Renato G. Bettiol, Emilio A. Lauret, Paolo Piccione
We provide explicit formulae for the first eigenvalue of the Laplace-Beltrami operator on a compact rank one symmetric space (CROSS) endowed with any homogeneous metric. As consequences, we prove that homogeneous metrics on CROSSes are isospectral if and only if they are isometric, and also discuss their stability (or lack thereof) as solutions to the Yamabe problem.
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The smallest Laplace eigenvalue of homogeneous 3-spheres
本文推导齐性3维球面（即带左不变度量的SU(2)）上Laplace–Beltrami算子最小非零特征值的显式公式，完整刻画Berger球面谱，并推广至SO(3)。
We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature surfaces immersed into certain 3-dimensional Riemannian spaces, in particular into homogeneous 3-manifolds. As an application we derive some consequences for strongly stable surfaces in such ambient spaces. Moreover, we also get a characterization of Hopf tori in certain Berger spheres.
We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type.
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.
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 $λ^{+}(t)=\max\{λ(t),0\}$.
We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem
We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schrödinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as…
We use spectral embeddings to give upper bounds on the spectral function of the Laplace--Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower bound of Peter Li for the smallest positive eigenvalue to all eigenvalues. We also improve Li's bound itself. Our bounds translate to explicit upper bounds on the heat kernel for both compact and noncompact homogeneous spaces.
We consider a family of manifolds with a class of degenerating warped product metrics $g_ε=ρ(ε,t)^{2a}dt^2 +ρ(ε,t)^{2b}ds_M^2$, with $M$ compact, $ρ$ homogeneous degree one, $a \le -1$ and $b > 0$. We study the Laplace operator acting on $L^{2}$ differential $p$-forms and give sharp accumulation rates for eigenvalues near the bottom of the essential spectrum of the limit manifold with metric $g_{0}$.
Diameter and Laplace eigenvalue estimates for compact homogeneous Riemannian manifolds
本文针对紧齐性黎曼流形，研究第一拉普拉斯特征值与直径平方的乘积在不变度量下的有界性，证明了迷向表示无重数时该泛函存在上界，推进了相关猜想的研究。
The Laplace-Beltrami spectrum on Naturally Reductive Homogeneous Spaces
本文研究了紧致自然红uctive齐性空间上拉普拉斯-贝叶斯算子的谱，通过广义Casimir算子的特征值和球形表示推导出公式，并应用于正常齐性度量的变体，首次分析了谱对度量变形参数的依赖性。
Hodge Laplacian on $1$-forms of homogeneous $3$-spheres
本文研究了SU(2)和SO(3)上的左不变度量下Hodge-Laplacian的谱，首次明确计算了Berger 3球面的特征值并提出并证明了齐性度量下第一个特征值的显式公式，展示了1形式谱决定度量的全等性。
In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cross curvature flow on locally homogeneous 3-manifolds. We show that, typically, the positive cross curvature flow on locally homogeneous 3-manifold produce an Heisenberg type sub-Riemannian geometry.
In this paper we consider the complex vector spaces of holomorphic cross-sections of homogeneous holomorphic vector bundles over elliptic adjoint orbits, and provide a sufficient condition for the vector spaces to be finite dimensional in view of root systems.
In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John Lott, allows us to investigate the long term behavior of collapsing solutions of the flow, producing soliton solutions in the limit. We also review Lott's results on the long term behavior of three-dimensional homogeneous solutions of Ricci flow, demonstrating the coordinates we choose and reviewing the groupoid technique. We find cross curvature soliton metrics on Sol and Nil, and show that the cross …
We show that any two left-invariant metrics on $S^3\cong\operatorname{SU}(2)$ which are isospectral for the associated classical Dirac operator $D$ must be isometric. In the case of left-invariant metrics of positive scalar curvature, we compute and use the smallest eigenvalue of $D^2$. We show analogous results for left-invariant metrics on $\operatorname{SO}(3)=S^3/\{\pm1\}$ for each of its two spin structures.
Given a reductive homogeneous space M=G/H endowed with a naturally reductive metric, we study the one-parameter family of connections joining the canonical and the Levi-Civita connection (t=0, 1/2). We show that the Dirac operator D^t corresponding to t=1/3 is the so-called ``cubic'' Dirac operator recently introduced by B. Kostant, and derive the formula for its square for any t, thus generalizing the classical Parthasarathy formula on symmetric spaces. Applications include the existence of a new G-invariant first order differential operator on spinors and an eigenvalue estimate for the first…
Deformation of Kähler Metrics and an Eigenvalue Problem for the Laplacian on a Compact Kähler Manifold
本文在固定体积的紧 Kähler 流形上研究 Laplace 算子第 k 个特征值作为度量泛函的极值问题，引入 λ_k-极值 Kähler 度量概念并推导其极值条件，同时给出乘积流形、齐性流形和平坦复环面等例子。
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