Let M^4 be a closed minimal hypersurface in S^5 with constant nonnegative scalar curvature. Denote by f_3 the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is prove...Let M^4 be a closed minimal hypersurface in S^5 with constant nonnegative scalar curvature. Denote by f_3 the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f_3 and g are constant,then M^4 is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form of M^4. This result provides another piece of supporting evidence to the Chern conjecture.展开更多
In this paper, we reprove a theorem of M. Anderson [Invent. Math., 69 （1982）, pp. 477-494] which established the existence of a minimal hypersurface in the hyperbolic space with prescribed asymptotic boundary with n...In this paper, we reprove a theorem of M. Anderson [Invent. Math., 69 （1982）, pp. 477-494] which established the existence of a minimal hypersurface in the hyperbolic space with prescribed asymptotic boundary with non-negative mean curvature in the non-parametric case. We use the mean curvature flow method.展开更多
We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S^n＋1 satisfying S f4 - f^2 3 ≤1/nS^3 where S is the squared norm of...We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S^n＋1 satisfying S f4 - f^2 3 ≤1/nS^3 where S is the squared norm of the second fundamental form of M, and fk = ∑λi^k and λi（1 ≤ i ≤ n） are the principal curvatures of M. We prove that there exists a positive constant δ（n）（≥ n/2） depending only on n such that if n ≤ S ≤ n ＋δ（n）, then S ≡ n, i.e., M is one of the Clifford torus S^K （√k/n） × S^n-k （V√n-k/n） for 1≤ k ≤ n - i. Moreover, we prove that if S is a constant, then there exists a positive constant T（n）（≥ n -2/3） depending only on n such that ifn ≤ S 〈 n ＋ τ（n）, then S ≡n, i.e.. M is a Clifford torus.展开更多
We study the global behavior of complete minimal δ-stable hypersurfaces in Rby using L~2-harmonic 1-forms.We show that a complete minimal δ-stable(δ >(n-1)~2/n~2)hypersurface in Rhas only one end.We also obtain ...We study the global behavior of complete minimal δ-stable hypersurfaces in Rby using L~2-harmonic 1-forms.We show that a complete minimal δ-stable(δ >(n-1)~2/n~2)hypersurface in Rhas only one end.We also obtain two vanishing theorems of complete noncompact quaternionic manifolds satisfying the weighted Poincar′e inequality.These results are improvements of the first author’s theorems on hypersurfaces and quaternionic K¨ahler manifolds.展开更多
基金supported by the National Natural Science Foundation of China(Nos.11471078,11622103)
文摘Let M^4 be a closed minimal hypersurface in S^5 with constant nonnegative scalar curvature. Denote by f_3 the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f_3 and g are constant,then M^4 is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form of M^4. This result provides another piece of supporting evidence to the Chern conjecture.
文摘In this paper, we reprove a theorem of M. Anderson [Invent. Math., 69 （1982）, pp. 477-494] which established the existence of a minimal hypersurface in the hyperbolic space with prescribed asymptotic boundary with non-negative mean curvature in the non-parametric case. We use the mean curvature flow method.
基金Supported by the National Natural Science Foundation of China (11071211)
文摘We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S^n＋1 satisfying S f4 - f^2 3 ≤1/nS^3 where S is the squared norm of the second fundamental form of M, and fk = ∑λi^k and λi（1 ≤ i ≤ n） are the principal curvatures of M. We prove that there exists a positive constant δ（n）（≥ n/2） depending only on n such that if n ≤ S ≤ n ＋δ（n）, then S ≡ n, i.e., M is one of the Clifford torus S^K （√k/n） × S^n-k （V√n-k/n） for 1≤ k ≤ n - i. Moreover, we prove that if S is a constant, then there exists a positive constant T（n）（≥ n -2/3） depending only on n such that ifn ≤ S 〈 n ＋ τ（n）, then S ≡n, i.e.. M is a Clifford torus.
基金The NSF(11471145,11371309)of China and Qing Lan Project
文摘We study the global behavior of complete minimal δ-stable hypersurfaces in Rby using L~2-harmonic 1-forms.We show that a complete minimal δ-stable(δ >(n-1)~2/n~2)hypersurface in Rhas only one end.We also obtain two vanishing theorems of complete noncompact quaternionic manifolds satisfying the weighted Poincar′e inequality.These results are improvements of the first author’s theorems on hypersurfaces and quaternionic K¨ahler manifolds.
