In this article, we introduce the Hausdorff convergence to derive a differentiable sphere theorem which shows an interesting rigidity phenomenon on some kind of manifolds.
Let Mn be a compact, simply connected n (≥3)-dimensional Riemannian manifold without bound-ary and Sn be the unit sphere Euclidean space Rn+1. We derive a differentiable sphere theorem whenever themanifold concerned ...Let Mn be a compact, simply connected n (≥3)-dimensional Riemannian manifold without bound-ary and Sn be the unit sphere Euclidean space Rn+1. We derive a differentiable sphere theorem whenever themanifold concerned satisfies that the sectional curvature KM is not larger than 1, while Ric(M)≥n+2 4 and the volume V (M) is not larger than (1 + η)V (Sn) for some positive number η depending only on n.展开更多
The precise inner solutions of gravity field equations of hollow and solid spheres are calculated in this paper. To avoid space curvature infinite at the center of solid sphere, we set an integral constant to be zero ...The precise inner solutions of gravity field equations of hollow and solid spheres are calculated in this paper. To avoid space curvature infinite at the center of solid sphere, we set an integral constant to be zero directly at present. However, according to the theory of differential equation, the integral constant should be determined by the known boundary conditions of spherical surface, in stead of the metric at the spherical center. By considering that fact that the volumes of three dimensional hollow and solid spheres in curved space are different from that in flat space, the integral constants are proved to be nonzero. The results indicate that no matter what the masses and densities of hollow sphere and solid sphere are, there exist space-time singularities at the centers of hollow sphere and solid spheres. Meanwhile, the intensity of pressure at the center point of solid sphere can not be infinite. That is to say, the material can not collapse towards the center of so-called black hole. At the center and its neighboring region of solid sphere, pressure intensities become negative values. There may be a region for hollow sphere in which pressure intensities may become negative values too. The common hollow and solid spheres in daily live can not have such impenetrable characteristics. The results only indicate that the singularity black holes predicated by general relativity are caused by the descriptive method of curved space-time actually. If black holes exist really in the universe, they can only be the Newtonian black holes, not the Einstein’s black holes. The results revealed in the paper are consistent with the Hawking theorem of singularity actually. They can be considered as the practical examples of the theorem.展开更多
In this paper,we prove several convergence theorems for the mean curvature flow of n-dimensional closed submanifolds in the unit sphere S^(n+k)under integral curvature pinching conditions.In particular,we prove that i...In this paper,we prove several convergence theorems for the mean curvature flow of n-dimensional closed submanifolds in the unit sphere S^(n+k)under integral curvature pinching conditions.In particular,we prove that if the L^(n)-norm of the second fundamental form of the initial submanifold is small enough,then the mean curvature flow either shrinks to a round point in finite time,or converges to a totally geodesic submanifold as the time tends to infinity.As a consequence of the smooth convergence theorems,we obtain several differentiable sphere theorems for certain submanifolds in S^(n+k).展开更多
基金Supported by the NNSF of China (10671066)the NSF of Shandong Province (Q2008A08)Scientific Research Foundation of QFNU
文摘In this article, we introduce the Hausdorff convergence to derive a differentiable sphere theorem which shows an interesting rigidity phenomenon on some kind of manifolds.
基金supported by National Natural Science Foundation of China (Grant No.10871069)the Youth Natural Science Foundation of Shandong Province (Grant No. Q2008A08)the Youth Foundation of Qufu Normal University
文摘Let Mn be a compact, simply connected n (≥3)-dimensional Riemannian manifold without bound-ary and Sn be the unit sphere Euclidean space Rn+1. We derive a differentiable sphere theorem whenever themanifold concerned satisfies that the sectional curvature KM is not larger than 1, while Ric(M)≥n+2 4 and the volume V (M) is not larger than (1 + η)V (Sn) for some positive number η depending only on n.
文摘The precise inner solutions of gravity field equations of hollow and solid spheres are calculated in this paper. To avoid space curvature infinite at the center of solid sphere, we set an integral constant to be zero directly at present. However, according to the theory of differential equation, the integral constant should be determined by the known boundary conditions of spherical surface, in stead of the metric at the spherical center. By considering that fact that the volumes of three dimensional hollow and solid spheres in curved space are different from that in flat space, the integral constants are proved to be nonzero. The results indicate that no matter what the masses and densities of hollow sphere and solid sphere are, there exist space-time singularities at the centers of hollow sphere and solid spheres. Meanwhile, the intensity of pressure at the center point of solid sphere can not be infinite. That is to say, the material can not collapse towards the center of so-called black hole. At the center and its neighboring region of solid sphere, pressure intensities become negative values. There may be a region for hollow sphere in which pressure intensities may become negative values too. The common hollow and solid spheres in daily live can not have such impenetrable characteristics. The results only indicate that the singularity black holes predicated by general relativity are caused by the descriptive method of curved space-time actually. If black holes exist really in the universe, they can only be the Newtonian black holes, not the Einstein’s black holes. The results revealed in the paper are consistent with the Hawking theorem of singularity actually. They can be considered as the practical examples of the theorem.
基金supported by National Natural Science Foundation of China(Grant Nos.11531012,12071424,12171423 and 12471051)。
文摘In this paper,we prove several convergence theorems for the mean curvature flow of n-dimensional closed submanifolds in the unit sphere S^(n+k)under integral curvature pinching conditions.In particular,we prove that if the L^(n)-norm of the second fundamental form of the initial submanifold is small enough,then the mean curvature flow either shrinks to a round point in finite time,or converges to a totally geodesic submanifold as the time tends to infinity.As a consequence of the smooth convergence theorems,we obtain several differentiable sphere theorems for certain submanifolds in S^(n+k).
