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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 $g=6$,…

In this paper, by meticulously constructing a minimizing sequence within a suitable Sobolev space and leveraging the variational principle, we establish that the first non-zero eigenvalue of the Laplace-Beltrami operator on an embedded…

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…

This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and…

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…

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…

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…

In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric…

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…

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…

Jakobson and Nadirashvili \cite{JN} constructed a sequence of eigenfunctions on $T^2$ with a bounded number of critical points, answering in the negative the question raised by Yau \cite{Yau1} which asks that whether the number of the…

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 paper, we provide the lower bounds of the first non-zero basic eigenvalue on a closed singular Riemannian manifold $(M,\mathcal{F})$ with basic mean curvature that depends on the given non-negative lower bound of the Ricci curvature…

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…

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…

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely…

We introduce nonlocal minimal surfaces on closed manifolds and establish a far-reaching Yau-type result: in every closed, $n$-dimensional Riemannian manifold we construct infinitely many nonlocal $s$-minimal surfaces. We prove that, when…

In this paper, we study the Calabi-Yau conjectures for complete minimal hypersurfaces $\Sigma^{n}\subset \mathbb{R}^{n+1}$ in dimensions $n\ge 3$. These conjectures ask whether a complete minimal hypersurface must be unbounded, and more…

In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus $\gamma$ and the area. The equality in Yang-Yau's…