A sharp estimate for the bottom of the spectrum of the Laplacian on K\"{a}hler manifolds

On a complete noncompact K\"{a}hler manifold we prove that the bottom of the spectrum for the Laplacian is bounded from above by $m^2$ if the Ricci curvature is bounded from below by $-2(m+1)$. Then we show that if this upper bound is achieved then the manifold has at most two ends. These results improve previous results on this subject proved by P. Li and J. Wang in \cite {L-W3} and \cite{L-W} under assumptions on the bisectional curvature.Comment: 26 page

The volume growth of complete gradient shrinking Ricci solitons

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature

On a characterization of the complex hyperbolic space

Consider a compact K\"{a}hler manifold $M^m$ with Ricci curvature lower bound $Ric_M\geq -2(m+1) .$ Assume that its universal cover $$ has maximal bottom of spectrum $\lambda_1(\widetilde{M}$ Then we prove that $\widetilde{M}$ is isometric to the complex hyperbolic space $\Bbb{CH}^m.

Improved Beckner-Sobolev inequalities on K\"ahler manifolds

We prove new Beckner-Sobolev type inequalities on compact K\"{a}hler manifolds with positive Ricci curvature. As an application, we obtain a diameter upper bound that improves the Bonnet-Myers bound

Structure at infinity for shrinking Ricci solitons

The paper mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. It is shown that for such a soliton with bounded curvature, if the round cylinder $\mathbb{R}\times \mathbb{S}^{n-1}/\Gamma$ occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic to the same round cylinder at infinity. The result is then applied to obtain structural results at infinity for four dimensional gradient shrinking Ricci solitons. It is previously known that such solitons with scalar curvature approaching zero at infinity must be smoothly asymptotic to a cone. For the case that the scalar curvature is bounded from below by a positive constant, we conclude that along each end the soliton is asymptotic to a quotient of $\mathbb{R}\times \mathbb{S}^{3}$ or converges to a quotient of $\mathbb{R}^{2}\times \mathbb{S}^{2}$ along each integral curve of the gradient vector field of the potential function. For four dimensional K\"{a}hler Ricci solitons, stronger conclusion holds. Namely, they either are smoothly asymptotic to a cone or converge to a quotient of $\mathbb{R}^{2}\times \mathbb{S}^{2}$ at infinity

Positively curved shrinking Ricci solitons are compact

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.Comment: submitte

Gradient estimate for harmonic functions on K\"ahler manifolds

We prove a sharp integral gradient estimate for harmonic functions on noncompact K\"ahler manifolds. As application, we obtain a sharp estimate for the bottom of spectrum of the p-Laplacian and prove a splitting theorem for manifolds achieving this estimate.Comment: 34pages, accepte

Conical structure for shrinking Ricci solitons

For a shrinking Ricci soliton with Ricci curvature convergent to zero at infinity, it is proved that it must be asymptotically conical.Comment: submitte

Analysis of weighted Laplacian and applications to Ricci solitons

We study both function theoretic and spectral properties of the weighted Laplacian $\Delta_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$ with its Bakry-\'{E}mery 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 $\Delta_f$ 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.Comment: Will appear in Comm. Anal. Geo

Geometry of shrinking Ricci solitons

The main purpose of this paper is to investigate the curvature behavior of four dimensional shrinking gradient Ricci solitons. For such soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\mathrm{Rm}$ of $M$ satisfies the estimate $|\mathrm{Rm}|\le c\,S$ for some constant $c$. Moreover, the curvature operator $\mathrm{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\mathrm{Rm}\geq -c\,\left(\ln r\right)^{-1/4},$ where $r$ is the distance function to a fixed point in $M$. As application, we prove that if the scalar curvature converges to zero at infinity, then the manifold must be asymptotically conical. 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.Comment: 28 pages, submitted, v2 has a new section about the conical structure of soliton