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Register a new userSome classifications of biharmonic hypersurfaces with constant scalar curvature
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We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and in a non-positively curved Einstein space. Our results provide additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture that a biharmonic submanifold in a sphere has constant mean curvature, and two more cases that support Chens conjecture on biharmonic hypersurfaces (Corollaries 2.2,2.7).
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We give a complete classification of local and global conformal biharmonic maps between any two space forms by proving that a conformal map between two space forms is proper biharmonic if and only if the dimension is 4, the domain is flat, and it is a restriction of a Mobius transformation. We also show that proper k-polyharmonic conformal maps between Euclidean spaces exist if and only if the dimension is 2k and they are precisely the restrictions of Mobius transformations. This provides infinitely many simple examples of proper k-polyharmonic maps with nice geometric structure.
In 1968, Simons introduced the concept of index for hypersurfaces immersed into the Euclidean sphere S^{n+1}. Intuitively, the index measures the number of independent directions in which a given hypersurface fails to minimize area. The earliest results regarding the index focused on the case of minimal hypersurfaces. Many such results established lower bounds for the index. More recently, however, mathematicians have generalized these results to hypersurfaces with constant mean curvature. In this paper, we consider hypersurfaces of constant mean curvature immersed into the sphere and give lower bounds for the index under new assumptions about the immersed manifold.
We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and *-scalar curvature.
In this article, we study hypersurfaces $Sigmasubset mathbb{R}^{n+1}$ with constant weighted mean curvature. Recently, Wei-Peng proved a rigidity theorem for CWMC hypersurfaces that generalizes Le-Sesum classification theorem for self-shrinker. More specifically, they showed that a complete CWMC hypersurface with polynomial volume growth, bounded norm of the second fundamental form and that satisfies $|A|^2H(H-lambda)leq H^2/2$ must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prove that under some conditions if the reverse inequality holds then the hypersurface must either be a hyperplane or a generalized cylinder. As an application of one of the results proved in this paper, we will obtain another version of the classification theorem obtained by the authors of this article, that is, we show that under some conditions, a complete CWMC hypersurface with $Hgeq 0$ must either be a hyperplane or a generalized cylinder.
A classical problem in constant mean curvature hypersurface theory is, for given $Hgeq 0$, to determine whether a compact submanifold $Gamma^{n-1}$ of codimension two in Euclidean space $R_+^{n+1}$, having a single valued orthogonal projection on $R^n$, is the boundary of a graph with constant mean curvature $H$ over a domain in $R^n$. A well known result of Serrin gives a sufficient condition, namely, $Gamma$ is contained in a right cylinder $C$ orthogonal to $R^n$ with inner mean curvature $H_Cgeq H$. In this paper, we prove existence and uniqueness if the orthogonal projection $L^{n-1}$ of $Gamma$ on $R^n$ has mean curvature $H_Lgeq-H$ and $Gamma$ is contained in a cone $K$ with basis in $R^n$ enclosing a domain in $R^n$ containing $L$ such that the mean curvature of $K$ satisfies $H_Kgeq H$. Our condition reduces to Serrins when the vertex of the cone is infinite.
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