Zhongyang Sun
In this paper, we partially solve Yau' Conjecture of the first eigenvalue of an embedded compact minimal hypersurface of unit sphere $\mathbb{S}^{n+1}(1)$, i.e., Corollary 1.2. In particular, Corollary 1.3 proves that the condition $\int_{Ω_{1}}|\nabla u|^{2}=(n+1)\int_{Ω_{1}}u^{2}$ is naturally true and meaningful in Corollary 1.2.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
A well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface $M^n$ in the unit sphere $S^{n+1}(1)$ is just its dimension $n$. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non-stable range.
In this note, we investigate the well-known Yau rigidity theorem for minimal submanifolds in spheres. Using the parameter method of Yau and the DDVV inequality verified by Lu, Ge and Tang, we prove that if $M$ is an $n$-dimensional oriented compact minimal submanifold in the unit sphere $S^{n+p}(1)$, and if $K_{M}\geq\frac{sgn(p-1)p}{2(p+1)},$ then $M$ is either a totally geodesic sphere, the standard immersion of the product of two spheres, or the Veronese surface in $S^4(1)$. Here $sgn(\cdot)$ is the standard sign function. We also extend the rigidity theorem above to the case where $M$ is a…
In this paper, we derive the CR Reilly's formula and its applications to studying of the first eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the first Dirichlet eigenvalue estimate in a compact pseudohermitian (2n+1)-manifold with boundary and the first eigenvalue estimate of the tangential sublaplacian on closed oriented embedded p-minimal hypersurfaces in a closed pseudohermitian (2n+1)-manifold of vanishing torsion.
The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture
本文证明单位球面中嵌入极小超曲面的第一非零特征值等于其维数，从而肯定解决 Yau 猜想，并由此推导出多个刚性定理。
Huisken-Yau-type uniqueness for area-constrained Willmore spheres
本文在渐近 Schwarzschild 且标量曲率非负的 Riemannian 3-流形中，证明满足面积半径远大于内半径的非负 Hawking 质量大面积约束 Willmore 球面不存在，回答了 Huisken 的问题。
Chung-Grigor'yan-Yau's inequality describes upper bounds of eigenvalues of Laplacian in terms of subsets ("input") and their volumes. In this paper we will show that we can reduce "input" in Chung-Grigor'yan-Yau's inequality in the setting of Alexandrov spaces satisfying CD$(0,\infty)$. We will also discuss a related conjecture for some universal inequality among eigenvalues of Laplacian.
This is a continuation of Tang and Yan, which investigated the first eigenvalues of minimal isoparametric hypersurfaces with $g=4$ distinct principal curvatures and focal submanifolds in unit spheres. For the focal submanifolds with $g=6$, the present paper obtains estimates on all the eigenvalues, among others, giving an affirmative answer in one case to the problem posed in Tang and Yan, which may be regarded as a generalization of Yau's conjecture. In two of the four unsettled cases in Tang and Yan for focal submanifolds $M_1$ of OT-FKM-type, we prove the first eigenvalues to be their dimen…
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…
This paper is a continuation of our paper math.AG/0205321 where we have built a combinatorial model for the torus fibrations of Calabi-Yau toric hypersurfaces. This part addresses the connection between the model torus fibration and the complex and Kähler geometry of the hypersurfaces.
We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitzéica equation as reductions of real forms of $SL(3, \C)$ anti--self--dual Yang--Mills equations by two translations, or equivalently as a special case of the Hitchin equation. We use the Loftin--Yau--Zaslow construction to give an explicit expression for a six--real dimensional semi--flat Calabi--Yau metric in terms of a solution to the affine-sphere equation and show how a subclass of such metrics arises from 3rd Painlevé transcendents.
Eigenvalues and eigenforms on Calabi-Yau threefolds
本文提出一种数值算法，通过输入近似度量计算Calabi-Yau流形上Laplace-de Rham算子的特征值与特征形式，并在Fermat五次三维流形和$\mathbb{P}^{3}$上验证了算法的有效性。
In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the Yau rigidity theory of submanifolds. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems for the mean curvature flow. It should be emphasized that Theorem 4.1 is an optimal convergence theorem for the mean curvature flow of arbitrary codimension, which implies the first optimal differentiable sphere theorem for sub…
In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point $p$ in a spacetime $N$, we consider a canonical family of surfaces approaching $p$ along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving an "optimal embedding equation" whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in [7]. Namely, there is a natural solution which is a local minimum, and the l…
Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they deviate from being deformable to a family of special Lagrangian deformations by examining in detail Joyce's criteria on this family. The result suggests a potential image-support rigidity of A-branes that wrap around a special Lagrangian 3-sphere in a Calabi-Yau 3-fold, which res…
We show how a Bochner type formula can be used to establish universal inequalities for the eigenvalues of the drifted Cheng-Yau operator on a bounded domain in a pinched Cartan-Hadamard manifold with the Dirichlet boundary condition. In the first theorem, the hyperbolic space case is treated in an independent way. For the more general setting, we first establish a Rauch comparison theorem for the Cheng-Yau operator and two estimates associated with the Bochner type formula for this operator. Next, we get some integral estimates of independent interest. As an application, we compute our univers…
Spectrum of a certain class of first order conformally invariant operators on the sphere is explicitly computed. The class contains the (elliptic verions of) Rarita-Schwinger operator and its higher spin analogues.
We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudoconvex pseudo-Hermitian structure $\θ$ on the CR sphere S 2n+1 $\subset$ C n+1 , achieves its maximum when $\θ$ is the standard contact form.
A Thomas-Yau-Joyce result for Lagrangian spheres in K3 surfaces
本文证明在满足同调镜像对称的K3曲面上，Bridgeland稳定性蕴含特殊Lagrangian球面存在，验证Thomas-Yau-Joyce猜想，并推广至乘积Calabi-Yau三维流形，为紧流形上首例。
We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly round surfaces including large constant mean curvature spheres in asymptotically flat 3-manifolds with positive mass; it is also positive definite on small geodesics spheres, whose centers do not have vanishing curvature, in Riemannian 3-manifolds of nonnegative scalar curvature. We …
Considering the almost rigidity of the Obata theorem, we generalize Petersen and Aubry's sphere theorem about eigenvalue pinching without assuming the positivity of Ricci curvature, only assuming $Ric\geq-Kg$ and $diam\leq D$ for some positive constants $K>0$ and $D>0$.
In this paper, we study the first eigenvalue of Jacobi operator on an $n$-dimensional non-totally umbilical compact hypersurface with constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$. We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature $H$ and the dimension $n$.
Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature
该研究针对围长为3且β>α的充分正则图，通过关联离散Ricci曲率与局部匹配性质的新方法，证明了其Lin-Lu-Yau曲率的精确下界，并由此推导出该类图的精确直径与特征值界。
In this short note we extend an estimate due to J. Simons on the first stability eigenvalue of minimal hypersurfaces in spheres to the singular setting. Specifically, we show that any singular minimal hypersurface in $S^{n+1}$, which is not totally geodesic and satisfies the α-structural hypothesis, has first stability eigenvalue at most -2n, with equality if and only if it is a product of two round spheres. The equality case was settled independently in the classical setting by Wu and Perdomo.
A remark on the first eigenvalue of the p-Laplacian on compact submanifolds in the unit sphere
本文针对3/2<p<2建立了奇异p-laplacian的积分不等式，并由此推导出单位球面中极小子流形和预定数量曲率子流形p-laplacian第一特征值的下界。
On the First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere
针对单位球面中闭嵌入极小超曲面，改进了其Laplace-Beltrami算子第一非零特征值的下界，结果略优于Duncan-Sire-Spruck的界。
We try to present an estimate relating the first Dirichlet and Neumann eigenvalues of a compact bordered Riemannian surface.
An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices
通过基本矩阵特征值，建立了欧氏球面中紧致极小子流形的最优刚性定理，不限制余维数，并给出广义克利福德环面和韦罗内塞流形的新刻画。
This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and non-existences will be considered. (2) The Yau conjecture on the first eigenvalues of the embedded minimal hypersurfaces in the unit spheres. The history and progress of the Yau conjecture on minimal isoparametric hypersurfaces will be stated.
In 1980 Yang and Yau~\cite{YY} proved the celebrated upper bound for the first eigenvalue on an orientable surface of genus $γ$. Later Li and Yau~\cite{LY} gave a simple proof of this bound by introducing the concept of conformal volume of a Riemannian manifold. In the same paper they proposed an approach for obtaining a similar estimate for non-orientable surfaces. In the present paper we formalize their argument and improve the bounds stated in~\cite{LY}.
$p$-Eigenvalue pinching sphere theorems
本文基于$ p $-拉普拉斯算子的特征值，证明了两个球面定理，表明在特定曲率条件下，流形与球面同胚或微分同胚。
In this article, we prove an eigenvalue pinching theorem for the first eigenvalue of the Laplacian on compact hypersurfaces in a sphere. Let $(M^n,g)$ be a closed, connected and oriented Riemannian manifold isometrically immersed by $ϕ$ into $§^{n+1}$. Let $q>n$ and $A>0$ be some real numbers satisfying $|M|^\frac{1}{n}(1+\|B\|_q)\leq A$. Suppose that $ϕ(M)\subset B(p_0,R)$, where $p_0$ is a center of gravity of $M$ and radius $R<\fracπ{2}$. We prove that there exists a positive constant $\e$ depending on $q$, $n$, $R$ and $A$ such that if $n(1+\|H\|_\infty^2)-\e\leq ł_1$, then $M$ is diffeomo…
The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices
本文利用基本矩阵特征值与 Lu 不等式，建立单位球中各维紧致极小 Legendrian 子流形的最优夹挤与刚性定理，并刻画 Calabi 环面。
The stability index and Yau's conjecture for Carlotto-Schulz minimal hypertori, part II
本文研究Carlotto-Schulz极小嵌入$X_{CS}^n$的稳定性算子特征值- $n$的重数，给出下界并猜想稳定性指标公式，数值验证前100个$n$的猜想及Yau猜想对$2\le n\le 260$成立。
In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in $\ir{3}$ obtained by Colding-Minicozzi under weaker conditions.
We prove Li-Yau-Kröger type bounds for Neumann-type eigenvalues of the poly-harmonic operator and of the biharmonic operator on bounded domains in a Euclidean space. We also prove sharp estimates for lower order eigenvalues of a biharmonic Steklov problem and of the Laplacian, which directly implies two sharp Reilly-type inequalities for the corresponding first nonzero eigenvalue.
We consider the Jacobi operator, defined on a closed oriented hypersurfaces immersed in the Euclidean space with the same volume of the unit sphere. We show a local generalization for the classical result of the Willmore functional for the Euclidean sphere. As a consequence, we prove that the first eigenvalue of the Jacobi operator in the Euclidean sphere is a local maximum and this result is a global one in the closed oriented surfaces space of $\mathbb{R}^3$ and genus zero.
We establish in this paper an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases.
Volume gap between the minimal submanifold and the unit sphere
本文通过改进Cheng-Li-Yau方法中的常数系数并利用Cheng-Yang的Laplacian特征值估计，扩大了极小子流形与单位球面之间的体积间隙。
Second Jacobi eigenvalues and spectral rigidity for surfaces in Berger spheres
本文为Berger球面中闭曲面第二Jacobi特征值建立上界，结合多种几何工具，并应用于非正欧拉示性数情形，在临界参数处刻画等式条件。
The CR-Volume of Horizontal Submanifolds of Spheres
本文针对奇数维球面中的 Legendrian 或水平子流形，提出在 CR-自同构下不变的 CR-体积，并将其用于相应的 Willmore 能量。
An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices II
针对单位球面中的紧致极小子流形，利用其基本矩阵的特征值，证明了满足特定挤压条件时该子流形必为全测地子球面、广义Clifford环面或Veronese流形，并完成了等号情形的分类。
n this paper we define an invariant of a pair of 6 dimensional symplectic %optional manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of the bounding cochains (solution of A infinity version of Maurer-Cartan equation of the filtered A infinity algebra associated to the Lagrangian submanifold). In the case when the Lagrangian submanifold is a rational homology sphere, it becomes a numerical invariant. This invariant depends on the choice of almost complex structure. The way…
Optimal lower estimate for the first eigenvalue of the p-Laplacian
本文推导了单位球面中紧致子流形的积分不等式，并以此刻画球面，从而给出了p-Laplacian第一特征值的最佳下估计。
By comparing the Laplace spectrum of the sphere $\mathbb{S}^n$ to its Weyl function $w(x) = \frac{ω_n}{(2π)^n}|\mathbb{S}^n|x^{n/2}$, we show that no analogue of Pólya's eigenvalue conjecture holds in general for Riemannian manifolds with positive sectional curvature.
We show a Gromov-Hausdorff approximation to the product of the standard spheres $S^{n-p}\times S^p$ for Riemannian manifolds with positive Ricci curvature under some pinching condition on the eigenvalues of the Laplacian acting on functions and forms.
Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of Aoo-algebras built of differential forms on the symplectic manifold. We show that these symplectic Aoo-algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these Aoo-algebras are equivalent to the standard de Rham differential graded algebra on certain odd-dimensional sphere bundles over the symplectic manifold. From this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yau's symplectic Aoo-algebras satisfy the Cal…
We attempt to define a new invariant I of (almost) Calabi-Yau 3-folds M, by counting special Lagrangian rational homology 3-spheres N in M in each 3-homology class, with a certain weight w(N) depending on the topology of N. This is motivated by the Gromov-Witten invariants of a symplectic manifold, which count the J-holomorphic curves in each 2-homology class. In order for this invariant to be interesting, it should either be unchanged by deformations of the underlying (almost) Calabi-Yau structure, or else transform according to some rigid set of rules as the periods of the almost Calabi-Yau …
New minimal surface doublings of the Clifford torus and contributions to questions of Yau
该研究利用前期一般性定理构造了三维球面中不受透镜空间提升限制的新型Clifford环面极小倍化曲面，得到指定亏格嵌入极小曲面数量的二次下界，并在两类倍化曲面情形下验证了丘成桐的第一特征值猜想。
We consider the higher order buckling eigenvalues of the following Dirichlet poly-Laplacian in the unit sphere $(-Δ)^p u=Λ(-Δ) u$ with order $p(\geq2)$. We obtain universal bounds on the $(k+1)$th eigenvalue in terms of the first $k$th eigenvalues independent of the domains. In particular, for $p=2$, our result is sharp than estimates on eigenvalues of the buckling problem obtained by Wang and Xia.
In this paper we exhibit deformations of the hemisphere $S^{n+1}_+$, $n\geq 2$, for which the ambient Ricci curvature lower bound $\text{Ric}\geq n $ and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the boundary decreases. The existence of these metrics suggests that any resolution of Yau's conjecture on the first eigenvalue of minimal hypersurfaces in spheres would likely need to consider more geometric data than a Ricci curvature lower bound.
Given a Riemmanian manifold, we provide a new method to compute a sharp upper bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of radius less than the injectivity radius of the manifold. This upper bound is obtained by transforming the metric tensor into a rotationally symmetric metric tensor that preserves the area of the geodesic spheres. The provided upper bound can be computed using only the area function of the geodesic spheres contained in the geodesic ball and it is sharp in the sense that the first eigenvalue of geodesic ball coincides with o…
We show that the action of conformal vector fields on functions on the sphere determines the spectrum of the Laplacian (or the conformal Laplacian), without further input of information. The spectra of intertwining operators (both differential and non-local) with principal part a power of the Laplacian follows as a corollary. An application of the method is the sharp form of Gross' entropy inequality on the sphere. The same method gives the spectrum of the Dirac operator on the sphere, as well as of a continuous family of nonlocal intertwinors, and an infinite family of odd-order differential …
A lower bound for the first eigenvalue of a minimal hypersurface in the sphere
本文研究了超球面中闭极小超曲面的 Laplace 算子第一特征值下界问题，通过特征函数调和延拓的 Rayleigh 商估计，证明了新的下界不等式，改进了已有结果。
On the eigenvalues of the Laplacian on ellipsoids obtained as perturbation of unit sphere
本文研究二维单位球面扰动所得椭球上的 Laplace 特征值，并在高斯曲率条件下建立其与球面特征值的比较，呼应 Lichnerowicz 定理。
As is known, the Blaschke tensor $A$ (a symmetric covariant $2$-tensor) is one of the fundamental Möbius invariants in the Möbius differential geometry of submanifolds in the unit sphere $\mathbb S^n$, and the eigenvalues of $A$ are referred to as the Blaschke eigenvalues. In this paper, we continue our job for the study on the submanifolds in $\bbs^n$ with parallel Blaschke tensors which we simply call {\em Blaschke parallel submanifolds} to find more examples and seek a complete classification finally. The main theorem of this paper is the classification of Blaschke parallel submanifolds in …
In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at spatial infinity of an asymptotically flat initial data set. Similar to the small sphere limit of the Wang-Yau quasi-local mass, we prove that the leading order term of the quasi-local mass recovers the stress-energy tensor. For a vacuum spacetime, the quasi-local mass decays faster and the leading order term is related to the Bel-Robinson tensor. Several new techniques of evaluating quasilocal mass are developed in this note.
Minimal surfaces with negative curvature in large dimensional spheres
本文通过构造具有大自同构群的闭黎曼曲面上的几乎双曲最小曲面，肯定回答了丘成桐关于大维球面中存在负诱导曲率闭最小曲面的问题。
The first eigenvalue of the Laplacian on orientable surfaces
本文推广Ros的论证方法，利用到任意n>0维复射影空间的全纯映射，对多数亏格大于3的可定向曲面，给出了Yang-Yau不等式的定量改进结果。
For an $n$-dimensional polytope $Ω$ in $\mathbb{R}^{n}$, we study lower bounds for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. In the asymptotic formula on the average of the first $k$ eigenvalues, Li and Yau (1983) obtained the first term with the order $k^{\frac2n}$, which is optimal. The next landmark goal is to give the second term with the order $k^{\frac1n}$ in the asymptotic formula. For this purpose, Kovař\'ık, Vugalter and Weidl (2009) have made an important breakthrough in the case of dimension 2. It is our purpose to study the $n$-dimensional case for arbitrary…
Eigenvalues on spheres
研究二维球面上高斯曲率下界为1的光滑黎曼度量的正拉普拉斯特征值，通过建立谱计数比较，得出特征值相关结论，并应用于完备三维流形上多项式增长调和函数空间，得到维数界和刚性结果。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。