Eigen-value monotonicity for the Ricci-Hamilton flow

Monotonicity of the first eigenvalue of the Laplace and the p-Laplace operators under a forced mean curvature flow

In this paper, we would like to give an answer to \textbf{Problem 1} below issued firstly in [J. Mao, Eigenvalue estimation and some results on finite topological type, Ph.D. thesis, IST-UTL, 2013]. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced mean curvature flow considered here, we can obtain that the first eigenvalues of the Laplace and the $p$-Laplace operators are monotonic under this flow. Surprisingly, during this process, we get an interesting byproduct, that is, without any complicate…

Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula

Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $σ$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in certain situations, the value distribution of $f$ under $σ$ is approximately Gaussian. Write $μ$ for the measure whose density with respect to $σ$ is $|\nabla f|^2$. We observe that the value distribution of $f$ under $μ$ admits a unimodal density attaining its maximum at the origin. Thus, in a sense, the zero set of an eigenfunction is the largest of all level sets.…

Kähler-Ricci流下的矩阵Li-Yau-Hamilton估计

Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow

该研究证明了Kähler-Ricci流耦合下热方程与反向共轭热方程正解的矩阵Li-Yau-Hamilton估计，并推导得到单调性公式，丰富了几何分析相关理论。

Eigenvalue monotonicity for the Ricci-Hamilton flow, revised version

This is a revised version of our short note [arxiv.math.DG/0403065] where we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We show that the eigenvalue of the Lapacian operator on a compact domain associated with the evolving Ricci flow is non-decreasing provided the scalar curvature having a non-negative lower bound and Einstein tensor being not too negative. This result will be useful in the study of blow-up models of the Ricci-Hamilton flow. We remark that using our idea, D.Kokotov…

A new Li-Yau-Hamilton estimate for Kahler-Ricci flow

In this paper we prove a new matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in \cite{Ch1}. This new inequality is shown to be connected with Perelman's entropy formula through a family of differential equalities. In the rest of the paper, We show several applications of this new estimate and its linear version proved earlier in \cite{CN}. These include a sharp heat kernel comparison theorem, generalizing the earlier result of Li and Tian, a manifold version of Stoll's theorem on the …

On the spectra of geometric operators evolving with geometric flows

In this work we generalise various recent results on the evolution and monotonicity of the eigenvalues of certain geometric operators under specified geometric flows. Given a closed, compact Riemannian manifold $\big(M^n,g(t)\big)$ and a smooth function $η\in C^{\infty}(M)$ we consider the family of operators $\mathbb{L}=Δ- g(\nablaη,\nabla\cdot)+cR$, where $R$ is the scalar curvature and $c$ is some real constant. We define a geometric flow on $M$ which encompasses the Ricci, the Ricci - Bourguignon and the Yamabe flows. Supposing that the metric $g(t)$ evolves along this general geometric fl…

Isoperimetric inequalities for eigenvalues by inverse mean curvature flow

By studying the monotonicity of the first nonzero eigenvalues of Laplace and p-Laplace operators on a closed convex hypersurface $M^n$ which evolves under inverse mean curvature flow in $\mathbb{R}^{n+1}$, the isoperimetric lower bounds for both eigenvalues were founded.

First Eigenvalues of Geometric Operators under the Ricci Flow

In this paper, we prove that the first eigenvalues of $-Δ+ cR$ ($c\geq \frac14$) is nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized flow for the case $c=1/4$, and $r\le 0$.

Hamilton type entropy formula along the Ricci flow on surfaces with boundary

In this article, we establish a monotonicity formula of Hamilton type entropy along Ricci flow on compact surfaces with boundary. We also study the relation between our entropy functional and the $\mathcal{W}$-functional of Perelman type.

Ricci流与抛物频率下的矩阵Li-Yau-Hamilton估计

Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency

该研究证明了与Ricci流耦合的热方程及反向共轭热方程正解的矩阵Li-Yau-Hamilton估计，借此确立带修正因子的抛物频率单调性，并得到曲率非负条件下的唯一延拓结果。

Variation of the first eigenvalue of $(p,q)$-Laplacian along the Ricci-harmonic flow

In this paper, we study monotonicity for the first eigenvalue of a class of $(p,q)$-Laplacian. We find the first variation formula for the first eigenvalue of $(p,q)$-Laplacian on a closed Riemannian manifold evolving by the Ricci-harmonic flow and construct various monotic quantities by imposing some conditions on initial manifold.

广义Ricci流下热方程正解的梯度估计与抛物频率单调性

Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow

本文在广义Ricci流和较弱Ricci曲率有界条件下建立热方程正解的Li-Yau型与Hamilton型梯度估计，并由此得到时空Harnack不等式及抛物频率单调性。

可变耦合参数下扩展 Ricci 流中几何常数的演化与单调性

Evolution and Monotonicity of Geometric Constants under Extended Ricci Flows with Variable Coupling Parameters

研究在含可变耦合参数的扩展 Ricci 流下几何常数 λ 的演化，通过修改演化方程系数推导其在扩展与归一化流下的公式，并给出单调性成立条件。

Eigenvalues and entropies under the harmonic-Ricci flow

In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eigenvalues of Laplacian under the harmonic-Ricci flow. Finally, we obtain the first variation of the shrinker and expanding entropies of the harmonic-Ricci flow.