This paper studies the prescribed contact angle boundary value problem of a certain type of mean curvature equation.Applying the maximum principle and the moving frame method and based on the location of the maximum p...This paper studies the prescribed contact angle boundary value problem of a certain type of mean curvature equation.Applying the maximum principle and the moving frame method and based on the location of the maximum point,the boundary gradient estimation of the solutions to the equation is obtained.展开更多
In this paper,we mainly study the global rigidity theorem of Riemannian submanifolds in space forms.Let Mn(n≥3)be a complete minimal submanifold in the unit sphere Sn+p(1).Forλ∈[0,n2−1/p),there is an explicit posit...In this paper,we mainly study the global rigidity theorem of Riemannian submanifolds in space forms.Let Mn(n≥3)be a complete minimal submanifold in the unit sphere Sn+p(1).Forλ∈[0,n2−1/p),there is an explicit positive constant C(n,p,λ),depending only on n,p,λ,such that,if∫MSn/2dM<∞,∫M(S−λ)n/2+dM<C(n,p,λ),then Mn is a totally geodetic sphere,where S denotes the square of the second fundamental form of the submanifold and∫+=max{0,f}.Similar conclusions can be obtained for a complete submanifold with parallel mean curvature in the Euclidean space Rn+p.展开更多
In the present paper we obtain the following result: Theorem Let M^R be a compact submanifold with parallel mean curvature vector in a locally symmetric and conformally flat Riemannian manifold N^(n+p)(p>1). If the...In the present paper we obtain the following result: Theorem Let M^R be a compact submanifold with parallel mean curvature vector in a locally symmetric and conformally flat Riemannian manifold N^(n+p)(p>1). If then M^n lies in a totally geodesic submanifold N^(n+1).展开更多
In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piece...In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.展开更多
Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of ...Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mn, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mn immersed in Sn +pwith parallel nor- malized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.展开更多
In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F : S1 × [0, T ) → R^2 which satisfies the followin...In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F : S1 × [0, T ) → R^2 which satisfies the following evolution equation δ^2F /δt^2 (u, t) = k(u, t)N(u, t)-▽ρ(u, t), ∨(u, t) ∈ S^1 × [0, T ) with the initial data F (u, 0) = F0(u) and δF/δt (u, 0) = f(u)N0, where k is the mean curvature and N is the unit inner normal vector of the plane curve F (u, t), f(u) and N0 are the initial velocity and the unit inner normal vector of the initial convex closed curve F0, respectively, and ▽ρ is given by ▽ρ Δ=(δ^2F /δsδt ,δF/δt) T , in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F , it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax) and when t goes to Tmax, either the solution convergesto a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R^1,1.展开更多
In this article, we obtain some results about the mean curvature integrals of the parallel body of a convex set in R^n. These mean curvature integrals are generalizations of the Santalo's results.
The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^(n+1)( 1) is considered. Let A = ∑i,j,k h(ijk)~2( λi+ nH)~2,B = ∑i,j,k h(ijk)~2( λi+ nH) ·( ...The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^(n+1)( 1) is considered. Let A = ∑i,j,k h(ijk)~2( λi+ nH)~2,B = ∑i,j,k h(ijk)~2( λi+ nH) ·( λj+ nH),S = ∑i( λi+ nH)~2, where h(ij)= λiδ(ij). Utilizing Lagrange's method, a sharper pointwise estimation of 3(A- 2B) in terms of S and |▽h|~2 is obtained, here |▽h|~2= ∑i,j,k h(ijk)~2. Then, with the help of this, it is proved that Mis isometric to the Clifford hypersurface if the square norm of the second fundamental form of Msatisfies certain conditions. Hence, the pinching result of the minimal hypersurface is extended to the hypersurface with constant mean curvature case.展开更多
Let Mn be a closed spacelike submanifold isometrically immersed in de Sitter space S^n+p _p(c).Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamen...Let Mn be a closed spacelike submanifold isometrically immersed in de Sitter space S^n+p _p(c).Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of Mn,respectively.Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for Mn immersed in ~S^n+p _p(c) with parallel normalized mean curvature vector field is proved.When n≥3, the pinching constant is the best.Thus,the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math,1998,95:499-505) is corrected.Moreover,the reduction of the codimension when Mn is a complete submanifold in S^n+p _p(c) with parallel normalized mean curvature vector field is investigated.展开更多
The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator:-div((1+|u| 2) p-22u)=f(x,u),\ x∈Ω, u∈W 1,p 0(Ω),is considered, where Ω is a bounded domain in ...The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator:-div((1+|u| 2) p-22u)=f(x,u),\ x∈Ω, u∈W 1,p 0(Ω),is considered, where Ω is a bounded domain in R n(n>p>1) with smooth boundary Ω.Under some natural conditions together with some conditions weaker than (AR) condition,we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if f(x,u)|u| p-2u→+∞ as u→∞.展开更多
In this paper, we consider the heat flow for the Hsystem with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. W...In this paper, we consider the heat flow for the Hsystem with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero in W 1,n with the decay rate t 2/(2-n) as time goes to infinity.展开更多
This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloc...This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity.展开更多
In this paper, the authors obtain the Backlund transformation on time-like surfaces with constant mean curvature in R2.1. Using this transformation, families of surfaces with constant mean curvature from known ones ca...In this paper, the authors obtain the Backlund transformation on time-like surfaces with constant mean curvature in R2.1. Using this transformation, families of surfaces with constant mean curvature from known ones can be constructed.展开更多
Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H(n+1)(c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that ...Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H(n+1)(c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that (1) if the multiplicities of the two distinct principal curvatures are greater than 1,then Mn is isometric to the Riemannian product Sk(r)×H(n-k)(-1/(r2 + ρ2)), where r 〉 0 and 1 〈 k 〈 n - 1;(2)if H2 〉 -c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product S(n-1)(r) × H1(-1/(r2 +ρ2)) or S1(r) × H(n-1)(-1/(r2 +ρ2)),r 〉 0, if one of the following conditions is satisfied (i) S≤(n-1)t22+c2t(-2)2 on Mn or (ii)S≥ (n-1)t21+c2t(-2)1 on Mn or(iii)(n-1)t22+c2t(-2)2≤ S≤(n-1)t21+c2t(-2)1 on Mn, where t_1 and t_2 are the positive real roots of (1.5).展开更多
This paper discusses the existence problem in the study of some partial differential equations. The author gets some bifurcation on the prescribed mean curvature problem on the unit ball, the scalar curvature problem ...This paper discusses the existence problem in the study of some partial differential equations. The author gets some bifurcation on the prescribed mean curvature problem on the unit ball, the scalar curvature problem on the n-sphere, and some field equations. The author gives some natural conditions such that the standard bifurcation or Thom-Mather theory can be used.展开更多
Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G.Huisken and C. Sinestra...Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G.Huisken and C. Sinestrari in. These apriori bounds are satisfied for mean convex hypersurfaces in locally symmetric Riemannian manifolds with nonnegative sectional curvature.展开更多
Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curva...Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of Le in the Euclidean case展开更多
Authors discover that a spacelike surface in Minkowski 3-space is related to a integrable system. They obtain a representation formula for spacelike surfaces with prescribed mean curvature. This representation formula...Authors discover that a spacelike surface in Minkowski 3-space is related to a integrable system. They obtain a representation formula for spacelike surfaces with prescribed mean curvature. This representation formula is equivalent to that obtained ly Akutagawa and Nishikawa.展开更多
In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stabil...In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.展开更多
A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n + p)-dimensional manifold N^n+p with negative sectional curvature is proved. For ...A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n + p)-dimensional manifold N^n+p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H 〉 1 there exists a negative number τ(n,p, H) ∈ (-1, 0) with the property that if the sectional curvature of N is pinched in [-1, τ-(n,p, H)], and if the squared length of the second fundamental form is in a certain interval, then N^n+p is isometric to the hyperbolic space H^n+P(-1). As a consequence, this submanifold M is congruent to S^n(1√H^2 - 1) or the Veronese surface in S^4(1/√H^2-1).展开更多
基金supported by the National Natural Science Foundation of China (No.12061078)。
文摘This paper studies the prescribed contact angle boundary value problem of a certain type of mean curvature equation.Applying the maximum principle and the moving frame method and based on the location of the maximum point,the boundary gradient estimation of the solutions to the equation is obtained.
基金supported by the National Natural Science Foundation of China(11531012,12071424,12171423)the Scientific Research Project of Shaoxing University(2021LG016)。
文摘In this paper,we mainly study the global rigidity theorem of Riemannian submanifolds in space forms.Let Mn(n≥3)be a complete minimal submanifold in the unit sphere Sn+p(1).Forλ∈[0,n2−1/p),there is an explicit positive constant C(n,p,λ),depending only on n,p,λ,such that,if∫MSn/2dM<∞,∫M(S−λ)n/2+dM<C(n,p,λ),then Mn is a totally geodetic sphere,where S denotes the square of the second fundamental form of the submanifold and∫+=max{0,f}.Similar conclusions can be obtained for a complete submanifold with parallel mean curvature in the Euclidean space Rn+p.
文摘In the present paper we obtain the following result: Theorem Let M^R be a compact submanifold with parallel mean curvature vector in a locally symmetric and conformally flat Riemannian manifold N^(n+p)(p>1). If then M^n lies in a totally geodesic submanifold N^(n+1).
基金supported in part by the NSFC(11801496,11926352)the Fok Ying-Tung Education Foundation(China)the Hubei Key Laboratory of Applied Mathematics(Hubei University).
文摘In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.
基金Project supported by the Stress Supporting Subject Foundation of Zhejiang Province, China
文摘Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mn, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mn immersed in Sn +pwith parallel nor- malized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.
基金Kong and Wang was supported in part by the NSF of China (10671124)the NCET of China (NCET-05-0390)the work of Liu was supported in part by the NSF of China
文摘In this paper we investigate the one-dimensional hyperbolic mean curvatureflow for closed plane curves. More precisely, we consider a family of closed curves F : S1 × [0, T ) → R^2 which satisfies the following evolution equation δ^2F /δt^2 (u, t) = k(u, t)N(u, t)-▽ρ(u, t), ∨(u, t) ∈ S^1 × [0, T ) with the initial data F (u, 0) = F0(u) and δF/δt (u, 0) = f(u)N0, where k is the mean curvature and N is the unit inner normal vector of the plane curve F (u, t), f(u) and N0 are the initial velocity and the unit inner normal vector of the initial convex closed curve F0, respectively, and ▽ρ is given by ▽ρ Δ=(δ^2F /δsδt ,δF/δt) T , in which T stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F , it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax) and when t goes to Tmax, either the solution convergesto a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time R^1,1.
基金Supported in part by NNSFC(10671159)Hong Kong Qiu Shi Science and Technologies Research Foundation
文摘In this article, we obtain some results about the mean curvature integrals of the parallel body of a convex set in R^n. These mean curvature integrals are generalizations of the Santalo's results.
文摘The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^(n+1)( 1) is considered. Let A = ∑i,j,k h(ijk)~2( λi+ nH)~2,B = ∑i,j,k h(ijk)~2( λi+ nH) ·( λj+ nH),S = ∑i( λi+ nH)~2, where h(ij)= λiδ(ij). Utilizing Lagrange's method, a sharper pointwise estimation of 3(A- 2B) in terms of S and |▽h|~2 is obtained, here |▽h|~2= ∑i,j,k h(ijk)~2. Then, with the help of this, it is proved that Mis isometric to the Clifford hypersurface if the square norm of the second fundamental form of Msatisfies certain conditions. Hence, the pinching result of the minimal hypersurface is extended to the hypersurface with constant mean curvature case.
文摘Let Mn be a closed spacelike submanifold isometrically immersed in de Sitter space S^n+p _p(c).Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of Mn,respectively.Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for Mn immersed in ~S^n+p _p(c) with parallel normalized mean curvature vector field is proved.When n≥3, the pinching constant is the best.Thus,the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math,1998,95:499-505) is corrected.Moreover,the reduction of the codimension when Mn is a complete submanifold in S^n+p _p(c) with parallel normalized mean curvature vector field is investigated.
基金Supported by the National Natural Science Foundation of China(1 0 1 71 0 3 2 ) and the GuangdongProvincial Natural Science Foundation(0 1 1 6 0 6 )
文摘The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator:-div((1+|u| 2) p-22u)=f(x,u),\ x∈Ω, u∈W 1,p 0(Ω),is considered, where Ω is a bounded domain in R n(n>p>1) with smooth boundary Ω.Under some natural conditions together with some conditions weaker than (AR) condition,we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if f(x,u)|u| p-2u→+∞ as u→∞.
文摘In this paper, we consider the heat flow for the Hsystem with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero in W 1,n with the decay rate t 2/(2-n) as time goes to infinity.
基金supported in part by a grant from China Scholarship Councilthe National Natural Science Foundation of China(11301006)the Anhui Provincial Natural Science Foundation(1408085MA01)
文摘This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity.
基金the NNSF(19971084,10101025 and 10231050)of China
文摘In this paper, the authors obtain the Backlund transformation on time-like surfaces with constant mean curvature in R2.1. Using this transformation, families of surfaces with constant mean curvature from known ones can be constructed.
基金supported by NSF of Shaanxi Province (SJ08A31)NSF of Shaanxi Educational Committee (2008JK484+1 种基金2010JK642)Talent Fund of Xi'an University of Architecture and Technology
文摘Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H(n+1)(c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that (1) if the multiplicities of the two distinct principal curvatures are greater than 1,then Mn is isometric to the Riemannian product Sk(r)×H(n-k)(-1/(r2 + ρ2)), where r 〉 0 and 1 〈 k 〈 n - 1;(2)if H2 〉 -c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product S(n-1)(r) × H1(-1/(r2 +ρ2)) or S1(r) × H(n-1)(-1/(r2 +ρ2)),r 〉 0, if one of the following conditions is satisfied (i) S≤(n-1)t22+c2t(-2)2 on Mn or (ii)S≥ (n-1)t21+c2t(-2)1 on Mn or(iii)(n-1)t22+c2t(-2)2≤ S≤(n-1)t21+c2t(-2)1 on Mn, where t_1 and t_2 are the positive real roots of (1.5).
基金This work is partially supported by NNSF of ChinaSRF for Returned Overseas Chinese Scholars+1 种基金State Education Commission a scientific grant of Tsinghua University at Beijing
文摘This paper discusses the existence problem in the study of some partial differential equations. The author gets some bifurcation on the prescribed mean curvature problem on the unit ball, the scalar curvature problem on the n-sphere, and some field equations. The author gives some natural conditions such that the standard bifurcation or Thom-Mather theory can be used.
基金supported partially by the National Natural Science Foundation of China (10871171)the Chinese-Hungarian Sci. and Tech. cooperation (for 2007-2009)
文摘Under the hypothesis of mean curvature flows of hypersurfaces, we prove that the limit of the smooth rescaling of the singularity is weakly convex. It is a generalization of the result due to G.Huisken and C. Sinestrari in. These apriori bounds are satisfied for mean convex hypersurfaces in locally symmetric Riemannian manifolds with nonnegative sectional curvature.
基金supported by the NSFC(11101267,11271132)the Innovation Program of Shanghai Municipal Education Commission(13YZ087)the Science and Technology Program of Shanghai Maritime University(20120061)
文摘Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry, moving by mean curvature flow, we show that at the first finite singular time of mean curvature flow, certain subcritical quantities concerning the second fundamental form blow up. This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao, but also extends the latest work of Le in the Euclidean case
文摘Authors discover that a spacelike surface in Minkowski 3-space is related to a integrable system. They obtain a representation formula for spacelike surfaces with prescribed mean curvature. This representation formula is equivalent to that obtained ly Akutagawa and Nishikawa.
基金supported by the King Saud University D.S.F.P program
文摘In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.
基金Research supported by the NSFC (10231010)Trans-Century Training Programme Foundation for Talents by the Ministry of Education of ChinaNatural Science Foundation of Zhejiang Province (101037).
文摘A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n + p)-dimensional manifold N^n+p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H 〉 1 there exists a negative number τ(n,p, H) ∈ (-1, 0) with the property that if the sectional curvature of N is pinched in [-1, τ-(n,p, H)], and if the squared length of the second fundamental form is in a certain interval, then N^n+p is isometric to the hyperbolic space H^n+P(-1). As a consequence, this submanifold M is congruent to S^n(1√H^2 - 1) or the Veronese surface in S^4(1/√H^2-1).