Related papers: Optimal rigidity estimates for varifolds almost mi…

In this note, we prove the following generalization of a theorem of Shi and Tam \cite{ShiTam02}: Let $(\Omega, g)$ be an $n$-dimensional ($n \geq 3$) compact Riemannian manifold, spin when $n>7$, with non-negative scalar curvature and mean…

We establish some sharp weighted trace inequalities $W^{1,2}(\rho^{1-2\sigma}, M)\hookrightarrow L^{\frac{2n}{n-2\sigma}}(\pa M)$ on $n+1$ dimensional compact smooth manifolds with smooth boundaries, where $\rho$ is a defining function of…

Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies…

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 study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere…

We obtain a generic regularity result for stationary integral $n$-varifolds with only strongly isolated singularities inside $N$-dimensional Riemannian manifolds, in absence of any restriction on the dimension ($n\geq 2$) and codimension.…

Suppose that $\Sigma=\partial\Omega$ is the $n$-dimensional boundary, with positive (inward) mean curvature $H$, of a connected compact $(n+1)$-dimensional Riemannian spin manifold $(\Omega^{n+1},g)$ whose scalar curvature $R\ge…

In this paper, we obtain a rigidity result of $2$-dimensional complete lagrangian self-shrinkers with constant squared norm $|\vec{H}|^{2}$ of the mean curvature vector in the Euclidean space $\mathbb{R}^{4}$. The same idea is also used to…

Let $\gamma$ be a Riemannian metric on $\Sigma = S^1 \times T^{n-2}$, where $3 \leq n \leq 7$. Consider $\Omega = B^2 \times T^{n-2}$ with boundary $\partial \Omega = \Sigma$, and let $g$ be a Riemannian metric on $\Omega$ such that the…

In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open $3$-manifolds with uniformly positive sectional…

In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces $\Sigma\subset\mathbb{R}^{n+1}$. We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on…

In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in $\mathbb{R}^n$ with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round…

Let $\tilde{\Sigma}$ be the universal cover of a closed surface $\Sigma$ of genus at least $2$. We characterize all equivariantly area-minimizing maps from $\tilde{\Sigma}$ to a Hilbert sphere, which are equivariant with respect to an…

We study the sharp constant in the embedding theorem $H^1(0,2\pi)\to L_q(0,2\pi)$ in the case where the norm in $H^1(0,2\pi)$ is defined via the energy of a 1D magnetic Schr\"odinger operator.

Given a surface $\Sigma$ in $\mathbb{R}^3$ diffeomorphic to $S^2$, Struwe (Acta Math., 1988) proved that for almost every $H$ below the mean curvature of the smallest sphere enclosing $\Sigma$, there exists a branched immersed disk which…

Suppose M_t is a smooth family of compact connected two dimensional submanifolds of Euclidean space E^3 without boundary varying isometrically in their induced Riemannian metrics. Then we show that the mean curvature integrals over M_t are…

We prove that if $M$ is a closed $n$-dimensional Riemannian manifold, $n \ge 3$, with ${\rm Ric}\ge n-1$ and for which the optimal constant in the critical Sobolev inequality equals the one of the $n$-dimensional sphere $\mathbb{S}^n$, then…

In this paper, we investigate the volume comparison theorem related to $\sigma_2$-curvature. In particular, we show that volume comparison theorem with respect to $\sigma_2$-curvature holds for metrics close to strictly stable positive…

The optimality of the integral inequality $\int\limits_\gamma\sqrt{k_1^2+k_2^2+k_3^2}ds>2\pi$ for closed curves with non-vanishing curvatures in $\mathbb R^4$ is discussed. We prove that an arbitrary closed curve of constant positive…

In this paper, we consider the rigidity for an $n(\geq 4)$-dimensional submanfolds $M^n$ with parallel mean curvature in the space form ${\mathbb M}^{n+p}_c$ when the integral Ricci curvature of $M$ has some bound. We prove that, if…