Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian

On the first eigenvalue of the laplacian on compact surfaces of genus three

For any compact riemannian surface of genus three $(Σ,ds^2)$ Yang and Yau proved that the product of the first eigenvalue of the Laplacian $λ_1(ds^2)$ and the area $Area(ds^2)$ is bounded above by $24π$. In this paper we improve the result and we show that $λ_1(ds^2)Area(ds^2)\leq16(4-\sqrt{7})π\approx 21.668\,π$. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value $\approx 21.414\,π$.

On the asymptotics of determinant of Laplacian at the principal boundary of the principal stratum of the moduli space of Abelian differentials

We study the asymptotics of the determinant of Laplacian on a translation surface (a compact Riemann surface equipped with a conformal flat conical metric with trivial holonomy) of genus g with 2g-2 conical points of angle 4πas two conical points collide.

大亏格模空间上拉普拉斯算子行列式平均值

Averages of determinants of Laplacians over moduli spaces for large genus

研究大亏格模空间上拉普拉斯算子行列式正则化值的平均性质，证明其期望值随亏格增长而衰减，并趋近于常数。

An optimal inequality between scalar curvature and spectrum of the Laplacian

For a Riemannian closed spin manifold and under some topological assumption (non-zero $\hat{A}$-genus or enlargeability in the sense of Gromov-Lawson), we give an optimal upper bound for the infimum of the scalar curvature in terms of the first eigenvalue of the Laplacian. The main difficulty lies in the study of the odd-dimensional case. On the other hand, we study the equality case for the closed spin Riemannian manifolds with non-zero $\hat{A}$-genus. This work improves an inequality which was first proved by K. Ono in 1988.

First eigenvalue of the Laplacian on compact surfaces for large genera

For any Riemannian metric $ds^2$ on a compact surface of genus $g$, Yang and Yau proved that the normalized first eigenvalue of the Laplacian $λ_1(ds^2)Area(ds^2)$ is bounded in terms of the genus. In particular, if $Λ_1(g)$ is the supremum for each $g$, it follows that the asymptotic growth of the sequence ${Λ_1(g)}$ is no larger than the one of $4πg$. In this paper we improve the result and we show that \[ \limsup_{g\, \rightarrow\, \infty} \, \frac{1}{g}Λ_1(g) \leq 4(3-\sqrt{5})π\approx 3.056π. \]

Maximization of the first nontrivial eigenvalue on the surface of genus two

The first nontrivial eigenvalue of the Laplacian can be considered as a functional on the space of all Riemannian metrics of unit volume on a fixed surface. In this paper we prove that for the surface of genus 2 the supremum of this functional is equal to $16π$. This provides a positive answer to the conjecture by Jakobson, Levitin, Nadirashvili, Nigam and Polterovich.

Green function and self-adjoint Laplacians on polyhedral surfaces

Using Roelcke formula for the Green function, we explicitly construct a basis in the kernel of the adjoint Laplacian on a compact polyhedral surface $X$ and compute the $S$-matrix of $X$ at the zero value of the spectral parameter. We apply these results to study various self-adjoint extensions of a symmetric Laplacian on a compact polyhedral surface of genus two with a single conical point. It turns out that the behaviour of the $S$-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.

A negative mass theorem for surfaces of positive genus

We define the "sum of squares of the wavelengths" of a Riemannian surface (M,g) to be the regularized trace of the inverse of the Laplacian. We normalize by scaling and adding a constant, to obtain a "mass", which is scale invariant and vanishes at the round sphere. This is an anlaog for closed surfaces of the ADM mass from general relativity. We show that if M has positive genus then on each conformal class, the mass attains a negative minimum. For the minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality.

Upper bounds for the first eigenvalue of the Laplacian on non-orientable surfaces

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}.

Klein quartic 最大化 Laplacian 的第一正特征值的重数

The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian

本文证明在亏格为3的闭双曲曲面中，Klein quartic 最大化了 Laplacian 第一正特征值的重数（等于8），并给出了亏格2的部分结果及特征值重数的一般上界，改进了现有研究。

Holomorphic extensions of determinants of Laplacians

The Teichmuller space Teich(S) of a surface S in genus g>1 is a real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det'(Delta) on Teich(S) has a unique holomorphic extension to QF(S).

Geometry of the smallest 1-form Laplacian eigenvalue on hyperbolic manifolds

We relate small 1-form Laplacian eigenvalues to relative cycle complexity on closed hyperbolic manifolds: small eigenvalues correspond to closed geodesics no multiple of which bounds a surface of small genus. We describe potential applications of this equivalence principle toward proving optimal torsion homology growth in families of hyperbolic 3-manifolds Benjamini-Schramm converging to $\mathbb{H}^3.$

Small eigenvalues of the Conformal Laplacian

We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the $α$-genus.

Variational Properties Of The Second Eigenvalue Of The Conformal Laplacian

Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. Assume $[g]$ is a conformal class for which the Conformal Laplacian $L_g$ has at least two negative eigenvalues. We show the existence of a (generalized) metric that maximizes the second eigenvalue of $L_g$ over all conformal metrics (the first eigenvalue is maximized by the Yamabe metric). We also show that a maximal metric defines either a nodal solution of the Yamabe equation, or a harmonic map to a sphere. Moreover, we construct examples of each possibility.

On the second Robin eigenvalue of the Laplacian

We study the Robin eigenvalue problem for the Laplace-Beltrami operator on Riemannian manifolds. Our first result is a comparison theorem for the second Robin eigenvalue on geodesic balls in manifolds whose sectional curvatures are bounded from above. Our second result asserts that geodesic balls in nonpositively curved space forms maximize the second Robin eigenvalue among bounded domains of the same volume.

关于Witten-Laplacian的本征函数和节点集

On eigenfunctions and nodal sets of the Witten-Laplacian

本文研究了Witten-Laplacian的本征函数节点集性质，建立了Courant型定理并给出了闭本征值的乘重上界。

Two balls maximize the third Neumann eigenvalue in hyperbolic space

We show that the third eigenvalue of the Neumann Laplacian in hyperbolic space is maximal for the disjoint union of two geodesic balls, among domains of given volume. This extends a recent result by Bucur and Henrot in Euclidean space, while providing a new proof of a key step in their argument

Fricke-Macbeath曲面上的拉普拉斯算子谱

Spectrum of the Laplacian on the Fricke-Macbeath surface

本文研究亏格7的唯一Hurwitz曲面Fricke-Macbeath曲面上的拉普拉斯算子谱，证明其第一特征值有七重性且落在[1.23,1.26]区间，数值确定了对应自同构群的7维表示，还给出了相关Fuchsian群以0为中心的Dirichlet域顶点的代数坐标。

Bounds on Multiplicities of Laplace Operator Eigenvalues on Surfaces

In the present paper several bounds on multiplicities of eigenvalues of the Laplacian operator on surfaces are generalized from the case of either closed surface or simply-connected planar domain to the case of a surface of positive genus with holes.

黎曼曲面上旋量丛的伪拉普拉斯算子行列式与$\zeta^{(\rm reg)}(1)$

Determinants of pseudo-laplacians and $ζ^{({\rm reg})}(1)$ for spinor bundles over Riemann surfaces

本文研究紧黎曼曲面上埃尔米特线丛中Dolbeault拉普拉斯算子的自伴扩张，引入Robin质量概念并推导其显式表达式，以此描述标量拉普拉斯正则化$\zeta(1)$的Ricci流演化，还给出Morpurgo零亏格曲面相关结论的新证法。

退化双曲曲面与大亏格下的谱隙

Degenerating hyperbolic surfaces and spectral gaps for large genus

该研究考察大亏格双曲曲面的相邻 Laplace 特征值间隙，证明模空间中谱隙上确界在亏格趋于无穷时的下极限不小于 1/4，并建立退化情形的极小极大原理。

可定向曲面上Laplace算子的第一特征值

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。

Spectral theory for the Weil-Petersson Laplacian on the Riemann moduli space

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric $g_{\mathrm{WP}}$ on $\mathcal M_γ$, the Riemann moduli space of surfaces of genus $γ> 1$. This space has a singular compactification with respect to $g_{\mathrm{WP}}$, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete. The second theorem is a Weyl asymptotic formula for the counting function…

Simons' cone and equivariant maximization of the first $p$-Laplace eigenvalue

We consider an optimization problem for the first Dirichlet eigenvalue of the $p$-Laplacian on a hypersurface in $\mathbb{R}^{2n}$, with $n \ge 2$. If $p \ge 2n-1$, then among hypersurfaces in $\mathbb{R}^{2n}$ which are $O(n) \times O(n)$-invariant and have one fixed boundary component, there is a surface which maximizes the first Dirichlet eigenvalue of the $p$-Laplacian. This surface is either Simons' cone or a $C^1$ hypersurface, depending on $p$ and $n$. If $n$ is fixed and $p$ is large, then the maximizing surface is not Simons' cone. If $p=2$ and $n \le 5$, then Simons' cone does not ma…

具有交换边界拉普拉斯算子与Dirichlet-to-Neumann映射的曲面

Surfaces with Commuting Boundary Laplacian and Dirichlet-to-Neumann Map

本文研究了$d=2$情形下边界拉普拉斯算子与Dirichlet-to-Neumann映射可交换的曲面，证明了存在单参数族子流形满足此条件，并对亏格为0或边界有$k\geq 3$个连通分支的此类黎曼曲面进行了分类。

旋量丛上的拉普拉斯算子：自共轭扩展和正则化行列式

Laplacians in spinor bundles over translation surfaces: self-adjoint extentions and regularized determinants

本文研究了平坦共形度量下自旋丛上自共轭扩展的正则化行列式，并提出新的正则化方法。

嵌入曲面的特征值比界的一个证明

A Proof of the Eigenvalue Ratio Bound for Embedded Surfaces

本文研究了闭合嵌入曲面的谱与有界域狄利克雷谱的关系，证明了特征值比的下界，并确定了 genus 与最优常数的关系。

On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions

Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singularity of $\mathsf m$ or $\{\infty\}$. The pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ has conical singularities of angles $4π$ at the critical points of $f$ and other conical singularities that are the preimages of those of $\mathsf m$. We study the $ζ$-regularized determina…

Systole and $λ_{2g-2}$ of a hyperbolic surface

We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface $S$ of genus $g$, we get a geometric lower bound on ${λ_{2g-2}}(S)$: ${λ_{2g-2}}(S) > 1/4 + {ε_0}(S)$, where ${ε_0}(S) > 0$ is an explicit constant which depends only on the systole of $S$

短测地线与双曲面特征值的多重性

Short geodesics and multiplicities of eigenvalues of hyperbolic surfaces

本文研究了双曲面中短测地线数量与特征值多重性之间的关系，证明了在特定条件下特征值的多重性具有亚线性增长特性。

在双曲空间中带有负边界参数的Robin拉普拉斯算子的下界估计

Sharp estimates for the Robin Laplacian under a perimeter constraint in hyperbolic space

本文研究了双曲空间中Robin拉普拉斯算子的特征值问题，证明了测地球在特定条件下最大化特征值，并解决了相关开放问题。

双曲面的线性规划界限

Linear programming bounds for hyperbolic surfaces

本研究通过线性规划方法为双曲面提供了新的上界和下界，涉及系绳长、kissing 数和谱特性，适用于低亏格和高亏格情况。

离散曲面上拉普拉斯算子行列式的极值

Extremals of determinants for Laplacians on discrete surfaces

本文研究 Osgood-Phillips-Sarnak 定理的离散类比，证明离散 cotan 拉普拉斯算子的行列式在常离散高斯曲率度量处驻定，并通过数值计算发现这些驻点为极小值，与光滑情形相反。

子黎曼接触流形上磁场的谱性质

Spectral properties of magnetic fields on sub-Riemannian contact manifolds

研究三维接触流形上磁水平拉普拉斯算子的第一个特征值及其对流形拓扑结构的影响

Sur la multiplicité des valeurs propres du laplacien de Witten

On any compact manifold of dimension greater than 4, we prescribe the volume and any finite part of the spectrum of the Witten Laplacian acting on $p$-form for $0<p<n$. In particular, we prescribe the multiplicity of the first eigenvalues. On 3-dimensional manifolds, we give examples of multiple first eigenvalue for 1-forms, whose multiplicity depends on the maximal genus of embedded surfaces all of whose 1-cohomology is induced by the cohomology of the manifold. In particular, this multiplicity is at least 3.

On largeness and multiplicity of the first eigenvalue of hyperbolic surfaces

We apply topological methods to study the smallest non-zero number $λ_1$ in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set $\{S \in {\mathcal{M}_2}: {λ_1}(S) > 1/4 \}$ is unbounded and disconnects the moduli space ${\mathcal{M}_2}$.

Extremal eigenvalues of the Laplacian in a conformal class of metrics : the "conformal spectrum"

Let $M$ be a compact connected manifold of dimension $n$ endowed with a conformal class $C$ of Riemannian metrics of volume one. For any integer $k\geq0$, we consider the conformal invariant $λ_k ^c (C)$ defined as the supremum of the $k$-th eigenvalue $λ_k (g)$ of the Laplace-Beltrami operator $Δ_g$, where $g$ runs over $C$. First, we give a sharp universal lower bound for $λ_k ^c (C)$ extending to all $k$ a result obtained by Friedlander and Nadirashvili for $k=1$. Then, we show that the sequence $ \{λ_k ^c (C) \}$, that we call "conformal spectrum", is strictly increasing and satisfies, $\f…

Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, $Σ$, with genus $γ$ and $b$ boundary components, we maximize $σ_j(Σ,g) \ L(\partial Σ, g)$ over a class of s…

Sharp inequalities for determinants of Toeplitz operators and dbar-Laplacians on line bundles

We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang motivated by Arakelov geometry is obtained; applications to SU(2)-invariant determinantal random point processes on the two-sphere are also discussed. The inequalities are obtained as corollaries of a general theorem about the maximizers of a certain non-local functional defined on the space of all positively curved Hermitian metrics on an ample line bundle ov…

Index bounds for free boundary minimal surfaces of convex bodies

In this paper, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in $\mathbb{R}^3$ tends to infinity as its genus or the number of boundary components tends to infinity.

Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality

Let Σbe a compact surface of type (g, n), n > 0, obtained by removing n disjoint disks from a closed surface of genus g. Assuming χ(Σ)<0, we show that on Σ, the set of flat metrics which have the same Laplacian spectrum of Dirichlet boundary condition is compact in the C^\infty topology. This isospectral compactness extends the result of Osgood, Phillips, and Sarnak \cite{OPS3} for type (0,n) surfaces, whose examples include bounded plane domains. Our main ingredients are as following. We first show that the determinant of the Laplacian is a proper function on the moduli space of geodesically …

$S^3$中嵌入极小曲面的第一特征值

First eigenvalue of embedded minimal surfaces in $S^3$

该研究针对$S^3$中嵌入极小曲面的拉普拉斯算子第一特征值问题，证明其满足$λ_1≥1+ε_g$（$ε_g$为仅依赖曲面亏格的正常数），改进了Choi-Wang的已有结果。

Minimal surfaces in S^3: a survey of recent results

In this survey, we discuss various aspects of the minimal surface equation in the three-sphere S^3. After recalling the basic definitions, we describe a family of immersed minimal tori with rotational symmetry. We then review the known examples of embedded minimal surfaces in S^3. Besides the equator and the Clifford torus, these include the Lawson and Kapouleas-Yang examples, as well as a new family of examples found recently by Choe and Soret. We next discuss uniqueness theorems for minimal surfaces in S^3, such as the work of Almgren on the genus 0 case, and our recent solution of Lawson's …

Eigenvalue estimate and compactness for closed $f$-minimal surfaces

Let $Ω$ be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted Laplacian for closed embedded $f$-minimal hypersurfaces contained in $Ω$. Using this estimate, we prove a compactness theorem for the space of closed embedded $f$-minimal surfaces with the uniform upper bounds of genus and diameter in a complete $3$-manifold with Bakry-Émery Ricci curvature bounded below by a positive constant and admitting an exhaustion by bounded doma…

Metric uniformization and spectral bounds for graphs

We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate "Riemannian" metric to uniformize the geometry of the graph. In many interesting cases, the existence of such a metric is shown by examining the combinatorics of special types of flows. This involves proving new inequalities on the crossing number of graphs. In particular, we use our method to show that for any positive integer k, the kth smallest eigenvalue of the Laplacian on an n-vertex, bounded-degree planar graph is O(k/n). This bound is asymptotically tight…

曲面上的图的高阶Steklov特征值

Higher Steklov eigenvalues of graphs on surfaces

本文通过概率流的度量变形，研究了曲面上图的高阶Steklov特征值，并给出了其上界，同时推导了高阶拉普拉斯特征值的上界。

双曲自旋曲面谱隙的界

Bounds on spectral gaps of Hyperbolic spin surfaces

本文提出一种基于无限谱恒等式族和半定规划的方法，用于约束二维紧致双曲自旋流形与轨形的 Laplacian 和 Dirac 谱隙，获得了严格上界并确定了所有轨形谱隙集合的上界。

Weil-Petersson随机曲面的多项式速率谱间隙

Spectral gap with polynomial rate for Weil-Petersson random surfaces

该研究证明Weil-Petersson随机采样的亏格$g$闭双曲曲面拉普拉斯谱间隙至少为$\frac{1}{4}-O(\frac{1}{g^c})$（$g\to\infty$时概率趋近1），拓展并新证Anantharaman和Monk的结果，适配随机矩阵强收敛多项式方法至该场景。