Guoxin Wei, Lingzhong Zeng
In this paper, we study eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an n-dimensional Euclidean space and obtain a lower bound for eigenvalues, which generalizes the results due to Cheng-Wei [5] and gives an improvement of results due to Cheng- Qi-Wei [3].
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Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues yield characterizations of harmonicity, totally geodesic maps and biconformal changes of metric preserving harmonicity. A Schwarz lemma for pseudo harmonic morphisms is proved, using the dilatation of the eigenvalues and, in dimension five, a Bochner technique method, involving the Laplacian of the difference of the eigenvalues, gives conditions forcing pseudo h…
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.
In this article, we prove an isoperimetric inequality for the harmonic mean of the first $(n-1)$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Hyperbolic space. Our approach to prove this result also gives a similar inequality for the first $n$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Euclidean space.
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.
Conformal and extrinsic upper bounds for the harmonic mean of Neumann and Steklov eigenvalues
本文研究了Neumann和Steklov特征值调和平均值的共形与内蕴上界，并给出了闭子流形在空间形式中的最优上界。
In this note, an isoperimetric inequality for the harmonic mean of lower order Steklov eigenvalues is proved on bounded domains in noncompact rank-$1$ symmetric spaces. This work extends result of \cite{BR.01} and \cite{V.21} proved for bounded domains in Euclidean space and Hyperbolic space, respectively.
Extrinsic upper bounds for the harmonic mean of eigenvalues in curved manifolds
该研究针对截面曲率有上界的黎曼流形中的m维闭定向子流形，得到其拉普拉斯算子前m个非零特征值调和均值的外蕴上界，推广了此前相关结果。
Submission was withdrawn by authors. arXiv:math/0305140
We prove that on a compact $n$-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue $λ$ of the Dirac operator satisfies the inequality $λ^2 \geq \frac{n-1}{4(n-2)}\inf_M Scal$. In the limiting case the universal cover of the manifold is isometric to $R\times N$ where $N$ is a manifold admitting Killing spinors.
We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact Kähler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of Kählerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a Kählerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact Kähler-Einstein man…
We study the tt*-geometry with vanishing endormorphism $\mathcal{U}$. Given an integrable harmonic Higgs bundle $(E, h, Φ, \mathcal{U},\mathcal{Q})$ on a complex manifold $M$, Firstly we prove that, under the \emph{IS} condition, vanishing $\mathcal{U}$ implies vanishing Higgs field $Φ$ and the Chern connection of the Hermitian Einstein metric $h$ is a holomorphic connection, so the metric $h$ and $\mathcal{Q}$ are invariant. Secondly, without the \emph{IS} condition, we show that vanishing $\mathcal{U}$ will imply vanishing Higgs field $Φ$ if we assume that the Chern connection of $h$ is a ho…
} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $Δ^2$ on a bounded smooth domain $\Om$ in the Euclidean $n$-space ${\bf R}^n$ ($n\ge2$) and then prove that the corresponding first non-zero eigenvalue $Υ_1(\Om)$ admits the isoperimetric inequality of Szegö-Weinberger type: $Υ_1(\Om)\le Υ_1(B_{\Om})$, where $B_{\Om}$ is a ball in ${\bf R}^n$ with the same volume of $\Om$. The isoperimetric inequality of Szegö-Weinberger type for the first nonzero Neumann eigenvalue of the even-multi-Laplacian operators $Δ^{2m}$ ($m\ge1$) on $\Om$ is also exploited.
We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold $(M^n,g)$ of dimension $n>2$ to any closed, non-aspherical manifold $N$ containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres $N=\mathbb{S}^k$, $k\geq 3$, we obtain a distinguished family of nonconstant harmonic maps $M\to \mathbb{S}^k$ of index at most $k+1$, with singular set of codimension at least …
This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some conical singularities and maximizes the first eigenvalue in the conformal class of the background metric. We also prove that the map associating a finite number of eigenvectors of the maximizing $λ_1$ into the sphere is harmonic, establishing a link between conformal spectrum and har…
Dirac Eigenvalue Optimisation and Harmonic Maps to Complex Projective Spaces
本文通过建立 Dirac 特征值临界度量与到复射影空间调和映射的联系，证明环面多类共形结构中平坦度量最小化首个非零 Dirac 特征值，并几何重证 Bär 球面定理。
We use the density function of a harmonic space to obtain estimates for the eigenvalues of the Jacobi operator; when these estimates are sharp, then the harmonic space is a symmetric Osserman space.
We present a unified description of extremal metrics for the Laplace and Steklov eigenvalues on manifolds of arbitrary dimension using the notion of $n$-harmonic maps. Our approach extends the well-known results linking extremal metrics for eigenvalues on surfaces with minimal immersions and harmonic maps. In the process, we uncover two previously unknown features of the Steklov eigenvalues. First, we show that in higher dimensions there is a unique normalization involving both the volume of the boundary and of the manifold itself, which leads to meaningful extremal eigenvalue problems. Second…
In this paper, we investigate the Dirchlet eigenvalue problems of poly-Laplacian with any order and quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first $k$ eigenvalues which improve the previous results.
Local structure of gradient almost Ricci solitons with harmonic Weyl tensor
本文研究具有调和Weyl张量的梯度几乎Ricci孤立子，证明其Ricci张量在正则点邻域内至多有三个互异特征值，并分类了恰好有两个特征值的情形，推广了Catino和Kim的已知结果。
The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean c…
In this paper, by extending the notions of harmonic transplantation and harmonic radius in the Heisenberg group, we give an upper bound for the first eigenvalue for the following Dirichlet problem: $$(P_Ω) \left\{ \begin{array}{lllll} -Δ_{\mathbb{H}^1} u & = & λu & \mbox{in} & Ωu & = & 0 & \mbox{on} & \partial Ω, \end{array} \right.$$ where $ Ω$ is a regular bounded domain of $ \mathbb{H}^1$ with smooth boundary and $Δ_{\mathbb{H}^1}$ is the Kohn-Laplace operator. Using the results of P.Pansu which give the relation between the volume of $Ω$ and the perimeter of its boundary. we prove the foll…
Deep estimates for higher eigenvalues of the poly-Laplacian
该研究针对有界区域上多重拉普拉斯算子的高阶特征值下界问题，推导低维情形的深度特征值不等式，建立带约束与无约束的任意维数改进下界，同时优化了Stokes特征值问题与广义Pólya猜想的相关下界。
We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenböck techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds.
Eigenvalue optimization in higher dimensions and $p$-harmonic maps
本文研究了闭合黎曼流形上第k个拉普拉斯特征值的优化问题，证明了在不同归一化条件下存在性结果，并表明绝对连续极值点由p-调和映射诱导，且在p接近m时具有Hölder连续性。
Min-max harmonic maps and a new characterization of conformal eigenvalues
本文将共形类中前两个非平凡 Laplacian 特征值的极大值刻画为球面值映射能量的极小极大量，并由此推出特征值界、极大度量存在性及极大 Radon 测度正则性结果。
Regularity estimates on harmonic eigenmaps with arbitrary number of coordinates
本文重新审视曲面上调和映射的正则性估计对目标流形维数的独立性，研究映射到目标椭球体的 Laplace 调和特征映射，并收集工具处理几乎临界度量的收敛，为特征值无限组合的临界点的一般正则性理论提供初步步骤。
We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the $k$th eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Our inequality is sharper than the known Payne-Pólya-Weinberg type inequality and also covers the important Yang inequality on eigenvalues of the Dirichlet Laplacian. We also prove universal inequalities for the lower order…
We consider the lower order eigenvalues of poly-Laplacian with any order on spherical domains. We obtain universal inequalities for them and show that our results are optimal.
In this paper, we study eigenvalues of the poly-Laplacian with arbitrary order on a bounded domain in an $n$-dimensional Euclidean space and obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter. In particular, the result of Melas is included here.
In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus $γ$ and the area. The equality in Yang-Yau's estimate is attained for $γ=0$ by an old result of J. Hersch and it was recently shown by S. Nayatani and T. Shoda that it is also attained for $γ=2$. In the present article we combine techniques from algebraic geometry and minimal surface theory to show that Yang-Yau's inequality is strict for all genera $γ> 2$. Previously this was only known for $γ=1$. In the se…
We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging to a union of k touching identical round spheres. This proves a conjecture posed by the second author in 2002 and yields a sharp isoperimetric inequality for all nonzero eigenvalues of the Laplacian on a sphere. Earlier, the result was known only for k=1 (J. Hersch, 1970), k=2 (N. Nadirashvili, 2002; R. Petrides, 2014) and k=3 (N. Nadirashvili and Y. Sire, …
Sharp harmonic-mean inequalities for Neumann and Aharonov--Bohm spectra
该文证明曲面上Neumann和Aharonov--Bohm谱的锐双特征值等周不等式，结合互反Rayleigh--Ritz原理与谱序定理，给出调和平均上界及磁类比结果。
Eigenvalue estimates for the poly-Laplace operator on lattice subgraphs
本文引入了具有 Dirichlet 边界条件的子图上的离散多重 Laplace 算子，推导了格图 $\mathbb{Z}^{d}$ 有限子图上其前 $k$ 个特征值之和的上下界，并证明了高阶特征值与低阶特征值平方之间的不等式关系。
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