Related papers: Some results on Chern's problem

In this note, we give sufficient conditions for the (semi)stability of a hypersurface $H$ of $\mathbb{P}^N_k$ in terms of its degree $d$, the maximal multiplicity $\delta$ of its singularities, and the dimension $s$ of its singular locus.…

Let $F^{n+p}(c)$ be an $(n+p)$-dimensional simply connected space form with nonnegative constant curvature $c$. We prove that if $M^n(n\geq4)$ is a compact submanifold in $F^{n+p}(c)$, and if $Ric_M>(n-2)(c+H^2),$ where $H$ is the mean…

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold…

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…

Let $X:M^n\to \mathbb{R}^{n+1}$ be a complete properly immersed self-shrinker. In this paper, we prove that if the squared norm of the second fundamental form $S$ satisfies $1\leq S< C$ for some constant $C$, then $S=1$. Further we classify…

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 initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar…

The Dehn function of a metric space measures the area necessary in order to fill a closed curve of controlled length by a disc. As a main result, we prove that a length space has curvature bounded above by $\kappa$ in the sense of…

In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere $S^{2n+1}$ moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a…

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean…

A classical result of Milman roughly states that every Lipschitz function on $\mathbb{S}^n$ is almost constant on a sufficiently high-dimensional sphere $\mathbb{S}^m\subset \mathbb{S}^n$. In this paper we extend the result by proving that…

In this article, we prove that for an embedded minimal hypersurface $\Sigma^{m}$ in $S^{m+1}$, the first eigenvalue $\lambda_1$ of the Laplacian operator on $\Sigma$ satisfies: $$\lambda_1> \frac{m}{2}+G(m, |A|_{\max}, |A|_{\min} ) ,$$…

In this paper we deal with some problems concerning minimal hypersurfaces in Carnot-Caratheodory (CC) structures. More precisely we will introduce a general calibration method in this setting and we will study the Bernstein problem for…

On a compact Riemannian manifold with boundary having positive mean curvature, a fundamental result of Shi and Tam states that, if the manifold has nonnegative scalar curvature and if the boundary is isometric to a strictly convex…

The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes $\alpha, \beta$ on the boundary of a hyperbolic knot manifold $M$ has been verified in all cases other than small Seifert…

Let $M^n$ be an $n$-dimensional complete and locally conformally flat hypersurface in the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature $n(n-1)$. We show that if the total curvature $\left( \int _ { M } | H | ^ { n } d v…

Given a smooth closed oriented manifold $M$ of dimension $n$ embedded in $\mathbb{R}^{n+2}$ we study properties of the `solid angle' function $\Phi\colon\mathbb{R}^{n+2}\setminus M\to S^1$. It turns out that a non-critical level set of…

Let $\Omega$ be a open bounded domain in $\mathbb{R}^n $ with smooth boundary $\partial\Omega$. We consider the equation $ \Delta u + u^{\frac{n-k+2}{n-k-2}-\varepsilon} =0\,\hbox{ in }\,\Omega $, under zero Dirichlet boundary condition,…

Let $C$ be a strictly convex domain in a $3$-dimensional Riemannian manifold with sectional curvature bounded above by a constant and let $\Sigma$ be a constant mean curvature surface with free boundary in $C$. We provide a pinching…

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. Suppose the union of non-principal orbits…