closed hypersurface
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2020 ◽  
Vol 2020 (758) ◽  
pp. 261-280
Author(s):  
Shengwen Wang

AbstractWe show that if the entropy of any closed hypersurface is close to that of a round hyper-sphere, then it is close to a round sphere in Hausdorff distance. This generalizes the result of [3] to higher dimensions.


2018 ◽  
Vol 167 (02) ◽  
pp. 345-353
Author(s):  
ABRAÃO MENDES

AbstractIn this paper we generalise the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimising closed hypersurface Σ of a Riemannian 5-manifold M with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of Σ. Furthermore, if Σ saturates the respective upper bound and M has nonnegative Ricci curvature, then Σ is isometric to 𝕊4 up to scaling and M splits in a neighbourhood of Σ. Also, we obtain a rigidity result for the Riemannian cover of M when Σ minimises the volume in its homotopy class and saturates the upper bound.


2017 ◽  
Vol 26 (02) ◽  
pp. 1740014 ◽  
Author(s):  
Jonathan A. Hillman

Let [Formula: see text] and [Formula: see text] be the complementary regions of a closed hypersurface [Formula: see text] in [Formula: see text]. We use the Massey product structure in [Formula: see text] to limit the possibilities for [Formula: see text] and [Formula: see text]. We show also that if [Formula: see text] then it may be modified by a 2-knot satellite construction, while if [Formula: see text] and [Formula: see text] is abelian then [Formula: see text] or [Formula: see text]. Finally we use TOP surgery to propose a characterization of the simplest embeddings of [Formula: see text].


2017 ◽  
Vol 17 (4) ◽  
Author(s):  
Jie Xiao

AbstractUsing the semidiameter (in connection to the mean radius and surface radius) of a convex closed hypersurface in ℝ


2011 ◽  
Vol 54 (1) ◽  
pp. 67-75 ◽  
Author(s):  
QIN ZHANG

AbstractLet Mn be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1(1), n ≤ 8 and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.


2011 ◽  
Vol 22 (01) ◽  
pp. 131-143 ◽  
Author(s):  
GANGYI CHEN ◽  
HAIZHONG LI

Let M be an n-dimensional closed hypersurface with constant mean curvature H in a unit sphere Sn+1, n ≤ 8, and S the squared length of the second fundamental form of M. If |H| ≤ ε(n), then there exists a positive constant α(n, H), which depends only on n and H, such that if S0 ≤ S ≤ S0 + α(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ε(n) is a positive constant depending only on n and [Formula: see text].


2009 ◽  
Vol 51 (2) ◽  
pp. 413-423 ◽  
Author(s):  
QING-MING CHENG ◽  
YIJUN HE ◽  
HAIZHONG LI

AbstractLet M be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1, n ≤ 7, and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H, such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.


2006 ◽  
Vol 49 (1) ◽  
pp. 241-249 ◽  
Author(s):  
Qiaoling Wang ◽  
Changyu Xia

AbstractThis paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an $n({\geq}\,2)$-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss–Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean $n$-sphere. We also show that an $n({\geq}\,2)$-dimensional complete connected orientable hypersurface immersed in a unit sphere $S^{n+1}$ whose Gauss image is contained in a closed geodesic ball of radius less than $\pi/2$ in $S^{n+1}$ is diffeomorphic to a sphere. Finally, we prove that an $n({\geq}\,2)$-dimensional connected closed orientable hypersurface in $S^{n+1}$ with constant scalar curvature greater than $n(n-1)$ and Gauss image contained in an open hemisphere is totally umbilic.


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