We give lower bounds for the first Dirichilet eigenvalues for domains in submanifolds with locally bounded mean curvatures. These bounds depend on the injectivity radius, sectional curvature (upperbound) of the ambient space and on the mean curvature of the submanifold. For submanifolds fo Hadamard manifolds these lower bounds depend only on the dimension and mean curvature of the submanifold.
We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is isospectral to the disk.
We obtain upper bounds for the Steklov eigenvalues $σ_k(M)$ of a smooth, compact, connected, $n$-dimensional submanifold $M$ of Euclidean space with boundary $Σ$ that involve the intersection indices of $M$ and of $Σ$. One of our main results is an explicit upper bound in terms of the intersection index of $Σ$, the volume of $Σ$ and the volume of $M$ as well as dimensional constants. By also taking the injectivity radius of $Σ$ into account, we obtain an upper bound that has the optimal exponent of $k$ with respect to the asymptotics of the Steklov eigenvalues as $k \to \infty$.
We establish inequalities for the eigenvalues of Schrödinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, Pólya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schrödinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as…
We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second, we consider complex submanifolds of the complex projective space $\mathbb{C} P^N$ instead of submanifolds of $\mathbb{R}^N$ and we obtain an eigenvalue upper bound depending only on the dimension of the submanifold which is sharp for the first non-zero eigenvalue.
In this note, we obtain the sharp estimates for the first eigenvalue of Paneitz operator for $4$-dimensional compact submanifolds in Euclidean space. Since unit spheres and projective spaces can be canonically imbedded into Euclidean space, the corresponding estimates for the first eigenvalue are also obtained.
We get optimal lower bounds for the eigenvalues of the submanifold Dirac operator on locally reducible Riemannian manifolds in terms of intrinsic and extrinsic expressions. The limiting-cases are also studied. As a corollary, one gets several known results in this direction.
We give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type $N^n \times_f Q^q$, where $f \in C^\infty(N)$. We also study the essential spectrum of these minimal submanifolds.
We prove Reilly-type upper bounds for divergence-type operators of the second order as well as for Steklov problems on submanifolds of Riemannian manifolds of bounded sectional curvature endowed with a weighted measure.
The purpose of the present paper is to study the differential geometric properties of a quaternion CR-submanifold in a locally conformal quaternion Kaehler manifold.
We introduce warped product skew semi-invariant submanifolds of order $1$ of a locally product Riemannian manifold. We give a necessary and sufficient condition for skew semi-invariant submanifold of order 1 to be a locally warped product. We also prove that the invariant distribution which is involved in the definition of the submanifold is integrable under some restrictions. Moreover, we find an inequality between the warping function and the squared norm of the second fundamental form for such submanifolds. Equality case is also discussed.
In the present paper, we study hemi-slant submanifolds of a locally product Riemannian manifold. We prove that the anti-invariant distribution which is involved in the definition of hemi-slant submanifold is integrable and give some applications of this result. We get a necessary and sufficient condition for a proper hemi-slant submanifold to be a hemi-slant product. We also study this type submanifolds with parallel canonical structures. Moreover, we give two characterization theorems for the totally umbilical proper hemi-slant submanifolds. Finally, we obtain a basic inequality involving Ric…
In this paper, we investigate submanifolds with locally bounded mean curvature in Hadamard manifolds, product manifolds $N\times\mathbb{R}$, submanifolds with bounded $φ$-mean curvature in the hyperbolic space, and successfully give lower bounds for the weighted fundamental tone and the first eigenvalue of the $p$-Laplacian.
In this paper we present not only some properties related to bi-warped product submanifolds of locally conformal almost cosymplectic manifolds, but also we show how the squared norm of the second fundamental form and the bi-warped product's warping functions are related when the bi-warped product submanifold has a proper slant submanifold as a base or fiber.
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…
By means of a family of counter-examples, it is shown that the Reilly upper bound for the first eigenvalue of the Laplace operator for a compact submanifold in Euclidean space does not work for $n$-dimensional compact spacelike submanifolds of Lorentz-Minkowski spacetime $\mathbb{L}^m$, $m\geq n+2$. We develop a new suitable technique, based on an integral formula on compact spacelike sections of the light cone in $\mathbb{L}^m$. Then, a family of extrinsic upper bounds for the first eigenvalue of the Laplace operator for a compact spacelike submanifold in $\mathbb{L}^m$ is proved. For each on…
We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model centers of the tube. The bounds are sharp on these models. Moreover, when the models used are $CP^q$ or the complex hyperquadric, these bounds also give gap phenomena and comparison results.
In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of the locally symmetric manifold.
We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $ N \times \mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} \leq κ$. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces has positive fundamental tone.
The purpose of the present paper is to show that the components of the unit normal of any minimal surface with free boundary in the unit ball, are eigenfunctions associated with the eigenvalue $-2$, for some (new) natural eigenvalue problem for the Jacobi operator; this fact has analytic (spectral) consequences for free boundary minimal surfaces in the unit $3$-ball of index $4$, and might prove useful in order to characterize these. This is also in strong analogy with the case of minimal surfaces in $S^3$.
In this paper, we estimate the eigenvalues of the twisted Dirac operator on Kähler submanifolds of the complex projective space $CP^m$ and we discuss the sharpness of this estimate for the embedding $CP^d \hookrightarrow CP^m$.
In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.
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 …
We obtain inequalities for all Laplace eigenvalues of Riemannian manifolds with an upper sectional curvature bound, whose rudiment version for the first Laplace eigenvalue was discovered by Berger in 1979. We show that our inequalities continue to hold for conformal metrics, and moreover, extend naturally to minimal submanifolds. In addition, we obtain explicit estimates for Laplace eigenvalues of minimal submanifolds in terms of geometric quantities of the ambient space.
Our paper is an attempt to to verify the Chen's conjecture on biharmonic submanifolds and to classify biconservative submanifolds. In doing so we provide an affirmative answer to Chen's conjecture on biharmonic submanifolds. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $\mathbb{E}_{1}^{n+1}$ having complex eigenvalues has constant mean curvature. Moreover, every biharmonic Lorentz hypersurface $M_{1}^{n}$ having complex eigenvalues in $\mathbb{E}_{1}^{n+1}$ must be minimal.
Given a connected, compact, totally geodesic submanifold Y^m of noncompact type inside a compact locally symmetric space of noncompact type X^n, we provide a sufficient condition that ensures that [Y^m] is nonzero in H_m(X^n; R); in low dimensions, our condition is also necessary. We provide conditions under which there exist a tangential map of pairs from a finite cover (X-bar,Y-bar) to the nonnegatively curved duals (X_u,Y_u).
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 shall prove a classification theorem for immersed umbilic-free submanifolds in $\mathbb S^n$ with a parallel Blaschke tensor. For proving this classification, some new kinds of examples are first defined.
In this paper, we study deformations of coisotropic submanifolds in a locally conformal symplectic manifold. Firstly, we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. Secondly, we prove that the formal deformation problem is governed by an $L_\infty$-structure which is a $\frak b$-deformation of strong homotopy Lie algebroids introduced in Oh and Park (2005) in the symplectic context. Then we study deformations of locally conformal symplectic…
Recall that Federer-Fleming defined the notion of flat convergence of submanifolds of Euclidean space to solve the Plateau problem. Here we prove the upper semicontinuity of Neumann eigenvalues of the submanifolds when they converge in the flat sense without losing volume. With an additional condition on the boundaries of the submanifolds we prove the Dirichlet eigenvalues are semicontinuous as well. We show this additional boundary condition is necessary as well as the condition that the volumes converge to the volume of the limit submanifold. As an application of our theorems we see that the…
We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally geodesic subvarieties. As a corollary, we obtain that on a complex surface which is a compact quotient of the bidisc or of the 2-ball, there is at most a finite number of totally geodesic curves with negative self intersection. More generally, we prove that there are only finitely many exceptional totally geodesic divisors in a compact Hermitian locally symmetric …
In this paper, a correspondence via duality is established between the set of locally strongly convex symmetric equiaffine hyperspheres and the set of minimal symmetric Lagrangian submanifolds in a certain complex space form. By using this correspondence theorem, we are able to provide an alternative proof of the classification theorem for the locally strongly convex equiaffine hypersurfaces with parallel Fubini-Pick forms, which has been established recently by Z.J. Hu etc in a totally different way.
In this paper, we firstly consider Dirichlet eigenvalue problem which is related to Xin-Laplacian on the bounded domain of complete Riemannian manifolds. By establishing the general formulas, combining with some results of Chen and Cheng type, we prove some eigenvalue inequalities. As some applications, we consider the eigenvalues on some Riemannian manifolds admitting with special functions, the translating solitons, minimal submanifolds on the Euclidean spaces, submanifolds on the unit spheres, projective spaces and so on. In particular, for the case of translating solitons, some eigenvalue …
We prove Reilly-type upper bounds for the first non-zero eigenvalue of the Steklov problem associated with the $p$-Laplace operator on submanifolds of manifolds with sectional curvature bounded form above by a non-negative constant.
In this paper, we study some properties of warped product pointwise bi-slant submanifolds in locally metallic Riemannian manifolds and we construct some examples in Euclidean spaces.
We give some fundamental properties of the induced structures on submanifolds immersed in almost product or locally product Riemannian manifolds. We study the induced structure by the composition of two isometric immersions on submanifolds in an almost product Riemannian manifold. We give an effective construction for some induced structures on submanifolds of codimension 1 or 2 in Euclidean space.
We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.
We establish an upper estimate for the small eigenvalues of the twisted Dirac operator on Kahler submanifolds in Kahler manifolds carrying Kahlerian Killing spinors. We then compute the spectrum of the twisted Dirac operator of the canonical embedding \CP^d \rightarrow \CP^n in order to test the sharpness of the upper bounds
For $n\geq 7$, we give the optimal estimate for the second eigenvalue of Paneitz operators for compact $n$-dimensional submanifolds in an $(n+p)$-dimensional space form.
For an $n$-dimensional compact submanifold $M^n$ in the Euclidean space $\mathbf R^{N}$, we study estimates for eigenvalues of the Paneitz operator on $M^n$. Our estimates for eigenvalues are sharp.
In this note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.
