Under a necessary topological assumption, two global results are established for complete three dimensional manifolds. The first one provides a sharp upper bound for the bottom spectrum in terms of the scalar curvature lower bound. The second one shows that such manifolds do not admit any positive Green's function if the scalar curvature is bounded from below by a positive constant.

The purpose of this paper is to derive volume and other geometric information for three-dimensional complete manifolds with positive scalar curvature. In the case that the Ricci curvature is nonnegative, it is shown that the volume of the manifold must be of linear growth when the scalar curvature is bounded from below by a positive constant. This answers a question of Gromov in the affirmative for dimension three. Volume growth estimates are also obtained for the case when scalar curvature decays to zero. In fact, results of similar nature are established for the more general case that the Ricci curvature is asymptotically nonnegative.

It is shown that the integral of the scalar curvature on a geodesic ball of radius R in a three-dimensional complete manifold with nonnegative Ricci curvature is bounded above by 8π R asymptotically for large R provided that the scalar curvature is bounded between two positive constants.

The classical Minkowski inequality implies that the volume of a bounded convex domain in the Euclidean space is controlled from above by the integral of the mean curvature of its boundary. In this note, an analogous inequality is established without assuming convexity, valid for all bounded smooth domains in a complete manifold whose bottom spectrum is suitably large relative to its Ricci curvature lower bound. An immediate consequence is the nonexistence of embedded closed minimal hypersurfaces in such manifolds. The same nonexistence issue is also addressed for steady and expanding Ricci solitons. The proofs are very much inspired by a sharp monotonicity formula, derived for positive harmonic functions on manifolds with positive spectrum.

A classical result of Cheng states that the bottom spectrum of complete manifolds of fixed dimension and Ricci curvature lower bound achieves its maximal value on the corresponding hyperbolic space. The paper establishes an analogous result for three-dimensional complete manifolds with scalar curvature lower bound subject to some necessary topological assumptions. The rigidity issue is also addressed and a splitting theorem is obtained for such manifolds with the maximal bottom spectrum.

Two sharp comparison results are derived for three-dimensional complete noncompact manifolds with scalar curvature bounded from below. The first one concerns the Green's function. When the scalar curvature is nonnegative, it states that the rate of decay of an energy quantity over the level set is strictly less than that of the Euclidean space, unless the manifold itself is isometric to the Euclidean space. The result is in turn converted into a sharp area comparison for the level set of the Green's function when in addition the Ricci curvature of the manifold is assumed to be asymptotically nonnegative at infinity. The second result provides a sharp upper bound of the bottom spectrum in terms of the scalar curvature lower bound, in contrast to the classical result of Cheng which involves a Ricci curvature lower bound.

The classical Minkowski inequality implies that the volume of a bounded convex domain is controlled from above by the integral of the mean curvature of its boundary. In this note, we establish an analogous inequality without the convexity assumption for all bounded smooth domains in a complete manifold with its bottom spectrum being suitably large relative to its Ricci curvature lower bound. An immediate implication is the nonexistence of embedded compact minimal hypersurfaces in such manifolds. This nonexistence issue is also considered for steady and expanding Ricci solitons.

The paper concerns three-dimensional complete manifolds with scalar curvature bounded from below. One of the purposes is to establish a sharp comparison theorem for the bottom spectrum in the spirit of the classical result of Cheng. Another purpose is to derive volume and other geometric information in terms of the scalar curvature when the Ricci curvature is asymptotically nonnegative and the scalar curvature is positive. If the scalar curvature decays no faster than linearly, then the manifold does not admit any positive Green's function. When the scalar curvature is bounded from below by a positive constant, it is shown that the volume of unit balls must be bounded from above by the lower bound of the scalar curvature at infinity. In particular, in the case that the Ricci curvature is nonnegative, the volume of the manifold must be of linear growth. This answers a question of Gromov in the affirmative for dimension three. Volume estimates are also established for the case when scalar curvature decays polynomially.

This paper mainly concerns the area growth and bottom spectrum of complete stable minimal surfaces in a three-dimensional manifold with scalar curvature bounded from below. When the ambient manifold is the Euclidean space, by an elementary argument, it is shown directly from the stability inequality that the area of such minimal surfaces grows exactly as the Euclidean plane. Consequently, such minimal surfaces must be flat, a well-known result due to Fisher-Colbrie and Schoen as well as do Carmo and Peng. In the case of general ambient manifold, explicit area growth estimate is also derived. For the bottom spectrum, a self-contained argument involving positive Green's function is provided for its upper bound estimates. The argument extends to stable minimal hypersurfaces in a complete manifold of dimension up to six with sectional curvature bounded from below.

Self-similar solutions to Ricci flows, called Ricci solitons, are important geometric objects. To address the question whether new solitons can be constructed from existing ones through connected sums, we are led to investigate the issue of connectedness at infinity for solitons. The paper provides a brief account of our work along this line as well as a new result. The new result says that an n-dimensional gradient shrinking Ricci soliton is necessarily connected at infinity if its scalar curvature is bounded above by n/3.

Abstract. This paper is a continuation of our previous work concerning three-dimensional complete manifolds with scalar curvature bounded from below. One of the purposes is to improve a sharp comparison theorem for the bottom spectrum by removing a volume assumption on unit balls. Another purpose is to derive geometric information when the scalar curvature is assumed to be bounded from below by a positive constant. When the Ricci curvature is asymptotically nonnegative, it is shown that such manifolds must be parabolic and that the lower bound of the scalar curvature is explicitly bounded by the volume of unit balls. In particular, in the case that the Ricci curvature is nonnegative, this implies that the volume of the manifold must grow linearly, which answers a question of Gromov for dimension three.

Abstract A variant of Li–Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds of Euclidean space is obtained.

We develop Green's function estimate for manifolds satisfying a weighted Poincare inequality together with a compatible lower bound on the Ricci curvature. The estimate is then applied to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete K\"ahler manifolds. Consequently, such K\"ahler manifolds must be connected at infinity.

This paper 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 R x S-n(-1)/Gamma occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptotic at infinity to the same round cylinder. This result is applied to obtain structural results at infinity for four dimensional gradient shrinking Ricci solitons. It was 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 R x S-3 or converges to a quotient of R-2 x E (2) along each integral curve of the gradient vector field of the potential function.

We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady gradient Ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above.

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

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.

We show that the norm of the Riemann curvature tensor of any smooth solution to the Ricci flow can be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensor, and the elapsed time. This provides a new, direct proof of a result of Šesǔm, which asserts that the curvature of a solution on a compact manifold cannot blow up while the Ricci curvature remains bounded, and extends its conclusions to the noncompact setting. We also prove that the Ricci curvature must blow up at least linearly along a subsequence at a finite time singularity.

We prove that any shrinking Kahler Ricci soliton has only one end, and that any expanding Kahler Ricci soliton with proper potential has only one end.

For a Kähler manifold endowed with a weighted measure $e^{-f}\,dv,$ the associated weighted Hodge Laplacian $\Delta _{f}$ maps the space of $(p,q)$-forms to itself if and only if the $(1,0)$-part of the gradient vector field $\nabla f$ is holomorphic. We use this fact to prove that for such $f$, a finite energy $f$ harmonic function must be pluriharmonic. Motivated by this result, we verify that the same also holds true for $f$-harmonic maps into a strongly negatively curved manifold. Furthermore, we demonstrate that such $f$-harmonic maps must be constant if $f$ has an isolated minimum point. In particular, this implies that for a compact Kähler manifold admitting such a function, there is no nontrivial homomorphism from its first fundamental group into that of a strongly negatively curved manifold.