Abstract
Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote \({\Delta_g=-{\rm div}_g\nabla}\) the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation
on (M, g). Here, p, q ≥ 0, A(x), B(x) and h(x) are smooth functions on (M, g). We also derive the Harnack differential inequality for the positive solutions of
on (M, g) with initial data u(x, 0) > 0.
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Song, X., Zhao, L. Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds. Z. Angew. Math. Phys. 61, 655–662 (2010). https://doi.org/10.1007/s00033-009-0047-6
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DOI: https://doi.org/10.1007/s00033-009-0047-6