Analysis of weighted Laplacian and applications to Ricci solitons
By Ovidiu Munteanu, Jiaping Wang. 2011
158 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/b86d04827f67343b09a00c9df8227339f42cfb13
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In Applied Mathematics
In Astronomy and Astrophysics
In Geometry and Topology
Citations in the corpus
- Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, background): “, [17], [19], [42]) as indicators of an enforcement of some broader principle of asymptotic rigidity, however, the catalog of nontrivial examples is still exceedingly slim.”
- Complete $$\lambda $$λ-hypersurfaces of weighted volume-preserving mean curvature flow (2014, background): “Furthermore, Munteanu and Wang [25] have proved that areas of geodesic balls for n-dimensional complete and non-compact gradient shrinking Ricci solutions have at least linear growth.”
- Stability and compactness for complete f -minimal surfaces (2012, unknown): “Munteanu-Wang [11] extended the estimate (14) to (M, g, e−fdμ) with Ricf ≥ 12 and |∇f |(2) ≤ f .”