Related papers: A Liouville-type theorem for cylindrical cones

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…

Let $M$ be a locally symmetric irreducible closed manifold of dimension $\ge 3$. A result of Borel [Bo] combined with Mostow rigidity imply that there exists a finite group $G = G(M)$ such that any finite subgroup of $\text{Homeo}^+(M)$ is…

Let M be a closed embedded minimal hypersurface in a Euclidean sphere of dimension n+1, we prove that it is strongly rigid. As applications we confirm the conjecture proposed by Choi and Schoen in [3] and the Chern conjecture for n less…

E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the…

We explore Liouville's theorem and the Strong Liouville Property (SLP) for harmonic functions on Riemannian cones and surfaces. Our approach recasts the classical Liouville property in terms of the growth of radial eigenfunctions (in the…

In this paper, we study hypersurfaces $M_{r}^{4}$ $(r=0, 1, 2, 3, 4)$ satisfying $\triangle \vec{H}=\lambda \vec{H}$ ($\lambda$ a constant) in the pseudo-Euclidean space $\mathbb{E}_{s}^{5}$ $(s=0, 1, 2, 3, 4, 5)$. We obtain that every such…

For a connected $n$-dimensional compact smooth hypersurface $M$ without boundary embedded in $\mathbb{R}^{n+1}$, a classical result of Aleksandrov shows that it must be a sphere if it has constant mean curvature. Li and Nirenberg studied a…

Let $(M,\bar{g}, e^{-f}d\mu)$ be a complete metric measure space with Bakry-\'Emery Ricci curvature bounded below by a positive constant. We prove that, in $M$, there is no complete two-sided $L_f$-stable immersed $f$-minimal hypersurface…

Let $M^n$ be an $n$-dimensional closed minimal submanifold immersed in the unit sphere $\mathbb{S}^{n+m}$. Denote by $S$ and $\rho^{\perp}$ the squared norm of the second fundamental form and the normal scalar curvature of $M^n$,…

We investigate the existence of minimal hypersurfaces in $\mathbb{S}^{n+1}$ that are generated by the isoparametric foliation of a subsphere $\mathbb{S}^n$. By considering a generalized rotational ansatz formed by the union of homothetic…

A nonempty closed convex set in ${\mathbb R}^n$, not containing the origin, is called a pseudo-cone if with every $x$ it also contains $\lambda x$ for $x\ge 1$. We consider pseudo-cones with a given recession cone $C$, called…

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…

The aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in $\mathbb{R}^4$, while they do not exist in positively curved closed…

In this paper we have proved several approximation theorems for the family of minimal surfaces in R^3 that imply, among other things, that complete minimal surfaces are dense in the space of all minimal surfaces endowed with the topology of…

For a smooth subvariety $X\subset\Bbb P^N$, consider (analogously to projective normality) the vanishing condition $H^1(\Bbb P^N,\Cal I^2_X(k))=0$, $k\ge3$. This condition is shown to be satisfied for all sufficiently large embeddings of a…

Let S be an arbitrary real surface, with or without boundary, contained in a hypersurface M of the complex euclidean space \C^2, with S and M of class C^{2, a}, where 0 < a < 1. If M is globally minimal, if S is totally real except at…

In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics(MHD) system in $\Bbb R^3$. Let $(v, B, p)$ be a smooth solution to the stationary MHD equations in $\Bbb R^3$. We show that if there exist smooth matrix…

We prove a Liouville type theorem for arbitrarily growing positive viscosity supersolutions of fully nonlinear uniformly elliptic equations in halfspaces. Precisely, let $M^-_{\lambda, \Lambda}$ be the Pucci's inf- operator, defined as the…

For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact"…

In this paper, we show that a closed manifold $M^{n+1} (n \geq 7)$ endowed with a $C^\infty$-generic (Baire sense) metric contains infinitely many singular minimal hypersurfaces with optimal regularity. Moreover, for $2 \leq n \leq 6$, our…