Comparison theorem for kähler manifolds and positivity of spectrum

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An integral formula for the volume entropy with applications to rigidity (2009, background): “ any s > 0 the function us (x) = ξ(p)+d(x,γ (s))−s is a support function for ξ from above at p. Moreover us is clearly smooth at p. Therefore we have at p D2ξ ≤ D2u s. By the comparison theorem in [LW2] we have ξ 11 ≤ (us) 11 ≤ cosh2s sinh2s , ξii ≤ (us)ii ≤ coshs sinhs for i ≥ 2. Taking limit as”
Weighted Poincaré inequality and rigidity of complete manifolds (2006, background): “We would also like to point out that a similar theory was proposed in [13] for Kähler manifolds where the assumption is on the holomorphic bisectional curvature instead.”