We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete K?hler manifolds. We derive some elliptic differential inequalities from Weitzenb?ck form...We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete K?hler manifolds. We derive some elliptic differential inequalities from Weitzenb?ck formulas for the traceless Ricci tensor of K?hler manifolds with constant scalar curvature and the Bochner tensor of K?hler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several L^p and L~∞ pinching results are established to characterize K?hler-Einstein manifolds among K?hler manifolds with constant scalar curvature and complex space forms among K?hler-Einstein manifolds.Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact K?hler manifolds and noncompact K?hler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge,these kinds of results have not been reported.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
From[J.Differential Geom.,1990,31(1):285-299],one can obtain that compact self-shrinking hypersufaces in R^(n+1) with constant scalar curvature must be the standard sphere S^(n)(√n)(cf.[Front.Math.,2023,18(2):417-430...From[J.Differential Geom.,1990,31(1):285-299],one can obtain that compact self-shrinking hypersufaces in R^(n+1) with constant scalar curvature must be the standard sphere S^(n)(√n)(cf.[Front.Math.,2023,18(2):417-430]).This result was generalized by Guo[J.Math.Soc.Japan,2018,70(3):1103-1110]with assumption of a lower or upper scalar curvature bound.In this paper,we will generalize the scalar curvature rigidity theorem of Guo to the case of λ-hypersurfaces.We will also give an alternative proof of the theorem(cf.[2014,arXiv:1410.5302]and[Proc.Amer.Math.Soc.,2018,146(10):4459-4471])that λ-hypersurfaces which are entire graphs must be hyperplanes.展开更多
Let Mn（n ≥ 4） be an oriented closed submanifold with parallel mean curvature in an （n ＋ p）-dimensional locally symmetric Riemannian manifold Nn＋p. We prove that if the sectional curvature of N is positively pin...Let Mn（n ≥ 4） be an oriented closed submanifold with parallel mean curvature in an （n ＋ p）-dimensional locally symmetric Riemannian manifold Nn＋p. We prove that if the sectional curvature of N is positively pinched in [5, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ= 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu [15].展开更多
In 1980,Simon proposed a quantization conjecture about the Gaussian curvature K of closed minimal surfaces in unit spheres:if K(s+1)≤K≤K(s)(K(s):=2/(s(s+1)),s∈N),then either K=K(s)or K=K(s+1).Notice that the surfac...In 1980,Simon proposed a quantization conjecture about the Gaussian curvature K of closed minimal surfaces in unit spheres:if K(s+1)≤K≤K(s)(K(s):=2/(s(s+1)),s∈N),then either K=K(s)or K=K(s+1).Notice that the surface must be one of Calabi’s standard minimal 2-spheres if the curvature is a positive constant.The cases s=1 and s=2 were proven in the 1980s by Simon and others.In this paper,we give a pinching theorem of the Simon conjecture in the case s=3 and also give a new proof of the cases s=1 and s=2 by some Simons-type integral inequalities.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
In this paper we show that, under some conditions, if M is a manifold with Bakry-émery Ricci curvature bounded below and with bounded potential function then M is compact. We also establish a volume comparison th...In this paper we show that, under some conditions, if M is a manifold with Bakry-émery Ricci curvature bounded below and with bounded potential function then M is compact. We also establish a volume comparison theorem for manifolds with nonnegative Bakry-émery Ricci curvature which allows us to prove a topolological rigidity theorem for such manifolds.展开更多
We first study the Grassmannian manifoldG n (Rn+p)as a submanifold in Euclidean space Λ n (R n+p). Then we give a local expression for each map from Riemannian manifoldM toG n (R n+p) ?Λ n (R n+p), and use the local...We first study the Grassmannian manifoldG n (Rn+p)as a submanifold in Euclidean space Λ n (R n+p). Then we give a local expression for each map from Riemannian manifoldM toG n (R n+p) ?Λ n (R n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG n (R n+p). As a consequence of this formula we get a rigidity theorem.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
In this paper,we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in Rn+1,and if the squared norm of the second fundamental form of M satisfies 0≤S-1≤1/18,then S≡1 ...In this paper,we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in Rn+1,and if the squared norm of the second fundamental form of M satisfies 0≤S-1≤1/18,then S≡1 and M is a round sphere or a cylinder.More generally,let M be a complete λ-hypersurface of codimension one with polynomial volume growth in Rn+1 with λ≠0.Then we prove that there exists a positive constant γ,such that if |λ|≤γ and the squared norm of the second fundamental form of M satisfies0≤S-βλ≤1/18,then S≡βλ,λ> 0 and M is a cylinder.Here βλ=1/2(2+λ2+|λ|(λ2+4)1/2).展开更多
By studying modular invariance properties of some characteristic forms, we get some generalized anomaly cancellation formulas on(4 r-1)-dimensional manifolds with no assumption that the 3 rd de-Rham cohomology of mani...By studying modular invariance properties of some characteristic forms, we get some generalized anomaly cancellation formulas on(4 r-1)-dimensional manifolds with no assumption that the 3 rd de-Rham cohomology of manifolds vanishes. These anomaly cancellation formulas generalize our previous anomaly cancellation formulas on(4 r-1)-dimensional manifolds. We also generalize our previous anomaly cancellation formulas on(4 r-1)-dimensional manifolds and the Han–Yu rigidity theorem to the(a, b) case.展开更多
In this paper,we give certain homotopy and diffeomorphism versions as a generalization to an earlier result due to W.S.Cheung,Bun Wong and Stephen S.T.Yau concerning a local rigidity problem of the tangent bundle over...In this paper,we give certain homotopy and diffeomorphism versions as a generalization to an earlier result due to W.S.Cheung,Bun Wong and Stephen S.T.Yau concerning a local rigidity problem of the tangent bundle over compact surfaces of general type.展开更多
In this paper, the authors give a survey about λ-hypersurfaces in Euclidean spaces. Especially, they focus on examples and rigidity of λ-hypersurfaces in Euclidean spaces.
Let x:M→S^(n+p)(1)be an n-dimensional submanifold immersed in an(n+p)-dimensional unit sphere S^(n+p)(1).In this paper,we study n-dimensional submanifolds immersed in S^(n+p)(1)which are critical points of the functi...Let x:M→S^(n+p)(1)be an n-dimensional submanifold immersed in an(n+p)-dimensional unit sphere S^(n+p)(1).In this paper,we study n-dimensional submanifolds immersed in S^(n+p)(1)which are critical points of the functional S(x)=∫_(M)S^(n/2)dv,where S is the squared length of the second fundamental form of the immersion x.When x:M→S^(2+p)(1)is a surface in S^(2+p)(1),the functional S(x)=∫_(M)S^(n/2)dv represents double volume of image of Gaussian map.For the critical surface of S(x),we get a relationship between the integral of an extrinsic quantity of the surface and its Euler characteristic.Furthermore,we establish a rigidity theorem for the critical surface of S(x).展开更多
基金supported by the Foundation for training Young Teachers in University of Shanghai(ZZegd16003)supported by National Natural Science Foundation of China(11271071,11771087)LMNS,Fudan University
文摘We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete K?hler manifolds. We derive some elliptic differential inequalities from Weitzenb?ck formulas for the traceless Ricci tensor of K?hler manifolds with constant scalar curvature and the Bochner tensor of K?hler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several L^p and L~∞ pinching results are established to characterize K?hler-Einstein manifolds among K?hler manifolds with constant scalar curvature and complex space forms among K?hler-Einstein manifolds.Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact K?hler manifolds and noncompact K?hler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge,these kinds of results have not been reported.
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From[J.Differential Geom.,1990,31(1):285-299],one can obtain that compact self-shrinking hypersufaces in R^(n+1) with constant scalar curvature must be the standard sphere S^(n)(√n)(cf.[Front.Math.,2023,18(2):417-430]).This result was generalized by Guo[J.Math.Soc.Japan,2018,70(3):1103-1110]with assumption of a lower or upper scalar curvature bound.In this paper,we will generalize the scalar curvature rigidity theorem of Guo to the case of λ-hypersurfaces.We will also give an alternative proof of the theorem(cf.[2014,arXiv:1410.5302]and[Proc.Amer.Math.Soc.,2018,146(10):4459-4471])that λ-hypersurfaces which are entire graphs must be hyperplanes.
基金Supported by the National Natural Science Foundation of China(11531012,11371315,11301476)the TransCentury Training Programme Foundation for Talents by the Ministry of Education of Chinathe Postdoctoral Science Foundation of Zhejiang Province(Bsh1202060)
文摘Let Mn（n ≥ 4） be an oriented closed submanifold with parallel mean curvature in an （n ＋ p）-dimensional locally symmetric Riemannian manifold Nn＋p. We prove that if the sectional curvature of N is positively pinched in [5, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ= 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu [15].
基金supported by National Natural Science Foundation of China(Grant No.12171037)the Fundamental Research Funds for the Central Universities+1 种基金supported by National Natural Science Foundation of China(Grant Nos.12171037 and 12271040)China Postdoctoral Science Foundation(Grant No.2022M720261).
文摘In 1980,Simon proposed a quantization conjecture about the Gaussian curvature K of closed minimal surfaces in unit spheres:if K(s+1)≤K≤K(s)(K(s):=2/(s(s+1)),s∈N),then either K=K(s)or K=K(s+1).Notice that the surface must be one of Calabi’s standard minimal 2-spheres if the curvature is a positive constant.The cases s=1 and s=2 were proven in the 1980s by Simon and others.In this paper,we give a pinching theorem of the Simon conjecture in the case s=3 and also give a new proof of the cases s=1 and s=2 by some Simons-type integral inequalities.
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）.
文摘In this paper we show that, under some conditions, if M is a manifold with Bakry-émery Ricci curvature bounded below and with bounded potential function then M is compact. We also establish a volume comparison theorem for manifolds with nonnegative Bakry-émery Ricci curvature which allows us to prove a topolological rigidity theorem for such manifolds.
文摘We first study the Grassmannian manifoldG n (Rn+p)as a submanifold in Euclidean space Λ n (R n+p). Then we give a local expression for each map from Riemannian manifoldM toG n (R n+p) ?Λ n (R n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG n (R n+p). As a consequence of this formula we get a rigidity theorem.
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National Natural Science Foundation of China (Grant Nos. 11531012, 11371315 and 11601478)the China Postdoctoral Science Foundation (Grant No. 2016M590530)。
文摘In this paper,we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in Rn+1,and if the squared norm of the second fundamental form of M satisfies 0≤S-1≤1/18,then S≡1 and M is a round sphere or a cylinder.More generally,let M be a complete λ-hypersurface of codimension one with polynomial volume growth in Rn+1 with λ≠0.Then we prove that there exists a positive constant γ,such that if |λ|≤γ and the squared norm of the second fundamental form of M satisfies0≤S-βλ≤1/18,then S≡βλ,λ> 0 and M is a cylinder.Here βλ=1/2(2+λ2+|λ|(λ2+4)1/2).
基金Supported by NSFC(Grant Nos.11271062,NCET–13–0721)
文摘By studying modular invariance properties of some characteristic forms, we get some generalized anomaly cancellation formulas on(4 r-1)-dimensional manifolds with no assumption that the 3 rd de-Rham cohomology of manifolds vanishes. These anomaly cancellation formulas generalize our previous anomaly cancellation formulas on(4 r-1)-dimensional manifolds. We also generalize our previous anomaly cancellation formulas on(4 r-1)-dimensional manifolds and the Han–Yu rigidity theorem to the(a, b) case.
文摘In this paper,we give certain homotopy and diffeomorphism versions as a generalization to an earlier result due to W.S.Cheung,Bun Wong and Stephen S.T.Yau concerning a local rigidity problem of the tangent bundle over compact surfaces of general type.
基金supported by JSPS Grant-in-Aid for Scientific Research(B)(No.16H03937)the fund of Fukuoka University(No.225001)+2 种基金the National Natural Science Foundation of China(No.12171164)the Natural Science Foundation of Guangdong Province(No.2019A1515011451)GDUPS(2018)。
文摘In this paper, the authors give a survey about λ-hypersurfaces in Euclidean spaces. Especially, they focus on examples and rigidity of λ-hypersurfaces in Euclidean spaces.
文摘Let x:M→S^(n+p)(1)be an n-dimensional submanifold immersed in an(n+p)-dimensional unit sphere S^(n+p)(1).In this paper,we study n-dimensional submanifolds immersed in S^(n+p)(1)which are critical points of the functional S(x)=∫_(M)S^(n/2)dv,where S is the squared length of the second fundamental form of the immersion x.When x:M→S^(2+p)(1)is a surface in S^(2+p)(1),the functional S(x)=∫_(M)S^(n/2)dv represents double volume of image of Gaussian map.For the critical surface of S(x),we get a relationship between the integral of an extrinsic quantity of the surface and its Euler characteristic.Furthermore,we establish a rigidity theorem for the critical surface of S(x).
