Analysis of weighted Laplacian and applications to Ricci solitons
A gradient Ricci soliton \((M,g,f)\) is a Riemannian manifold \((M,g)\) with a function \(f\) on \(M\) such that \(\text{Ric}_g +\text{Hess}(f)=\lambda g\) for some scalar \(\lambda\): this class of manifolds, which extends Einstein manifolds, appears in Ricci flow analysis both as a self-similar solution and as a singularity model. By rescaling the metric, the scalar \(\lambda\) can be taken equal to \(-1/2,0\) or \( 1/2\), the soliton being called expanding, steady or shrinking resp.NEWLINENEWLINENEWLINEFor an expanding soliton with lower bound \(S_g\geq(1-n)/2\) of the scalar curvature \(S_g\), the authors prove that \(M\), if with more than one end, is isometric to the product \(G\times N\) where \(N\) is a compact Einstein manifold and \(G\) is the Gaussian expanding Ricci soliton \((\mathbb R, dx^2,x^2/4)\). This result can be compared to \textit{J. Cheeger} and \textit{D. Gromoll}'s split theorem for non-negative Ricci curvature [J. Differ. Geom. 6, 119--128 (1971; Zbl 0223.53033)], Li-Wang's wrapped product theorem for Riemannian manifolds with Ricci curvature bounded below and the authors' result for steady Ricci solitons [Commun. Anal. Geom. 19, No. 3, 451--486 (2011; Zbl 1246.53056)].NEWLINENEWLINENEWLINEFurthermore, the authors prove a linear growth rate for the geodesic ball volume on any non-compact shrinking gradient Ricci soliton: \(\text{Vol}\, B(m,r)\geq c r\); the upper bound \(\text{Vol}\,B(m,r)\leq C r^n\), with \(n=\dim M\), has been proved by \textit{H. D. Cao} and \textit{D. Zhou} [J. Differ. Geom. 85, No. 2, 175--186 (2010; Zbl 1246.53051)]. These two bounds correspond to the bounds (upper by Bishop, lower by Yau) for Riemannian manifolds with \(\text{Ric}_g \geq 0\).NEWLINENEWLINENEWLINEThese geometrical results are consequences of analytical results on smooth metric spaces \((M,g,\text{e}^{-f}d\text{vol}_g)\) with deformed Laplace-Beltrami operator \(\Delta_{g,f}=\Delta_g-\langle\nabla_gf,\nabla_g\rangle\) and Bakry-Émery curvature \(\text{Ric}_{g,f}=\text{Ric}_g+\text{Hess}(f)\). By combining pseudo-Poincaré inequalities, a Bochner identity and DeGiorgi-Nash-Moser theory, the authors prove a general gradient estimate for positive \(\Delta_{g,f}\)-harmonic functions on a manifold with Bakry-Émery curvature lower bound. With the same curvature lower bound assumption, they prove also for the infimum \(\lambda_{g,f}\) of the spectrum of the selfadjoint Laplacian \(\Delta_{g,f}\) on \(L^2(M,\text{e}^{-f}d\text{vol}_g)\) an upper bound \(\lambda_{g,f}\leq(n-1+a)^2/4\) where \(a\) is an upper bound for the linear growth of \(f\) (e.g. given by an asymptotic upper bound for the gradient \(\nabla f\)): Equality of this infimum with its upper bound imposes for the manifold either one end or a splitting as warped product \((M,g)=(\mathbb R\times N,dt^2+h(t)^2dg_N)\) with \(N\) compact and \(h(t)=\text{e}^t\) or \(\cosh(t)\).
scientific article; zbMATH DE number 6052980