The geometry of CR-submanifolds of Kaehler manifolds was initiated by Bejancu 1 and has been developed by 2–5 and others. They studied the geometry of CR-submanifolds with positive definite metric. Thus, this geometry may not be applicable to the other branches of mathematics and physics, where the metric is not necessarily definite. Moreover, because of growing importance of lightlike submanif...

The sphere S contains a simple family of constant mean curvature (CMC) hypersurfaces of the form Λp,q a ≡ S p(a)×Sq( √ 1− a) for p+ q+1 = n and a ∈ (0, 1) called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tori connected to each other by small catenoidal bridges at a ...

We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous ...

The problem of operating or designing a system with robust stability with respect to many parameters can be viewed as a geometric problem in multidimensional parameter space of finding the position of nominal parameters λ0 relative to hypersurfaces at which stability is lost in a bifurcation. The position of λ0 relative to these hypersurfaces may be quantified by numerically computing the bifur...

We present some general properties of biharmonic and biconservative submanifolds then survey recent results on such hypersurfaces in space forms. also propose an alternative version a well-known result Nomizu Smyth for hypersurfaces, by replacing the CMC hypothesis with more condition biconservativity.

Let $$M^{n+1}$$ be a closed manifold of dimension $$3\le n+1\le 7$$ . We show that for $$C^\infty $$ -generic metric g on M, to any connected, closed, embedded, 2-sided, stable, minimal hypersurface $$S\subset (M,g)$$ corresponds sequence hypersurfaces $$\{\Sigma _k\}$$ scarring along S, in the sense area and Morse index $$\Sigma _k$$ both diverge infinity and, when properly renormalized, conve...

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

In this paper, through making careful analysis of Gauss and Codazzi equations, we prove that four dimensional biharmonic hypersurfaces in nonzero space form have constant mean curvature. Our result gives the positive answer to conjecture proposed by Balmus-Montaldo-Oniciuc 2008 for hypersurfaces.

For a hypersurface V n−1 of a conformal space, we introduce a conformal differential invariant I = h 2 g , where g and h are the first and the second fundamental forms of V n−1 connected by the apolarity condition. This invariant is called the conformal quadratic element of V n−1. The solution of the problem of conformal rigidity is presented in the framework of conformal differential geometry ...