Geodesic Hypersphere Research Articles

Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery

We study mean curvature flow in \({\mathbb {S}}_K^{n+1}\), the round sphere of sectional curvature \(K>0\), under the quadratic curvature pinching condition \(|A|^{2} < \frac{1}{n-2} H^{2} + 4 K\) when \(n\ge 4\) and \(|A|^{2} < \frac{3}{5}H^{2}+\frac{8}{3}K\) when \(n=3\). This condition is related to a famous theorem of Simons (Ann Math 2(88):62–105, 1968), which states that the only minimal hypersurfaces satisfying \(\vert A\vert ^2<nK\) are the totally geodesic hyperspheres. It is related to but distinct from “two-convexity”. Notably, in contrast to two-convexity, it allows the mean curvature to change sign. We show that the pinching condition is preserved by mean curvature flow, and obtain a “cylindrical” estimate and corresponding pointwise derivative estimates for the curvature. As a result, we find that the flow becomes either uniformly convex or quantitatively cylindrical in regions of high curvature. This allows us to apply the surgery apparatus developed by Huisken and Sinestrari (Invent Math 175(1):137–221, 2009) (cf. Haslhofer and Kleiner, Duke Math J 166(9):1591–1626, 2017). We conclude that any smoothly, properly immersed hypersurface \({\mathcal {M}}\) of \({\mathbb {S}}_K^{n+1}\) satisfying the pinching condition is diffeomorphic to \({\mathbb {S}}^n\) or to the connected sum of a finite number of copies of \({\mathbb {S}}^1\times {\mathbb {S}}^{n-1}\). If \({\mathcal {M}}\) is embedded, then it bounds a 1-handlebody. The results are sharp when \(n\ge 4\).

𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms

Abstract Kaimakamis and Panagiotidou in [Taiwanese J. Math. 18(6) (2014), 1991–1998] proposed an open question: are there real hypersurfaces in nonflat complex space forms whose ∗-Ricci tensor satisfies the condition of 𝔻-parallelism? In this short note, we present an affirmative answer and prove that a three-dimensional real hypersurface in a nonflat complex space form has 𝔻-parallel ∗-Ricci tensor if and only if it is locally congruent to either a geodesic hypersphere of radius r in ℂ H 2(c) with tanh ⁡ ( | c | 2 r ) = 1 2 $\begin{array}{} \displaystyle \tanh(\frac{\sqrt{|c|}}{2}r) = \frac{1}{2} \end{array}$ or a ruled real hypersurface.

New stability results for spheres and Wulff shapes

Abstract We prove that a closed convex hypersurface of the Euclidean space with almost constant anisotropic first and second mean curvatures in the Lp -sense is W 2 ,p -close (up to rescaling and translations) to the Wulff shape. We also obtain characterizations of geodesic hyperspheres of space forms improving those of [10] and [11].

Classification of differential symmetry breaking operators for differential forms

We give a complete classification of conformally covariant differential operators between the spaces of differential i-forms on the sphere Sn and j-forms on the totally geodesic hypersphere Sn−1 by analyzing the restriction of principal series representations of the Lie group O(n+1,1). Further, we provide explicit formulæ for these matrix-valued operators in the flat coordinates and find factorization identities for them.

Ricci solitons on CR submanifolds of maximal CR dimension in complex projective space

We study parallel and symmetric second order tensor fields on CR submanifolds of maximal CR dimension of the complex projective space. Under some natural conditions, these tensors are scalar multiples of the metric; an example of submanifold satisfying these assumptions is the geodesic hypersphere. The result is used for Ricci solitons on these CR submanifolds.

A characterization of certain geodesic hyperspheres in complex projective space

We characterize geodesic hyperspheres of radius r such that cot^2(r)=\frac{1}{2} as the unique real hypersurfaces in complex projective space whose structure Jacobi operator satisfies a pair of conditions.

CHARACTERISATIONS OF GEODESIC HYPERSPHERES IN A NON-FLAT COMPLEX SPACE FORM

AbstractTotally η-umbilic real hypersurfaces are the simplest examples of real hypersurfaces in a non-flat complex space form. Geodesic hyperspheres in this ambient space are typical examples of such real hypersurfaces. We characterise every geodesic hypersphere by observing the extrinsic shapes of its geodesics and using the derivative of its contact form.

Some modifications of the theorem of Beltrami

The two main topics of this text are as follows: Firstly, three modifications of the theorem of Beltrami will be presented for diffeomorphisms between Riemannian manifolds and a space form which preserve the geodesic circles, the geodesic hyperspheres, or the minimal surfaces, respectively. Secondly, it is defined what it means for an infinitesimal deformation of a metric to preserve the geodesics up to first order, and a corresponding infinitesimal version of Beltrami’s theorem is given.

Invariant Submanifolds of Real Hypersurfaces of Complex Manifolds

Real hypersurfaces of a complex manifold admit a naturally induced almost contact structure F′ from the almost complex structure of the ambient manifold. We prove that for any F′-invariant submanifold M of a geodesic hypersphere in a non-flat complex space form and of a horosphere in a complex hyperbolic space, its second fundamental form h satisfies the condition \({h(FX,Y ) - h(X, FY) = g(FX, Y )\eta, X,Y \in T(M), 0 \ne \eta \in {T^\perp}(M)}\), which has been considered in [2] and [3].

On submanifolds with parallel mean curvature vector

We consider Mn, n ≥ 3, a complete, connected submanifold of a space form $\tilde{M}^{n+p}(\tilde{c})$, whose non vanishing mean curvature vector H is parallel in the normal bundle. Assuming the second fundamental form h of M satisfies the inequality <h>2 ≤ n2 |H|2/(n - 1), we show that for $\tilde{c}$ ≥ 0 the codimension reduces to 1. When M is a submanifold of the unit sphere, then Mn is totally umbilic. For the case $\tilde{c}$ < 0, one imposes an additional condition that is trivially satisfied when $\tilde{c}$ ≥ 0. When M is compact and has non-negative Ricci curvature then it is a geodesic hypersphere in the hyperbolic space. An alternative additional condition, when $\tilde{c}$ < 0, reduces the codimension to 3.

The nullity of a compact minimal hypersurface in a compact symmetric space of rank one

We determine a compact minimal hypersuface with the least nullity in the Cayley projective plane. Combining this with the preceding results, we conclude the following: Let X be a compact symmetric space of rank one and M a compact minimal hypersuface in X. Then the nullity of M is bounded from below by the dimension of X. When the nullity of M is equal to the dimension of X, M must be a minimal geodesic hypersphere in X. Conversely,the nullity of a minimal geodesic hypersphere in X is equal to the dimension of X.

Classifications of real hypersurfaces in complex space forms by means of curvature conditions

It has been proved there are no semi-parallel real hypersurfaces in the complex projective space CP n , n 3, and in any non-at complex space form of complex dimension 2. Also, characterizations of geodesic hyperspheres and ruled real hypersurfaces in CP n , n 3, have been obtained by considering some other curvature conditions. We generalize these results by studying two new conditions for real hypersurfaces in non-at complex space forms. As a corollary, we extend the known characterizations to real hypersurfaces of type A0 and A1 and ruled real hypersurfaces in non-at complex space forms. In particular, we prove that there are no semi-parallel real hypersurfaces in non-at complex space forms of complex dimension at least 2.

CR Submanifolds of Maximal CR Dimension in a Complex Hopf Manifold

We study CR submanifolds M in a Hopf manifold (C H N (λ), J 0, g 0) with the Boothby metric g 0,of maximal CR dimension. Any such M is a CR manifold ofhypersurface type, although embedded in higher codimension, and itsanti-invariant distribution H(M)⊥ is spanned by a unit vectorfield U. We classify the CR submanifolds M for which ξ = −J 0 Uis parallel in the normal bundle under assumptions on thespectrum of the Weingarten operator a ξ. We show that (1) ifa ξ(U) = (1/2)A (where A is the anti-Lee vector) andM fibres in tori over a CR submanifold of the complex projectivespace, then M lies on the (total space of the) pullback of the Hopf fibration via S ⊂ C P N − 1, for some geodesic hypersphere S, and (2) if a ξ(U)= 0 and Spec(a ξ) = {0, c}, for some c ∈ R ∖ {0}, then M is locally a Riemannian product of totally geodesicsubmanifolds.

On the global structure of Hopf hypersurfaces in a complex space form

It is known that a tube over a Kähler submanifold in a complex space form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed $C^{2n-1}$ regular Hopf hypersurface in the complex projective space is a tube over an irreducible algebraic variety. In the complex hyperbolic space a connected compact generic immersed $C^{2n-1}$ regular Hopf hypersurface is a geodesic hypersphere.

Application of an integral formula to CR‐submanifolds of complex hyperbolic space

The purpose of this paper is to study n‐dimensional compact CR‐submanifolds of complex hyperbolic space CH(n+p)/2, and especially to characterize geodesic hypersphere in CH(n+1)/2 by an integral formula.

The Nullity of a Compact Minimal Real Hypersurface in a Quaternion Projective Space

We will prove that the nullities of compact minimal real hypersurfaces in a quaternion projective space B Pn are bounded from below by 4n, and those with nullity 4n must be minimal geodesic hyperspheres.

Characterizations of geodesic hyperspheres in a complex projective space by observing the extrinsic shape of geodesics

Characterizations of geodesic hyperspheres in a complex projective space by observing the extrinsic shape of geodesics

Almost complex curves and Hopf hypersurfaces in the nearly K�hler 6-sphere

We characterize Hopf hypersurfaces inS6 as open parts of geodesic hyperspheres or of tubes around almost complex curves ofS6.

Geodesic hyperspheres in complex projective space

Geodesic hyperspheres in complex projective space

Complete 2-transnormal hypersurfaces in a Kaehler manifold of negative constant holomorphic sectional curvature

The idea of constant width has been developed in a somewhat differentspirit, as a topic in differentialgeometry, and the concept of transnormality has been introduced as the generalized one of constant width in a Riemannian manifold. Let M be a connected complete hypersurface of a connected complete Riemannian manifold M. For each igM, there exists,up to parametrization, a unique geodesic zx of M which intersectsM orthogonally at x. M is called a transnormal hypersurface of M it, for each pair x,y<E.M, the relationyr for xjgM, x~y means that yE.rx. Then we can consider the quotient space M―Mj~ with the quotient topology with respect to thisrelation. We call M an r-transnormal hypersurface if the natural projection of M onto M is an r-fold covering map. Topological structures of transnormal submanifolds are full of interest and have been investigated from various angles (for example, see [3]). On the other hand, differentialgeometric structures of 2-transnormal hypersurfaces in a space form have been given in [2] and [4]. Recentry, the author has studied in [5] differentialgeometric structures of compact 2-transnormal hypersurfaces in a complex space form. The purpose of this paper is to generalize the result in [5] to the case where 2-transnormal hypersurfaces are complete. Namely we shall prove that 2-transnormal hypersurfaces in a Kaehler manifold of negative constant holomorphic sectionalcurvature are tubes over some submanifolds or geodesic hyperspheres if any principal curva-