Related papers: Biharmonic hypersurfaces with bounded mean curvatu…

In R^3, let M be the infinite union of unit spheres whose centers lie at even integers on the x-axis; every pair of consecutive spheres touches at (2m+1, 0, 0). Desingularizing these point contacts yields Delaunay's classical constant mean…

In this short note we study nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that $\phi:(M,g)\to (N, h)$ is a biharmonic map, where $(M, g)$…

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere…

We consider biharmonic maps $\phi:(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $\alpha$ satisfies $1<\alpha<\infty$. If for such an $\alpha$,…

We study the existence problem for achronal hypersurfaces $M \hookrightarrow \overline{M}$ in a globally hyperbolic spacetime, whose mean curvature is a prescribed -- possibly singular -- source, and whose boundary is a given smooth…

In the Minkowski space, we consider a compact, spacelike hypersurface with boundary, which can be written as a graph on a spacelike hyperplane. We prove that, if its $k$-th mean curvature is constant, and its boundary is on the hyperplane…

In this paper we study triharmonic hypersurfaces immersed in a space form $N^{n+1}(c)$. We prove that any proper CMC triharmonic hypersurface in the sphere $\mathbb S^{n+1}$ has constant scalar curvature; any CMC triharmonic hypersurface in…

Let $H^p(L^2(M))$ be the space of all $L^2$-harmonic $p$-forms $(2\leq p\leq n-2)$ on complete submanifolds $M$ with flat normal bundle in spheres. In this paper, we first show that $H^p(L^2(M))$ is trivial if the total curvature of $M$ is…

We develop a global theory for complete hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in…

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over…

We prove that in Euclidean space $R^{n+1}$ any compact immersed nonnegatively curved hypersurface $M$ with free boundary on the sphere $S^n$ is an embedded convex topological disk. In particular, when the $m^{th}$ mean curvature of $M$ is…

We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant…

For a Riemannian manifold $M^{n+1}$ and a compact domain $\Omega \subset M^{n+1}$ bounded by a hypersurface $\partial \Omega$ with normal curvature bounded below, estimates are obtained in terms of the distance from $O$ to $\partial \Omega$…

In this work, complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with total absolute curvature at most 4 pi are classified. This classification suggests that the Cohn-Vossen inequality can be sharpened for surfaces…

We prove that a hemisphere in the Euclidean space $R^{n+1}$, viewed as the graph of a function, admits no smooth perturbations as graphs with mean curvature $H\ge 1$ whose boundary equator is fixed up to $C^2$. This is an extension of the…

We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrodinger operator $-L=-(\Delta +q)$ on a parabolic Riemannian manifold M, then u is either identically zero or it has no zeros on M, and the linear space of…

CMC surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h\in (0,1)$ there exist CMC…

In this paper, close surfaces are considered in 3-dimensional harmonic conformally flat space in point of the variation. It is shown that if the conformal vector field be tangent to surface and the sign of the mean curvature does not change…

An expression for the first variation of the area functional of the second fundamental form is given for a hypersurface in a semi-Riemannian space. The concept of the "mean curvature of the second fundamental form" is then introduced. Some…

Let $M$ be an $n(\geq3)$-dimensional oriented compact submanifold with parallel mean curvature in the simply connected space form $F^{n+p}(c)$ with $c+H^2>0$, where $H$ is the mean curvature of $M$. We prove that if the Ricci curvature of…