Explicit constants for Riemannian inequalities

We prove versions of various standard inequalities in which the dependence of the constant on the metric is explicit. 1 Technical results Definition. Let (M, g) be a smooth Riemannian n-manifold and let x ∈ M . Given Q > 1, k ∈ N, and p > n, the (Q, k, p)-harmonic radius at x, rH(Q, k, p)(x), is the supremum of reals r such that, on the geodesic ball Bx(r) of center x and radius r, there is a harmonic co-ordinate chart such that if gij are the components of g in these co-ordinates, then 1. Qδij ≤ gij ≤ Qδij as bilinear forms; 2. ∑ 1≤|β|≤k r ||∂βgij ||Lp ≤ Q− 1. The (Q, k, p)-harmonic radius of M is rH(Q, k, p)(M) := inf x∈M rH(Q, k, p)(x). Theorem 1.1 ([HH97], Theorem 11). Let n ∈ N, Q > 1, p > n, i > 0. Suppose (M, g) is a Riemannian n-manifold with injrad(M, g) ≥ i. 1. Let λ ∈ R. There exists C = C(n,Q, p, i, λ), such that if Ric ≥ λg, then the harmonic radius rH(Q, 1, p)(M) is ≥ C. 2. Let k ≥ 2, and let (C(j))0≤j≤k−2 be positive constants. There exists C = C(n,Q, p, i, (C(j))0≤j≤k−2), such that if for each 0 ≤ j ≤ k − 2 we have |∇Ric| ≤ C(j), then the harmonic radius rH(Q, k, p)(M) is ≥ C.