Pure Appl. Math., 33 (1980), 199–211.
Friedman, A., On the uniqueness of the Cauchy problem for parabolic equations.Amer. J. Math., 81 (1959), 503–511.
Gallot, S. &Meyer, D., Opérateur de courbure et Laplacien des formes différentielles d'une variété Riemannienne.J. Math. Pures Appl., 54 (1975), 259–284.
Gromov, M., Paul Levy's isoperimetric inequality. IHES preprint.
—, Curvature, diameter, and Betti numbers.Comment Math. Helv., 56 (1981), 179–197.
Karp, L., & Li, P., The heat equation on complete Riemannian manifolds. Preprint.
Li, P. &Yau, S. T., Estimates of eigenvalues of a compact Riemannian manifold.Proc. Sympos. Pure Math., 36 (1980), 205–239.
Maurey, B. & Meyer, D., Un lemma de géométrie Hilbertienne et des applications à la géométrie Riemannienne. Preprint.
Meyer, D., Un lemme de géométrie Hilbertienne et des applications à la géométrie Riemannienne.C. R. Acad. Sci. Paris, 295 (1982), 467–469.
—, Sur les hypersurfaces minimales des variétés Riemanniennes a courbure de Ricci positive ou nulle.Bull. Soc. Math. France, 111 (1983), 359–366.
Moser, J., A Harnack inequality for parabolic equations.Comm. Pure Appl. Math., 17 (1964), 101–134.
Serrin, J. B., A uniqueness theorem for the parabolic equationu 1 =a(x)u xx +b(x)u x +c(x)u.Bull. Amer. Math. Soc., 60 (1954), 344.
Simon, B., Instantons, double wells and large deviations.Bull. Amer. Math. Soc., 8 (1983), 323–326.
Varopoulos, N., The Poisson kernel on positively curved manifolds.J. Funct. Anal., 44 (1981), 359–380.
— Green's functions on positively curved manifolds.J. Funct. Anal., 45 (1982), 109–118.
Widder, D. V., Positive temperature on the infinite rod.Trans. Amer. Math. Soc., 55 (1944), 85–95.
Yau, S. T., Harmonic functions on complete Riemannian manifolds.Comm. Pure Appl. Math., 28 (1975), 201–228.
—, Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry.Indiana Univ. Math. J., 25 (1976), 659–670.
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Research partially supported by a Sloan fellowship and an NSF grant.
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Li, P., Yau, S.T. On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986). https://doi.org/10.1007/BF02399203
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DOI: https://doi.org/10.1007/BF02399203