Topology of Kahler Ricci solitons
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.
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arXiv subjects
Ovidiu Munteanu
Publications and source records attributed to Ovidiu Munteanu.
Holomorphic functions on certain Kähler manifolds
We investigate Liouville theorems and dimension estimates for the space of exponentially growing holomorphic functions on complete Kähler manifolds. While our work is motivated by the study of gradient Ricci solitons in the theory of Ricci flow, the most general results we prove here do not require any knowledge of curvature.
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Geometry of manifolds with densities
We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtain some splitting type results by analyzing the Busemann functions. In particular, we show that a complete manifold with nonnegative Bakry-Emery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions. While some of our results are even new for the gradient Ricci solitons, the novelty here is that only a lower bound of the Bakry-Emery curvature is involved in our analysis.
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Analysis of weighted Laplacian and applications to Ricci solitons
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.
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On gradient Ricci solitons
In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of 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 and show that if the total energy of a harmonic function on such a manifold is finite then the function is constant. Consequences to the structure of the manifold at infinity are also discussed.
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On the gradient estimate of Cheng and Yau
We improve the well known local gradient estimate of Cheng and Yau in the case when Ricci curvature has a negative lower bound.
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Smooth metric measure spaces with non-negative curvature
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.
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The Curvature of Gradient Ricci Solitons
We study integral and pointwise bounds on the curvature of gradient shrinking Ricci solitons. As applications we discuss gap and compactness results for gradient shrinkers.
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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.
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On convergence of the Kähler-Ricci flow
We study the convergence of the Kähler-Ricci flow on a Fano manifold under some stability conditions. More precisely we assume that the first eingenvalue of the $\bar\partial$-operator acting on vector fields is uniformly bounded along the flow, and in addition the Mabuchi energy decays at most logarithmically. We then give different situations in which the condition on the Mabuchi energy holds.
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The Poisson equation on complete manifolds with positive spectrum and applications
In this paper we investigate the existence of a solution to the Poisson equation on complete manifolds with positive spectrum and Ricci curvature bounded from below. We show that if a function $f$ has decay $f=O(r^{-1-\varepsilon}) $ for some $\varepsilon >0,$ where $r$ is the distance function to a fixed point, then the Poisson equation $Δu=f$ has a solution $u$ with at most exponential growth. We apply this result on the Poisson equation to study the existence of harmonic maps between complete manifolds and also existence of Hermitian Einstein metrics on holomorphic vector bundles over complete manifolds, thus extending some results of Li-Tam and Ni. Assuming that the manifold is simply connected and of Ricci curvature between two negative constants, we can prove that in fact the Poisson equation has a bounded solution and we apply this result to the Ricci flow on complete surfaces.
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On a characterization of the complex hyperbolic space
Consider a compact Kähler manifold $M^m$ with Ricci curvature lower bound $Ric_M\geq -2(m+1) .$ Assume that its universal cover $% \widetilde{M}$ has maximal bottom of spectrum $λ_1(\widetilde{M}%) =m^2.$ Then we prove that $\widetilde{M}$ is isometric to the complex hyperbolic space $\Bbb{CH}^m.$
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A sharp estimate for the bottom of the spectrum of the Laplacian on Kähler manifolds
On a complete noncompact Kähler 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.
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