Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary

X iv :0 90 5. 13 66 v2 [ m at h. SP ] 2 1 M ay 2 00 9 Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary Yiqian Shi ∗† and Bin Xu∗⋆ Abstract. Let eλ(x) be an eigenfun tion with respe t to the Lapla e-Beltrami operator ∆M on a ompa t Riemannian manifoldM without boundary: ∆Meλ = λ2eλ. We show the following gradient estimate of eλ: for every λ≥ 1, there holds λ‖eλ‖∞/C≤ ‖∇eλ‖∞ ≤ Cλ‖eλ‖∞ , where C is a positive onstant depending only on M. Mathematics Subject Classification (2000): Primary 35P20; Se ondary 35J05