Related papers: Stable 3-spheres in $\mathbb{C}^3$

A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean…

In this article we study surfaces in $\mathbb{S}^3(1) \times \mathbb{R}$ for which the $\mathbb{R}$-direction makes a constant angle with the normal plane. We give a complete classification for such surfaces with parallel mean curvature…

We obtain the explicit representation of Legendre surfaces in the unit $5$-sphere with harmonic mean curvature vector field, under the condition that the mean curvature function is constant along a certain special direction.

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition…

In this paper we construct generalizations to spheres of the well known Levi-Civita, Kustaanheimo-Steifel and Hurwitz regularizing transformations in Euclidean spaces of dimensions 2, 3 and 5. The corresponding classical and quantum…

In this paper, we prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere $\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere…

We consider a steady state $v_{0}$ of the Euler equation in a fixed bounded domain in $\mathbf{R}^{n}$. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center-stable subspaces. By rewriting the Euler…

Let $M$ be a convex body and let $K$ be a closed convex surface $K$ both contained in the Euclidean space $\mathbb{E}^3$. What can we say about $M$ if $K$ encloses $M$ and if from all the points in $K$ the body $M$ looks the same? In this…

In this paper, we study $4$-dimensional complete hypersurfaces with $w$-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for $4$-dimensional complete hypersurfaces with $w$-constant mean curvature.…

Llarull's theorem characterizes the round sphere $S^n$ among all spin manifolds whose scalar curvature is bounded from below by $n(n-1)$. In this paper we show that if the scalar curvature is bounded from below by $n(n-1)-\varepsilon$, the…

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the…

In this paper, we prove that any $C^{1}$-regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with…

In this paper we prove that the only algebraic constant mean curvature (cmc) surfaces in R^3 of order less than four are the planes, the spheres and the cylinders. The method used heavily depends on the efficiency of algorithms to compute…

We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector…

We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood.

In this article we establish $C^{3,\alpha}$-regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain $\Omega \subset \mathbb{R}^n$. In particular, stationary points satisfy the corresponding…

We prove a semi-global result on the existence of conformal embeddings of the two-sphere into the round three-sphere S^3(1) with prescribed mean curvature.

Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant "higher order mean curvatures". Here a $k$-th order mean curvature $Q_k$ ($k\geq1$) of a hypersurface $M^n$ is…

We show that the following three properties of a diffeomorphism $f$ of a smooth closed manifold are equivalent: (i) $f$ belongs to the $C^1$-interior of the set of diffeomorphisms having periodic shadowing property; (ii) $f$ has Lipschitz…

This is a preliminary note on a family of minimal surfaces in the 3-sphere defined by a compatible fourth order equation. The minimal surfaces are geometrically characterized either by having a surface of revolution like induced metric, or…