In this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into $H^m \times R$. As an application, in dimension 2, we prove that a complete minimal surface with finite total extrinsic curvature has finite index. On the other hand, for stable, minimal surfaces in $H^3$ or in $H^2 \times \R$, we give an upper bound on the infimum of the spectrum of the Laplacian and on the volume growth.
In this paper, we prove new pinching theorems for the first eigenvalue of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for the first eigenvalue in terms of higher order mean curvatures. We show that under a suitable pinching condition, the hypersurface is diffeomorpic and almost isometric to a standard sphere. Moreover, as a corollary, we show that a hypersurface of the Euclidean space which is almost Einstein is diffeomorpic and almost isometric to a standard sphere.
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.
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 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.
In this work we characterize certain immersed closed hypersurfaces of some ambient manifolds via the second eigenvalue of the Jacobi operator. First, we characterize the Clifford torus as the surface which maximizes the second eigenvalue of the Jacobi operator among all closed immersed orientable surfaces of $\mathbb S^3$ with genus bigger than zero. After, we characterize the slices of the warped product $I\times_h\mathbb S^n$, under a suitable hypothesis on the warping function $h:I\subset\mathbb R\to\mathbb R$, as the only hypersurfaces which saturate a certain integral inequality involving…
In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on the compact hypersurfaces of ambient spaces with bounded sectional curvature. As application we deduce rigidity results for stable constant mean curvature hypersurfaces $M$ of these spaces $N$. Indeed, we prove that if $M$ is included in a ball of radius small enough then the Hausdorff-distance between $M$ and a geodesic sphere $S$ of $N$ is small. Moreover $M$ is diffeomorphic and quasi-isometric to $S$. As other application, we give rigidity results for almost umbilic hypersurfaces.
The purpose of this paper is to study a complete orientable minimal hypersurface with finite index in an $(n+1)$-dimensional Riemannian manifold $N$. We generalize Theorems 1.5-1.6 (\cite{Seo14}). In 1976, Schoen and Yau proved the Liouville type theorem on stable minimal hypersurface, i.e., Theorem 1.7 (\cite{SchoenYau1976}). Recently, Seo (\cite{Seo14}) generalized Theorem 1.7 (\cite{SchoenYau1976}). Finally, we generalize Theorems 1.7 (\cite{SchoenYau1976}) and 1.8 (\cite{Seo14})
Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on the p-th Jacobi eigenvalue for p greater or equal than 2 one gains strong convergence to a smooth limit submanifold away from at most p-1 points.
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.
Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $\int_MH dv$ of the mean curvature $H$. In this paper, we derive an optimal upper bound for the second eigenvalue of the Jacobi operator $J_s$ of $M$. Moreover, when $r>1$, the bound is attained if and only if $M$ is totally umbilical and non-totally geodesic, when $r=1$, the bound is attained if $M$ is the Riem…
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 prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to $r$-stability as well as to almost-Einstein hypersurfaces.
In this paper, we give pinching Theorems for the first nonzero eigenvalue $λ$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is 1 then, for any $ε>0$, there exists a constant $C\_ε$ depending on the dimension $n$ of $M$ and the $L\_{\infty}$-norm of the mean curvature $H$, so that if the $L\_{2p}$-norm $\|H\|\_{2p}$ ($p\geq 2$) of $H$ satisfies $n\|H\|\_{2p}-C\_ε<λ$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/λ)^{1/2}$ is smaller than $ε$. Furthermore, we prove that if $C$ is a small enough constant…
We consider a solution f of a certain Dirichlet Problem on a domain in $S^{(n+1)}$ whose boundary is a minimal hypersurface and we prove a Poincare type inequality for f. One have equality iff Yau's conjecture about the first non-zero eigenvalue of closed minimal hypersurfaces of $S^{(n+1)}$ is true.
Given a smooth compact hypersurface $M$ with boundary $Σ=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $σ_k(M_j)$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_j$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.
The eigenvalue problem for the Sen--Witten operator on closed spacelike hypersurfaces is investigated. The (square of its) eigenvalues are shown to be given exactly by the 3-surface integral appearing in the expression of the total energy-momentum of the matter+gravity systems in Witten's energy positivity proof. A sharp lower bound for the eigenvalues, given in terms of the constraint parts of the spacetime Einstein tensor, i.e. the energy and momentum densities of the matter fields, is given.
Let $M$ be a closed hypersurface in a simply connected rank-1 symmetric space $\olm$. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of $M$ in terms of the Ricci curvature of $\olm$ and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of $M$.
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.
Let $M^n$ be a closed convex hypersurface lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close and almost isometric to a geodesic sphere $S(p_0,R_0)$ in $N$, but also its enclosed domain is $C^{1,α}$-close to a geodesic ball of constant curvature.
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.
In this paper, we investigate the first eigenvalues of two closed eigenvalue problems of the bi-Beltrami-Laplacian on minimal embedded isoparametric hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$. Although many mathematicians want to derive the corresponding results for the first eigenvalues of bi-Beltrami-Laplacian, they encountered great difficulties in proving the limit theorem of the version of bi-Beltrami-Laplacian along with the strategy due to I. Chavel and E. A. Feldman(Journal of Functional Analysis, 30 (1978), 198-222) and S. Ozawa (Duke Mathematics Journal, 48 (1981),767-778)…
Let $M^n$ be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose $\mathfrak{a}$ is a symmetric $(0,2)$ tensor field whose dual $(1,1)$ tensor $\mathcal{A}$ has $n$ distinct eigenvalues, and $\mathrm{tr}(\mathcal{A}^k)$ are constants for $k=1,\cdots, n-1$. We show that all the eigenvalues of $\mathcal{A}$ are constants, generalizing a theorem of de Almeida and Brito \cite{dB90} to higher dimensions. As a consequence, a closed hypersurface $M^n$ in $S^{n+1}$ is isoparametric if one takes $\mathfrak{a}$ above to be the second fundamental form, giving…
Let $M^{2n-1}$ be the smooth boundary of a bounded strongly pseudo-convex domain $Ω$ in a complete Stein manifold $V^{2n}$. Then (1) For $n \ge 3$, $M^{2n-1}$ admits a pseudo-Eistein metric; (2) For $n \ge 2$, $M^{2n-1}$ admits a Fefferman metric of zero CR Q-curvature; and (3) for a compact strictly pseudoconvex CR embeddable 3-manifold $M^3$, its CR Paneitz operator $P$ is a closed operator.
We prove a new upper bound for the first eigenvalue of the Dirac operator of a compact hypersurface in any Riemannian spin manifold carrying a non-trivial twistor spinor without zeros on the hypersurface. The upper bound is expressed as the first eigenvalue of a drifting Schrödinger operator on the hypersurface. Moreover, using a recent approach developed by O. Hijazi and S. Montiel, we completely characterize the equality case when the ambient manifold is the standard hyperbolic space.
In this paper, we would like to give an answer to \textbf{Problem 1} below issued firstly in [J. Mao, Eigenvalue estimation and some results on finite topological type, Ph.D. thesis, IST-UTL, 2013]. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced mean curvature flow considered here, we can obtain that the first eigenvalues of the Laplace and the $p$-Laplace operators are monotonic under this flow. Surprisingly, during this process, we get an interesting byproduct, that is, without any complicate…
This is Part 1 of two papers where we develop the basic potential theory of elliptic operators on posssibly singular almost minimzers using their hyperbolic unfoldings. We can establish surprisingly robust boundary Harnack inequalities along the singular set. We apply them to derive a Martin theory and solve classical boundary value problems and we also consider eigenvalue problems and variational problems on such singular spaces.
We obtain sharp estimates involving the mean curvatures of higher order of a complete bounded hypersurface immersed in a complete Riemannian manifold. Similar results are also given for complete spacelike hypersurfaces in Lorentzian ambient spaces.
In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface $M$ in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on $M$. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.
We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface $M$ has sufficiently small total scalar curvature then $M$ has only one end. We also obtain a vanishing theorem for $L^2$ harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minim…
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…
We prove that the index is bounded from below by a linear function of its first Betti number for any compact free boundary $f$-minimal hypersurface in certain positively curved weighted manifolds.
Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for the usual Dirac operator. The limiting cases are also investigated.
We study the isometric spacelike embedding problem in scaled de Sitter space, and obtain Weyl-type estimates and the corresponding closedness in the space of embeddings.
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.
We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.
For a convex domain $D$ bounded by the hypersurface $\partial D$ in a space of constant curvature we give sharp bounds on the width $R-r$ of a spherical shell with radii $R$ and $r$ that can enclose $\partial D$, provided that normal curvatures of $\partial D$ are pinched by two positive constants. Furthermore, in the Euclidean case we also present sharp estimates for the quotient $R/r$. From the obtained estimates we derive stability results for almost umbilical hypersurfaces in the constant curvature spaces.
We prove $ε$-closeness of hypersurfaces to a sphere in Euclidean space under the assumption that the traceless second fundamental form is $δ$-small compared to the mean curvature. We give the explicit dependence of $δ$ on $ε$ within the class of uniformly convex hypersurfaces with bounded volume.
