In this paper we establish the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity. This extends a previous result by Q. Zhang where a C 1 bound on the whole Riemann tensor was assumed. When Ricci curvature is also bounded from above we get exponential decay at infinity of the extremals. As a consequence of these analytical results we establish, under the same assumptions, that non-trivial shrinking Ricci solitons support a gradient Ricci soliton structure. On the way, we prove two results of independent interest: the existence of a distance-like function with uniformly controlled gradient and Hessian on complete non-compact manifolds with bounded Ricci curvature and strictly positive injectivity radius and a general growth estimate for the norm of the soliton vector field. This latter is based on a new Toponogov type lemma for manifolds with bounded Ricci curvature, and represents the first known growth estimate for the whole norm of the soliton field in the non-gradient case.

Rimoldi, M., Veronelli, G. (2019). Extremals of Log Sobolev inequality on non-compact manifolds and Ricci soliton structures. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 58(2) [10.1007/s00526-019-1518-z].

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

Veronelli, G
2019

Abstract

In this paper we establish the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity. This extends a previous result by Q. Zhang where a C 1 bound on the whole Riemann tensor was assumed. When Ricci curvature is also bounded from above we get exponential decay at infinity of the extremals. As a consequence of these analytical results we establish, under the same assumptions, that non-trivial shrinking Ricci solitons support a gradient Ricci soliton structure. On the way, we prove two results of independent interest: the existence of a distance-like function with uniformly controlled gradient and Hessian on complete non-compact manifolds with bounded Ricci curvature and strictly positive injectivity radius and a general growth estimate for the norm of the soliton vector field. This latter is based on a new Toponogov type lemma for manifolds with bounded Ricci curvature, and represents the first known growth estimate for the whole norm of the soliton field in the non-gradient case.
Articolo in rivista - Articolo scientifico
Log-Sobolev inequalities; Ricci solitons; distance-like functions
English
2019
58
2
66
reserved
Rimoldi, M., Veronelli, G. (2019). Extremals of Log Sobolev inequality on non-compact manifolds and Ricci soliton structures. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 58(2) [10.1007/s00526-019-1518-z].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/230172
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