Songbo Hou — Neckpinchhttps://neckpinch.com/authors/songbo-hou/New arXiv papers by Songbo Hou, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT Eigenvalues of the Laplace operator with potential under the backward Ricci flow on locally homogeneous 3-manifoldshttps://neckpinch.com/papers/2101.12464/arxiv:2101.12464Fri, 29 Jan 2021 00:00:00 GMTSongbo Hou, Shusen Yang — Let λ(t) be the first eigenvalue of −Δ+aR (a>0) under the backward Ricci flow on locally homogeneous 3-manifolds, where R is the scalar curvature. In the Bianchi case, we get the upper and lower bounds of λ(t). In particular, we show that when the the backward Ricci flow converges to a sub-Riemannian geometry after a proper re-scaling, λ⁺(t) approaches zero, where λ⁺(t)=max\λ(t),0\.Ricci flow3-manifoldsScalar curvatureHomogeneous spacesEigenvalues & spectrum Eigenvalues under the backward Ricci flow on locally homogeneous closed 3-manifoldshttps://neckpinch.com/papers/1602.07824/arxiv:1602.07824Thu, 25 Feb 2016 00:00:00 GMTSongbo Hou — In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.Ricci flow3-manifoldsHomogeneous spacesEigenvalues & spectrum Eigenvalues under the Ricci flow of model geometrieshttps://neckpinch.com/papers/1602.04597/arxiv:1602.04597Mon, 15 Feb 2016 00:00:00 GMTSongbo Hou — In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized Ricci flow of model geometries. In every Bianchi class, we estimate the derivative of the eigenvalue. Then we construct monotonic quantities under the Ricci flow and obtain upper and lower bounds for the eigenvalue.Ricci flowEigenvalues & spectrum