By Berestovskii V. N.
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Extra resources for A. D. Alexandrovs length manifolds with one-sided bounded curvature
We next consider a situation in which the Ricci curvature satisﬁes some Lp -integrability conditions, and describe upper bounds for the volume growth of balls obtained by P. Petersen and G. Wei in , who consider the slightly less general case where the function G below is a non-negative constant and M is compact. Previous related results have been obtained by S. Gallot, , Li and Yau,  and D. Yang, . As above, we assume that G is non-negative and continuous on [0, +∞) and that h (t) ∈ C 2 ([0, +∞)) is the non-negative solution of the problem h (t) − G (t) h (t) = 0, h (0) = 0 h (0) = 1.
Next, we compute the Laplacian of u. We observe that du = uk ϕk + uk ϕk with, according to the previous formulas, α α B i Bik . 5. Weitzenb¨ ock-type formulas 25 Hence, with the aid of the above calculations, ukt ϕt + ukt ϕt = duk − ut ϕtk with α α α α ukt = Bik B it + B i Bikt . By the deﬁnition of the Laplacian on the K¨ ahler manifold M , we have ukk ∆u = 4 k α α α α Bik B ik + B i Bikk =4 α,i,k α α α,i,k α j B i Bjα Hikk −4 α Bik B ik + 4 =4 α,i,k δ α B i Biβ Btγ B t Kβγδ . 27. Let (M, , , JM ), (N, (, ) , JN ) be K¨ ahler manifolds and let f : M → N be a holomorphic map.
61). Indeed, observe that BR (x) ⊂ BR+d (y) \ Bd−R (y) .
A. D. Alexandrovs length manifolds with one-sided bounded curvature by Berestovskii V. N.