[C] Calabi, E. An extension of E. Hopf 's maximum principle with an application to Riemannian geometry. Duke Math. J. 25 (1957) 45-56.

[H] Hamilton, Richard S. A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1 (1993), no.1, 113-126.

[KL] Leon Karp and Peter Li. The heat equation on complete riemannian manifolds. Unpublished notes, 1982.

[LY] Li, Peter and Yau, S.-T. On the parabolic kernel of the Schr¨odinger operator. Acta Math. 156 (1986), no. 3-4, 153-201.

[N] Ni, Lei. A note on Perelman's LYH inequality. arXiv:math.DG/0602337 [N2] Ni, Lei. The entropy formula for linear heat equation. J. Geom Anal. 14 (2004), no. 1, 87-100.

[NT] Ni, Lei and Tam, L. F. K¨ahler-Ricci flow and the Poincar`e-Lelong equation. Comm. Anal. Geom 12 (2004), no. 1-2, 111-141.

[P] Perelman, Grisha. The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159

[S] Shi, Wan-Xiong Deforming the metric on complete Riemannian manifolds. J. Differential Geom. 30 (1989), no. 1, 223-301.

[SZ] Souplet, P. and Zhang, Q.S. Sharp gradient estimate and Yau's Liouville theorem for the heat equation on non-compact manifolds, arXiv:math.DG/0502079