For a closed hypersurface Mn?Sn+1(1)with constant mean curvature and constant non-negative scalar curvature,we show that if tr(Ak)are constants for k=3,...,n-1 and the shape operator A,then M is isoparametric.The resu...For a closed hypersurface Mn?Sn+1(1)with constant mean curvature and constant non-negative scalar curvature,we show that if tr(Ak)are constants for k=3,...,n-1 and the shape operator A,then M is isoparametric.The result generalizes the theorem of de Almeida and Brito(1990)for n=3 to any dimension n,strongly supporting the Chern conjecture.展开更多
Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constan...Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).展开更多
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
基金supported by National Natural Science Foundation of China (Grant Nos.11722101,11871282 and 11931007)Beijing Natural Science Foundation (Grant No.Z190003)Nankai Zhide Foundation。
文摘For a closed hypersurface Mn?Sn+1(1)with constant mean curvature and constant non-negative scalar curvature,we show that if tr(Ak)are constants for k=3,...,n-1 and the shape operator A,then M is isoparametric.The result generalizes the theorem of de Almeida and Brito(1990)for n=3 to any dimension n,strongly supporting the Chern conjecture.
基金supported by National Natural Science Foundation of China(Grant No.11531012)China Postdoctoral Science Foundation(Grant No.BX20180274)Natural Science Foundation of Zhejiang Province(Grant No.LY20A010024)。
文摘Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).
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.
