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Register a new userNormal scalar curvature inequality on the focal submanifolds of isoparametric hypersurfaces
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An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines $C_A$, the set of points with Condition A in focal submanifolds. It turns out that the points in $C_A$ reach an upper bound of the normal scalar curvature $rho^{bot}$ (sharper than that in DDVV inequality [GT08], [Lu11]). We also determine the sets $C_P$ (points with parallel second fundamental form) and $C_E$ (points with Einstein condition), which achieve two lower bounds of $rho^{bot}$.
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A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametric condition, develops singularities in finite time, and converges in finite time to a smooth submanifold of lower dimension. We also give a precise description of the collapsing.
Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied by the authors in [LT]. In this paper, we will show that all these solutions are ancient solutions. We also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the Cherns conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres.
We find many examples of compact Riemannian manifolds $(M,g)$ whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the metric $g$ is replaced with $g$ in a neighbourhood of $g$. Our examples $(M,g)$ consist of certain minimal isoparametric hypersurfaces of spheres; their focal manifolds; the Lie groups $SU(n)$ for $nleq 17$, and $Sp(n)$ for all $n$; and all quaternionic Grassmannians.
For a closed hypersurface $M^nsubset S^{n+1}(1)$ with constant mean curvature and constant non-negative scalar curvature, the present paper shows that if $mathrm{tr}(mathcal{A}^k)$ are constants for $k=3,ldots, n-1$ for shape operator $mathcal{A}$, then $M$ is isoparametric. The result generalizes the theorem of de Almeida and Brito cite{dB90} for $n=3$ to any dimension $n$, strongly supporting Cherns conjecture.
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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