Related papers: First stability eigenvalue of singular minimal hyp…

Given $k\in \mathbb{R},$ $v,$ $D>0,$ and $n\in \mathbb{N},$ let $\left\{ M_{\alpha }\right\} _{\alpha =1}^{\infty }$ be a Gromov-Hausdorff convergent sequence of Riemannian $n$--manifolds with sectional curvature $\geq k,$ volume $>v,$ and…

We study the uniqueness of horospheres and equidistant spheres in hyperbolic space under different conditions. First we generalize the Bernstein theorem by Do Carmo and Lawson to the embedded hypersurfaces with constant higher order mean…

In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a…

We study topological lower bounds on the number of small Laplacian eigenvalues on hyperbolic surfaces. We show there exist constants $a,b>0$ such that when $(g+1)<a\frac{n}{\log n}$, any hyperbolic surface of genus-$g$ with $n$ cusps has at…

In recent decades qualitative inverse scattering methods with eigenvalues as target signatures received much attention. To understand those methods a knowledge on the properties of the related eigenvalue problems is essential. However, even…

Following the strategy developed by Chodosh, Li, Minter and Stryker, and using the volume estimate of Antonelli and Xu, we prove that, in $\mathbb R^6$, a complete, two-sided, stable minimal hypersurfaces is flat.

We show several results comparing sharp eigenvalue bounds for the first Steklov eigenvalue on surfaces under change of the topology. Among others, we obtain strict monotonicity in the genus. Combined with results of the second named author…

In a previous work we apply lattice point theorems on hyperbolic spaces obtaining asymptotic formulas for the number of integral representations of negative integers by quadratic and hermitian forms of signature (n,1) lying in Euclidean…

For three dimensional complete Riemannian manifolds with scalar curvature no less than one, we obtain the sharp upper bound of complete stable minimal surfaces' diameter.

An upper bound on the first S^1 invariant eigenvalue of the Laplacian for invariant metrics on the 2-sphere is used to find obstructions to the existence of isometric embeddings of such metrics in (R^3,can). As a corollary we prove: If the…

We introduce a new flatness index for the boundary of an open subset $\Omega$ of $\mathbb{R}^n$, $n\ge 2$. This index provides a necessary condition for $\partial\Omega$ to be a harmonic pseudosphere and sufficient conditions for a harmonic…

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent…

In this paper, we study $n$-dimensional hypersurfaces with constant $m^{\text{th}}$ mean curvature $H_m$ in a unit sphere $S^{n+1}(1)$ and prove that if the $m^{\text{th}}$ mean curvature $H_m$ takes value between $\dfrac{1}{(\tan…

We investigate stability of (2+1)-dimensional ring solitons of the nonlinear Schrodinger equation with focusing cubic and defocusing quintic nonlinearities. Computing eigenvalues of the linearised equation, we show that rings with spin…

In this paper, continuing our previous work, we investigate the third gap problem in the Simon conjecture for closed minimal surfaces in the unit sphere. By developing refined third-order Simons-type integral identities and establishing new…

Given a compact surface with boundary, we introduce a family of functionals on the space of its Riemannian metrics, defined via eigenvalues of a Steklov-type problem. We prove that each such functional is uniformly bounded from above, and…

This work deals with theoretical and numerical aspects related to the behavior of the Steklov-Lam\'e eigenvalues on variable domains. After establishing the eigenstructure for the disk, we prove that for a certain class of Lam\'e…

Let $ \Omega \subsetneq \mathbf{R}^n\,(n\geq 2)$ be an unbounded convex domain. We study the minimal surface equation in $\Omega$ with boundary value given by the sum of a linear function and a bounded uniformly continuous function in $…

For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here ``eigenvalue''…

We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eingenvalue problem. In order to improve the stability of the plate, it seems…