In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work in [14] classify complete gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic functions on gradient shrinking Kähler and gradient steady Ricci solitons. Consequences to the structure of shrinking and steady solitons at infinity are also discussed.for some constant C 0 > 0, independent of r. Let us denote u (t) := J ′ J (r) .
We study both function theoretic and spectral properties of the weighted Laplacian Δ f on complete smooth metric measure space (M, g, e −f dv) with its Bakry-Émery curvature Ric f bounded from below by a constant. In particular, we establish a gradient estimate for positive f -harmonic functions and a sharp upper bound of the bottom spectrum of Δ f in terms of the lower bound of Ric f and the linear growth rate of f. We also address the rigidity issue when the bottom spectrum achieves its optimal upper bound under a slightly stronger assumption that the gradient of f is bounded.Applications to the study of the geometry and topology of gradient Ricci solitons are also considered. Among other things, it is shown that the volume of a noncompact shrinking Ricci soliton must be of at least linear growth. It is also shown that a nontrivial expanding Ricci soliton must be connected at infinity provided its scalar curvature satisfies a suitable lower bound.
In this paper we study both function theoretic and spectral properties on complete noncompact smooth metric measure space (M, g, e −f dv) with nonnegative Bakry-Émery Ricci curvature. Among other things, we derive a gradient estimate for positive f -harmonic functions and obtain as a consequence the strong Liouville property under the optimal sublinear growth assumption on f. We also establish a sharp upper bound of the bottom spectrum of the f -Laplacian in terms of the linear growth rate of f. Moreover, we show that if equality holds and M is not connected at infinity, then M must be a cylinder. As an application, we conclude steady Ricci solitons must be connected at infinity. ∆ f := ∆ − ∇f · ∇ is more naturally associated with such a smooth metric measure space than the classical Laplacian as it is symmetric with respect to the measure e −f dv. That is,for any ϕ, ψ ∈ C ∞ 0 (M ) . Again, we point out that the operator f -Laplacian is very much related to the Laplacian of a suitable conformal change of the background Riemannian metric. It also appears as the generator of a class of stochastic diffusion processes, the Brownian motion with drifts.The Bakry-Émery Ricci tensor [1] of the metric measure space (M, g, e −f dv) is defined byRic f := Ric + Hess(f ),
Abstract. The main purpose of this paper is to investigate the curvature behavior of four dimensional shrinking gradient Ricci solitons. For such a soliton M with bounded scalar curvature S, it is shown that the curvature operator Rm of M satisfies the estimate |Rm| ≤ c S for some constant c. Moreover, the curvature operator Rm is asymptotically nonnegative at infinity and admits a lower bound Rm ≥ −c (ln r) −1/4 , where r is the distance function to a fixed point in M. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.
Abstract. We show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.
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