Jarolim Bures, Vladimir Soucek
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.
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We show that under some non-degeneracy assumption the only submersive harmonic morphism on a conformally flat $3-$sphere is the Hopf fibration. The proof involves an appropriate use the Chern-Simons functional.
In this paper we provide a sharp characterization of the smooth four-dimensional sphere. The assumptions of the theorem are conformally invariant, and can be reduced to an L^2 inequality of the Weyl tensor and positivity of the Yamabe invariant.
Universal inequalities for eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds
本文研究了共形平坦黎曼流形上Dirichlet Laplacian的特征值问题，并在双曲空间$\mathbb{H}^n(-1)$上建立了特征值的普遍不等式。
In this paper we prove that under a lower bound on the Ricci curvature and an asymptotic assumption on the scalar curvature, a complete conformally compact manifold $(M^{n+1},g)$, with a pole $p$ and with the conformal infinity in the conformal class of the round sphere, has to be the hyperbolic space.
The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere. The conformal structure of generic conformally flat (local-)hypersurfaces is characterized as conformally flat (local-)3-metrics with the Guichard condition. Then, there is a certain class of orthogonal analytic (local-)Riemannian 2-metrics with constant Gauss curvature -1 such that any 2-metric of the class gives rise to a one-parameter family of conformally flat 3-metrics with the Guichard condition. In this paper, w…
We give explicit formulas for all odd order differential intertwinors on the subbundle of the bundle of spinor-$k$-forms that are annihilated by the Clifford multiplication over the odd dimensional standard sphere. The Dirac and Rarita-Schwinger operators appear in the case of $k=0$ and $k=1$, respectively.
Let (M,g) be a compact Riemannian manifold of dimension >2. We show that there is a metric h conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with repect to h is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension >1.
We generalized Xiang, Qi and Wei's results on the M-eigenvalues of Riemann curvature tensor to higher dimensional conformal flat manifolds. The expression of M-eigenvalues and M-eigenvectors are found in our paper. As a special case, M-eigenvalues of conformal flat Einstein manifold have also been discussed, and the conformal the invariance of M-eigentriple has been found. We also discussed the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold. We proved that the M-eigenvalue can determine the Riemann curvature tensor uniquely and generalize the real M-eigenval…
We study some conformally invariant integral equations using the method of moving spheres.
E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in S^4 is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in S^4 and their loop group symmetries. We also generalise these results to conformally flat n-immersions in (2n-2)-spheres with flat normal bundle and co…
Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $Σ$ be a compact connected and orientable surface immersed in $M$ which is a stable constant mean curvature (CMC) surface or an index one minimal surface. We prove that $Σ$ is homeomorphic either to a sphere or to a torus. Moreover, in case $Σ$ is homeomorphic to a torus, then it is embedded, minimal, conformal to a flat square torus and Ric$(N)=0$ where $N$ is a unit …
Biharmonic conformal immersions into a 3-dimensional conformally flat space
本文研究曲面到共形平坦3-空间的双调和共形浸入，给出全脐曲面的刻画与构造方法，分类一族共形浸入中的双调和映射，并构造包括Hopf柱面在内的大量新例子。
We construct pairs of conformally equivalent isospectral Riemannian metrics $ϕ_1 g$ and $ϕ_2 g$ on spheres $S^n$ and balls $B^{n+1}$ for certain dimensions $n$, the smallest of which is $n=7$, and on certain compact simple Lie groups. In the case of Lie groups, the metric $g$ is left-invariant. In the case of spheres and balls, the metric $g$ is not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds $(M,g)$ we also show that the functions $ϕ_1$ and $ϕ_2$ are isospectral potentials for the Schrödinger operator $\hbar^2Δ+ϕ$. To our knowledge, thes…
Let $(M,g)$ be a compact Riemannian manifold and $P_g$ an elliptic, formally self-adjoint, conformally covariant operator of order $m$ acting on smooth sections of a bundle over $M$. We prove that if $P_g$ has no rigid eigenspaces (see Definition 2.2), the set of functions $f \in C^\infty(M, R)$ for which $P_{e^fg}$ has only simple non-zero eigenvalues is a residual set in $C^\infty(M,R)$. As a consequence we prove that if $P_g$ has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics in the $C^m$-topology. We also prove that t…
In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poin-car{é}-Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof of the rigidity of the hyper-bolic space as the only conformally compact Poincar{é}-Einstein manifold with the round sphere as its conformal infinity.
Conformally critical metrics and optimal bounds for Dirac eigenvalues on spin surfaces
本文研究了在固定共形类下Dirac算子本征值的极小化问题，推导了闭合流形上极小解存在的条件，并推导了二维球面上Dirac算子的等周不等式及其共形谱特征。
We prove the Hijazi inequality, an estimate for Dirac eigenvalues, for complete manifolds of finite volume. Under some additional assumptions on the dimension and the scalar curvature, this inequality is also valid for elements of the essential spectrum. This allows to prove the conformal version of the Hijazi inequality on conformally parabolic manifolds if the spin analog to the Yamabe invariant is positive.
In this paper we study bounds for the first eigenvalue of the Paneitz operator $P$ and its associated third-order boundary operator $B^3$ on four-manifolds. We restrict to orientable, simply connected, locally confomally flat manifolds that have at most two umbilic boundary components. The proof is based on showing that under the hypotheses of the main theorems, the considered manifolds are confomally equivalent to canonical models. This equivalence is proved by showing the injectivity of suitable developing maps. Then the bounds on the eigenvalues are obtained through explicit computations on…
We study integral curvature conditions for a Riemannian metric $g$ on $S^4$ that quantify the best bilipschitz constant between $(S^4,g)$ and the standard metric on $S^4$. Our results show that the best bilipschitz constant is controlled by the $L^2$-norm of the Weyl tensor and the $L^1$-norm of the $Q$-curvature, under the conditions that those quantities are sufficiently small, $g$ has a positive Yamabe constant and the $Q$-curvature is mean-positive. The proof of the result is achieved in two steps. Firstly, we construct a quasiconformal map between two conformally related metrics in a posi…
In this paper, we prove a classification theorem of 4-manifolds according to some conformal invariants, which generalizes the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}. Moreover, it provides a four-dimensional analogue of the well-known classification theorem of Schoen-Yau \cite{SY2} on 3-manifolds with positive Yamabe invariants.
The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in $\R^{n+2}$ and those which are not conformally equivalent to a minimal surface in $\R^{n+2}$. On the level of their conformal Gauss maps into $Gr_{1,3}(\R^{1,n+3})=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ these two classes of Willmore immersions into $S^{n+2}$ correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of $\R^{1,n+3}$, contains a fixed lightlik…
We consider the generalization of classical Blaschke's Problem to higher codimension case, characterizing Darboux pair of isothermic surfaces and dual S-Willmore surfaces as the only non-trivial surface pairs that envelop a 2-sphere congruence and conformally correspond to each other. When the sphere congruence is the mean curvature spheres of one envelop surface, it must be a cmc-1 surface in hyperbolic 3-space, or a S-Willmore surface. A study of conformally immersed surface pairs is indicated with discussion on the geometric meaning of new invariants.
The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this problem has been recently achieved by Nadirashvili-Sire and Petrides using related, though different methods. In particular, it was shown that for a given $k$, the maximum of the $k$-th Laplace eigenvalue in a conformal class on a surface is either attained on a metric which is smooth except possibly at a finite number of conical singularities, or it is attained in the limit while a "bubble tree" is formed on a surfa…
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.
This paper is devoted to the study of minimal immersions of flat $n$-tori into spheres, especially those immersed by the first eigenfunctions (such immersion is called $λ_1$-minimal immersion), which also play important roles in spectral geometry. It is known that there are only two non-congruent $λ_1$-minimal $2$-tori in spheres, which are both flat. For higher dimensional case, the Clifford $n$-torus in $\mathbb{S}^{2n-1}$ might be the only known example in the literature. In this paper, by discussing the general construction of homogeneous minimal flat $n$-tori in spheres, we construct many…
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 $n$.
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.
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。
The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions, mutatis mutandis, in the standard construction of the KP hierarchy equations and solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies its relationship with the theory of quaternionic holomorphic 2-tori in HP^1 (which are equivalent to conformally immersed 2-tori in S^4). After describing how the Sato-Segal-Wilson construction of KP solutions (particularly solutions of finite type) carries over to this quaternio…
Let $M^n$ be an $n$-dimensional complete and locally conformally flat hypersurface in the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature $n(n-1)$. We show that if the total curvature $\left( \int _ { M } | H | ^ { n } d v \right) ^ { \frac { 1 } { n } }$ of $M$ is sufficiently small, then $M^n$ is totally geodesic.
We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincaré manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship generalizes the result for hypersurfaces in ${\H}^{n+1}$ and their connection to the conformal geometry of ${\SS}^n$ as exhibited in [7] and gives a correspondence between Weingarten hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant equations on the conformal inf…
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。
The defining equations for Killing vector fields and conformal Killing vector fields are overdetermined systems of PDE. This makes it difficult to solve the systems numerically. We propose an approach which reduces the computation to the solution of a symmetric eigenvalue problem. The eigenvalue problem is then solved by finite element techniques. The formulation itself is valid in any dimension and for arbitrary compact Riemannian manifolds. The numerical results which validate the method are given in two dimensional case.
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。
We extend the sphere theorem of \cite{CGY03} to give a conformally invariant characterization of $(\mathbb{CP}^2, g_{FS})$. In particular, we introduce a conformal invariant $β(M^4,[g]) \geq 0$ defined on conformal four-manifolds satisfying a `positivity' condition; it follows from \cite{CGY03} that if $0 \leq β(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result of this paper is a `gap' result showing that if $b_2^{+}(M^4) > 0$ and $4 \leq β(M^4,[g]) < 4(1 + ε)$ for $ε> 0$ small enough, then $M^4$ is diffeomorphic to $\mathbb{CP}^2$. The Ricci flow is used in a crucial way to…
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。
$p$-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…
We study the non-embddability property for a class of real hypersurfaces, called real hypersurfaces of involution type, into the sphere in the low codimensional case, by making use of property of a naturally related Gauss curvature. We also study rigidity problems for conformal maps between a class of Kähler manifolds with pseud-conformally flat metrics.
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。
Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds
本文研究局部共形平坦4-流形到球面的共形调和映射，利用共形不变能量泛函的凸性证明了定量唯一性结果，并为此建立了流形上的二阶 Hardy 不等式。
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.
In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with. This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$…
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.
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。
Conformally covariant differential symmetry breaking operators for a vector bundle of rank 3 over S^3
本文研究了三维球面上的三维向量丛与二维球面线丛之间微分对称破缺算子的构造与分类，确定了参数(λ, ν, m)的存在条件。
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。
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。
We present explicit formulas for the spectra of higher spin operators on the subbundle of the bundle of spinor-valued trace free symmetric tensors that are annihilated by the Clifford multiplication over the standard sphere in odd dimension. In even dimensional case, we give the spectra of the square of such operators. The Dirac and Rarita-Schwinger operators are zero-form and one-form cases, respectively. We also give eigenvalue formulas for the conformally invariant differential operators of all odd orders on the subbundle of the bundle of spinor-valued forms that are annihilated by the Clif…
In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is non-negative definite.
We consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the Ricci tensor to be a constant. It is related to the notion of minimal volumes in comparison geometry. Such a metric with the smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric of volume in a suitable sense in the conformal class. The problem is reduced to solve a Pucci type equation with respect to the Schouten tensor. We establish a local gradient estimate for this type of conformally invariant fully nonlinear uniform elliptic equations. Combining it with the the…
We give a complete classification of tangential bidifferential operators of total order at most $n$ which are expressed purely in terms of the Laplacian on the ambient space of an $n$-dimensional manifold. This gives a curved analogue of the classification, due to Ovsienko--Redou and Clerc, of conformally invariant bidifferential operators on the sphere. As an application, we construct a large class of formally self-adjoint conformally invariant differential operators.
Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does, and that their values explode if a knot degenerates to a singular knot with double points.
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.
Conformal Loxodromes
本文在莫比乌斯几何中定义了共形斜航线的曲率模拟，并推导了其满足的五阶不变常微分方程。
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 …
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