In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S^(6), which also yields to a description of such surfaces in terms of the loop group language. Moreover, appl...In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S^(6), which also yields to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, he also obtains an algorithm to construct totally isotropic Willmore two-spheres in S^(6). This allows him to derive new examples of geometric interests. He first obtains a new, totally isotropic Willmore two-sphere which is not S-Willmore(i.e., has no dual surface) in S^(6). This gives a negative answer to an open problem of Ejiri in 1988. In this way he also derives many new totally isotropic, branched Willmore two-spheres which are not S-Willmore in S^(6).展开更多
Let M^n be a compact Willmore submanifold in the unit sphere Sn＋p. In this note, we investigate the first eigenvalue of the SchrSdinger operator L = -△ - q on M, where q is some potential function on M, and present...Let M^n be a compact Willmore submanifold in the unit sphere Sn＋p. In this note, we investigate the first eigenvalue of the SchrSdinger operator L = -△ - q on M, where q is some potential function on M, and present a gap estimate for the first eigenvalue of L.展开更多
Let M be a closed Willmore hypersurface in the sphere S^n＋1（1） （n ≥ 2） with the same mean curvature of the Willmore torus Wm,n-m, if SpecP（M） = Spec^P（Wm,n-m ） （p = 0, 1,2）, then M is Wm,n-m.
Let M be a compact convex hypersurface of class C2, which is assumed to bound a nonempty convex body K in the Euclidean space Rn and H be the mean curvature of M. We obtain a lower bound of the total square of mean cu...Let M be a compact convex hypersurface of class C2, which is assumed to bound a nonempty convex body K in the Euclidean space Rn and H be the mean curvature of M. We obtain a lower bound of the total square of mean curvature fM H2dA The bound is the Minkowski quermassintegral of the convex body K. The total square of mean curvature attains the lower bound when M is an （n - 1）-sphere.展开更多
基金supported by the National Natural Science Foundation of China(Nos.11971107,11571255).
文摘In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S^(6), which also yields to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, he also obtains an algorithm to construct totally isotropic Willmore two-spheres in S^(6). This allows him to derive new examples of geometric interests. He first obtains a new, totally isotropic Willmore two-sphere which is not S-Willmore(i.e., has no dual surface) in S^(6). This gives a negative answer to an open problem of Ejiri in 1988. In this way he also derives many new totally isotropic, branched Willmore two-spheres which are not S-Willmore in S^(6).
基金supported by the National Natural Science Foundation of China(Grant No.11701565)the Research Project of National University of Defense Technology(Grant No.ZK17-03-29)。
基金Supported by the National Natural Science Foundation of China(11071211)the Zhejiang Natural Science Foundation of China
文摘Let M^n be a compact Willmore submanifold in the unit sphere Sn＋p. In this note, we investigate the first eigenvalue of the SchrSdinger operator L = -△ - q on M, where q is some potential function on M, and present a gap estimate for the first eigenvalue of L.
文摘Let M be a closed Willmore hypersurface in the sphere S^n＋1（1） （n ≥ 2） with the same mean curvature of the Willmore torus Wm,n-m, if SpecP（M） = Spec^P（Wm,n-m ） （p = 0, 1,2）, then M is Wm,n-m.
文摘Let M be a compact convex hypersurface of class C2, which is assumed to bound a nonempty convex body K in the Euclidean space Rn and H be the mean curvature of M. We obtain a lower bound of the total square of mean curvature fM H2dA The bound is the Minkowski quermassintegral of the convex body K. The total square of mean curvature attains the lower bound when M is an （n - 1）-sphere.
