Unit Sphere Sn Research Articles

Some applications of Bochner formula to submanifolds of a unit sphere

The aims of this paper are to give estimates for the eigenvalue of the Laplacian on submanifolds in a unit sphere Sn+p(1) and to give some new characterizations of spheres in Sn+p(1).

SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN UNIT SPHERES

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].

EMBEDDED HYPERSURFACES WITH CONSTANT mTH MEAN CURVATURE IN A UNIT SPHERE

In this paper, we study n-dimensional hypersurfaces with constant mth mean curvature in a unit sphere Sn+1(1) and construct many compact nontrivial embedded hypersurfaces with constant mth mean curvature Hm > 0 in Sn+1(1), for 1 ≤ m ≤ n-1. Moreover, if the 2nd mean curvature H2 takes value between [Formula: see text] and [Formula: see text] for any integer k ≥ 2 and n ≥ 3, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H2 (i.e. constant scalar curvature) in Sn+1(1); If the 4th mean curvature H4 takes value between [Formula: see text] and [Formula: see text] for any integer k ≥ 3 and n ≥ 5, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H4 in Sn+1(1).

New finite-gap solutions for the coupled Burgers equations engendered by the Neumann systems

On the tangent bundle TSN−1 of the unit sphere SN–1, this paper reduces the coupled Burgers equations to two Neumann systems by using the nonlinearization of the Lax pair, whose Liouville integrability is displayed in the scheme of the r-matrix technique. Based on the Lax matrix of the Neumann systems, the Abel–Jacobi coordinates are appropriately chosen to straighten out the restricted Neumann flows on the complex torus, from which the new finite-gap solutions expressed by Riemann theta functions for the coupled Burgers equations are given in view of the Jacobi inversion.

Finite-gap solutions of 2+1 dimensional integrable nonlinear evolution equations generated by the Neumann systems

Each soliton equation in the Korteweg–de Vries (KdV) hierarchy, the 2+1 dimensional breaking soliton equation, and the 2+1 dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation are reduced to two or three Neumann systems on the tangent bundle TSN−1 of the unit sphere SN−1. The Lax–Moser matrix for the Neumann systems of degree N−1 is deduced in view of the Mckean–Trubowitz identity and a bilinear generating function, whose favorite characteristic accounts for the problem of the genus of Riemann surface matching to the number of elliptic variables. From the Lax–Moser matrix, the constrained Hamiltonians in the sense of Dirac–Poisson bracket for all the Neumann systems are written down in a uniform recursively determined by integrals of motion. The involution of integrals of motion and constrained Hamiltonians is completed on TSN−1 by using a Lax equation and their functional independence is displayed over a dense open subset of TSN−1 by a direct calculation, which contribute to the Liouville integrability of a family of Neumann systems in a new systematical way. We also construct the hyperelliptic curve of Riemann surface and the Abel map straightening out the restricted Neumann flows that naturally leads to the Jacobi inversion problem on the Jacobian with the aid of the holomorphic differentials, from which some finite-gap solutions expressed by Riemann theta functions for the 2+1 dimensional breaking soliton equation, the 2+1 dimensional CDGKS equation, the KdV, and the fifth-order KdV equations are presented by means of the Riemann theorem.

A Form of Alexandrov-Fenchel Inequality

There are many important integral formulae and integral inequalities for convex bodies (see [4], [18]). The Brunn-Minkowski inequality and Alexandrov-Fenchel inequality are among the most important integral inequalities in the theory of convex bodies, and the Minkowski type integral formulae and more general formulae of Chern are very useful in the global geometry of convex hypersurfaces. Most of these formulae and inequalities can be stated in the integral forms on the unit sphere Sn with the convexity assumption. It seems of interest to establish similar results without the convexity assumption. For a convex body, the polar of the body is also convex. The support function of the convex body corresponds to the gauge function of its polar body. In other words, the geometry of a convex body can be reflected from its polar dual. With this relation, we will introduce a class of domains in Rn+1 called k∗-convex (see Definition 4.2) as a natural generalization of convex bodies. We will derive a form of the Alexandrov-Fenchel

Estimate for index of closed minimal hypersurfaces in spheres

The aim of this work is to deal with index of closed orientable non-totally geodesic minimal hypersurface Σn of the Euclidean unit sphere Sn+1 whose second fundamental form has squared norm bounded from below by n. In this case we shall show that the index of stability, denoted by IndΣ, is greater than or equal to n + 3, with equality occurring at only Clifford tori $\mathbf{S}^k(\frac{k}{n})\times\mathbf{S}^{n-k}(\sqrt{\frac{(n-k)}{n}})$. Moreover, we shall prove also that, besides Clifford tori, we have the following gap: IndΣ ≥ 2n + 5.

SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A SPHERE

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 $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.

Constant mean curvature hypersurfaces with two principal curvatures in a sphere

In this paper we consider a compact oriented hypersurface Mn with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere Sn+1. Denote by \({\phi_{ij}}\) the trace free part of the second fundamental form of Mn, and Φ be the square of the length of \({\phi_{ij}}\) . We obtain two integral formulas by using Φ and the polynomial \({P_{H,m}(x)=x^{2}+ \frac{n(n-2m)}{\sqrt{nm(n-m)}}H x -n(1+H^{2})}\) . Assume that BH,m is the square of the positive root of PH,m(x) = 0. We show that if Mn is a compact oriented hypersurface immersed in the sphere Sn+1 with constant mean curvatures H having two distinct principal curvatures λ and μ then either \({\Phi=B_{H,m}}\) or \({\Phi=B_{H,n-m}}\) . In particular, Mn is the hypersurface \({S^{n-m}(r)\times S^{m}(\sqrt{1-r^{2}})}\) .

First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature

Let M M be an n n -dimensional compact hypersurface with constant scalar curvature n ( n − 1 ) r n(n-1)r , r > 1 r> 1 , in a unit sphere S n + 1 ( 1 ) S^{n+1}(1) . We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral ∫ M H d M \int _MHdM of the mean curvature H H . In this paper, we first study the eigenvalue of the Jacobi operator J s J_s of M M . We derive an optimal upper bound for the first eigenvalue of J s J_s , and this bound is attained if and only if M M is a totally umbilical and non-totally geodesic hypersurface or M M is a Riemannian product S m ( c ) × S n − m ( 1 − c 2 ) S^m(c)\times S^{n-m}(\sqrt {1-c^2}) , 1 ≤ m ≤ n − 1 1\leq m\leq n-1 .

THE SCALAR CURVATURE OF MINIMAL HYPERSURFACES IN A UNIT SPHERE

In this paper, we study n-dimensional compact minimal hypersurfaces in a unit sphere Sn+1(1) and give an answer for S. S. Chern's conjecture. We have shown that [Formula: see text] if S > n, and prove that an n-dimensional compact minimal hypersurface with constant scalar curvature in Sn+1(1) is a totally geodesic sphere or a Clifford torus if [Formula: see text], where S denotes the squared norm of the second fundamental form of this hypersurface.

Compact embedded rotation hypersurfaces of Sn+1

In this paper, we prove that $$ S^{{n - 1}} {\left( {{\sqrt {\frac{{n - m}} {n}} }} \right)} \times S^{1} {\left( {{\sqrt {\frac{m} {n}} }} \right)} $$ and round geodesic spheres are the only n-dimensional compact embedded rotation hypersurfaces with H m = 0 (1 ≤ m ≤ n − 1) in a unit sphere Sn+1(1). When m = 1, our result reduces to the result of T. Otsuki [O1], [O2], Brito and Leite [BL].

Immersed Hypersurfaces in the Unit Sphere S m+1 with Constant Blaschke Eigenvalues

For an immersed submanifold x: Mm → Sn in the unit sphere Sn without umbilics, an eigenvalue of the Blaschke tensor of x is called a Blaschke eigenvalue of x. It is interesting to determine all hypersurfaces in Sn with constant Blaschke eigenvalues. In this paper, we are able to classify all immersed hypersurfaces in Sm+1 with vanishing Mobius form and constant Blaschke eigenvalues, in case (1) x has exact two distinct Blaschke eigenvalues, or (2) m = 3. With these classifications, some interesting examples are also presented.

The Bargmann transform and regularization of the 2, 3, 5-dimensional Kepler problem

We introduce a Bargmann transform for the space L2(Sn) of square integrable functions on the n=2,3,5 dimensional unit sphere Sn inmersed in Rn+1. This is done on base of the Hopf fibration for the spheres Sk↦Sd with (k,d)=(1,1),(3,2),(7,4) and a suitable canonical transformation relating two different ways to regularize the n=2,3,5 dimensional Kepler problem (with fixed negative energy) involving the null complex quadric Qn inmersed in Cn+1. We prove the unitarity of the Bargmann transform onto a suitable space of analytical functions. We give reproducing kernels for these spaces of analytical functions which, for the cases n=3,5, are defined as the kernel of quantizations of restrictions for regularizations of the classical Kepler problem. We give an inversion formula for our Bargmann transform. We also give a set of coherent states for L2(Sn), n=2,3,5 and their semiclassical asymptotics (ℏ→0). Our Bargmann transform is actually a coherent states transform. Additionally, we use the moment map technique in order to construct a map with values in Qn that gives the base to define our canonical transformation.

Complete Hypersurfaces with Constant Mean Curvature in a Unit Sphere

By investigating hypersurfaces M n in the unit sphere S n+1(1) with constant mean curvature and with two distinct principal curvatures, we give a characterization of the torus S 1(a) × $S^{n-1}(\sqrt{1-a^2})$ , where $a^2=\frac{2+nH^2\pm\sqrt{n^2H^4+4(n-1)H^2}}{2n(1+H^2)}$ . We extend recent results of Hasanis et al. [5] and Otsuki [10].

Compact hypersurfaces in a unit sphere

We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

SOME RESULTS ON HARMONIC MAPS FOR FINSLER MANIFOLDS

By simplifying the first and the second variation formulas of the energy functional and generalizing the Weitzenböck formula, we study the stability and the rigidity of harmonic maps between Finsler manifolds. It is proved that any nondegenerate harmonic map from a compact Einstein Riemannian manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature is totally geodesic and there is no nondegenerate stable harmonic map from a Riemannian unit sphere Sn (n > 2) to any Finsler manifold.

Turbulence Phenomena in Real Analysis

The purpose of this paper is first to show that if X is any locally compact but not compact perfect Polish space and Open image in new window stands for the one-point compactification of X, while EX is the equivalence relation which is defined on the Polish group C(X,R+*) by Open image in new window where f, g are in C(X,R+*), then EX is induced by a turbulent Polish group action. Second we show that given any Open image in new window if we identify the n-dimensional unit sphere Sn with the one-point compactification of Rn via the stereographic projection, while En,r is the equivalence relation which is defined on the Polish group Cr(Rn,R+*) by Open image in new window where f, g are in Cr(Rn,R+*), then En,r is also induced by a turbulent Polish group action.

A rigidity theorem for submanifolds inSn+p with constant scalar curvature

LetM1 be a closed submanifold isometrically immersed in a unit sphereSn+p. Denote byR, H andS, the normalized scalar curvature, the mean curvature, and the square of the length of the second fundamental form ofM1, respectively. SupposeR is constant and ≥1. We study the pinching problem onS and prove a rigidity theorem forM1 immersed inSn+p with parallel normalized mean curvature vector field. Whenn≥8 or,n=7 andp≤2, the pinching constant is best.

Hypersurfaces of prescribed mean curvature in central projection-II

This work, which is a direct continuation of a preceding paper, is dedicated to interior regularity results with respect to a “singular variational integral” with linear growth. The Dirichlet Problem associated with the corresponding variational problem describes surfaces of prescribed mean curvature that admits of a central projection onto an open domain of the unit sphere S n . Our results are accomplished by using methods from geometric measure theory together with certain L∞ bounds that we shall prove here as well.