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Register a new userBiharmonic hypersurfaces in a product space $L^mtimes mathbb{R}$
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In this paper, we study biharmonic hypersurfaces in a product of an Einstein space and a real line. We prove that a biharmonic hypersurface with constant mean curvature in such a product is either minimal or a vertical cylinder generalizing a result of cite{OW} and cite{FOR}. We derived the biharmonic equation for hypersurfaces in $S^mtimes mathbb{R}$ and $H^mtimes mathbb{R}$ in terms of the angle function of the hypersurface, and use it to obtain some classifications of biharmonic hypersurfaces in such spaces. These include classifications of biharmonic hypersurfaces which are totally umbilical or semi-parallel for $mge 3$, and some classifications of biharmonic surfaces in $S^2times mathbb{R}$ and $H^2times mathbb{R}$ which are constant angle or belong to certain classes of rotation surfaces.
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Given a hypersurface $M$ of null scalar curvature in the unit sphere $mathbb{S}^n$, $nge 4$, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in $Rr^{n+1}$ as a normal graph over a truncated cone generated by $M$. Furthermore, this graph is 1-stable if the cone is strictly 1-stable.
In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove the non-existence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chens conjecture on biharmonic submanifolds.
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).
A diagonal metric sum_{i=1}^n g_{ii} dx_i^2 is termed Guichard_k if sum_{i=1}^{n-k}g_{ii}-sum_{i=n-k+1}^n g_{ii}=0. A hypersurface in R^{n+1} is isothermic_k if it admits line of curvature co-ordinates such that its induced metric is Guichard_k. Isothermic_1 surfaces in R^3 are the classical isothermic surfaces in R^3. Both isothermic_k hypersurfaces in R^{n+1} and Guichard_k orthogonal co-ordinate systems on R^n are invariant under conformal transformations. A sequence of n isothermic_k hypersurfaces in R^{n+1} (Guichard_k orthogonal co-ordinate systems on R^n resp.) is called a Combescure sequence if the consecutive hypersurfaces (orthogonal co-ordinate systems resp.) are related by Combescure transformations. We give a correspondence between Combescure sequences of Guichard_k orthogonal co-ordinate systems on R^n and solutions of the O(2n-k,k)/O(n)xO(n-k,k)-system, and a correspondence between Combescure sequences of isothermic_k hypersurfaces in R^{n+1} and solutions of the O(2n+1-k,k)/O(n+1)xO(n-k,k)-system, both being integrable systems. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries.
The aim of this paper is to classify cohomogeneity one isometric actions on the 4-dimensional Minkowski space $mathbb{R}^{3,1}$, up to orbit equivalence. Representations, up to conjugacy, of the acting groups in $O(3,1)ltimes mathbb{R}^{3,1}$ are given in both cases, proper and non-proper actions. When the action is proper, the orbits and the orbit spaces are determined.
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