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Register a new userExamples of scalar-flat hypersurfaces in $mathbb{R}^{n+1}$
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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.
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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.
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.
As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo cite{Guo} proved that $n$-dimensional compact self-shrinkers in $mathbb{R}^{n+1}$ with scalar curvature bounded from above or below by some constant are isometric to the round sphere $mathbb{S}^n(sqrt{n})$, which implies that $n$-dimensional compact self-shrinkers in $mathbb{R}^{n+1}$ with constant scalar curvature are isometric to the round sphere $mathbb{S}^n(sqrt{n})$(see also cite{Hui1}). Complete classifications of $n$-dimensional translating solitons in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature and of $n$-dimensional self-expanders in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature were given by Mart{i}n, Savas-Halilaj and Smoczykcite{MSS} and Ancari and Chengcite{AC}, respectively. In this paper we give complete classifications of $n$-dimensional complete self-shrinkers in $mathbb{R}^{n+1}$ with nonnegative constant scalar curvature. We will also give alternative proofs of the classification theorems due to Mart{i}n, Savas-Halilaj and Smoczyk cite{MSS} and Ancari and Chengcite{AC}.
We prove a theorem of Hadamard-Stoker type: a connected locally convex complete hypersurface immersed in $H^n times R$ (n>1), where $H^n$ is n-dimensional hyperbolic space, is embedded and homeomorphic either to the n-sphere or to $R^n$. In the latter case it is either a vertical graph over a convex domain in $H^n$ or has what we call a simple end.
In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $mathbb{R}^{n+1}_{1}$ along the inverse Gauss curvature flow (i.e., the evolving speed equals the $(-1/n)$-th power of the Gaussian curvature) with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of the spacelike graph of a positive constant function defined over the piece of $mathscr{H}^{n}(1)$ as time tends to infinity.
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