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In this paper we establish the existence of extremals for the Log Sobolevfunctional on complete non-compact manifolds with Ricci curvature bounded frombelow and strictly positive injectivity radius, under a condition nearinfinity. When Ricci curvature is also bounded from above we get exponentialdecay at infinity of the extremals. As a consequence of these analyticalresults we establish, under the same assumptions, that non-trivial shrinkingRicci solitons support a gradient Ricci soliton structure. On the way, we provetwo results of independent interest: the existence of a distance-like functionwith uniformly controlled gradient and Hessian on complete non-compactmanifolds with bounded Ricci curvature and strictly positive injectivity radiusand a general growth estimate for the norm of the soliton vector field onmanifolds with bounded Ricci curvature.

Extremals of Log Sobolev inequality on non-compact manifolds and Ricci soliton structures