Related papers: Partial result of Yau's Conjecture of the first ei…

After Chern's conjecture on the discreteness of the constant scalar curvatures of compact minimal submanifolds $M^n$ in unit spheres $\mathbb{S}^{n+q}$, Z. Q. Lu proposed a conjecture regarding the second gap, based on his ingenious…

A well-known conjecture of Yau states that the area of one of Clifford minimal hypersurfaces $S^k\big{(}\sqrt{\frac{k}{n}}\, \big{)}\times S^{n-k}\big{(}\sqrt{\frac{n-k}{n}}\, \big{)}$ gives the lowest value of area among all non-totally…

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 $\phi$…

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…

A minimal immersion from a surface to $S^3$ can be viewed both as a critical point of the area and of the energy. Although no difference appears at first order, looking at the respective second variations unveils significant differences. It…

In 1980 Yang and Yau~\cite{YY} proved the celebrated upper bound for the first eigenvalue on an orientable surface of genus $\gamma$. Later Li and Yau~\cite{LY} gave a simple proof of this bound by introducing the concept of conformal…

This is a survey on the recent progress in several applications of isoparametric theory, including an affirmative answer to Yau's conjecture on the first eigenvalue of Laplacian in the isoparametric case, a negative answer to Yau's 76th…

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…

Upper bounds of the first non-trivial eigenvalue $\lambda_1$ of the Laplace operator of a compact submanifold $M^n$ of Euclidean space $\R^{m+1}$, by means of a new technique, are obtained. Each of the upper bounds of $\lambda_1$ depends on…

In this article we prove that entire critical points $(u,\nabla)$ of the self-dual $U(1)$-Yang-Mills-Higgs functional $E_1$, with energy $$E_1(u,\nabla;B_R):=\int_{B_R}\left[|\nabla…

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…

We derive a sharp upper bound for the first eigenvalue $\lambda_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<\infty$. We then prove that a particular class of conformally compact submanifolds within…

Let $M\subset S^{n+1}\subset\mathbb{R}^{n+2}$ be a compact minimal hypersurface of the $n$-dimensional Euclidean unit sphere. Let us denote by $|A|^2$ the square of the norm of the second fundamental form and $J(f)=-\Delta f-nf-|A|^2f$ the…

In this paper, we give pinching Theorems for the first nonzero eigenvalue $\lambda$ 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 $\epsilon>0$, there…

It was proved by Montiel and Ros that for each conformal structure on a compact surface there is at most one metric which admits a minimal immersion into some unit sphere by first eigenfunctions. We generalize this theorem to the setting of…

We consider the eigenvalue problem $\Delta^{\mathbb{S}^2} \xi + 2 \xi=0 $ in $ \Omega $ and $\xi = 0 $ along $ \partial \Omega $, being $\Omega$ the complement of a disjoint and finite union of smooth and bounded simply connected regions in…

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…

Let $S^2$ be the 2-dimensional unit sphere and let $J_\alpha $ denote the nonlinear functional on the Sobolev space $H^{1,2}(S^2)$ defined by $$ J_\alpha(u) = \frac{\alpha}{4}\int_{S^2}|\nabla u|^2 d\omega + \int_{S^2} u d\omega -\ln…

In this article, we establish a geometric lower bound for the first positive eigenvalue $\lambda^{(1)}_{1}$ of the rough Laplacian acting on $1$-forms for closed $2n$-dimensional Riemannian manifolds with nonvanishing Euler characteristic.…

Let $\Sigma$ be a closed, embedded, oriented hypersurface in a closed oriented Riemannian manifold $N$. Under a lower bound on the Ricci curvature and an upper bound on the sectional curvature of $N$, we establish a lower bound for the…