In this paper, let n ≥ 2 be an integer, P = diag（-In-k,In-k,Ik） for some integer κ∈[0, n）, and ∑∪→R^2n be a partially symmetric compact convex hypersurface, i.e., x ∈∑ implies Px∈∑. We prove that if ∑ is...In this paper, let n ≥ 2 be an integer, P = diag（-In-k,In-k,Ik） for some integer κ∈[0, n）, and ∑∪→R^2n be a partially symmetric compact convex hypersurface, i.e., x ∈∑ implies Px∈∑. We prove that if ∑ is （r, R）-pinched with R/r〈 √2, then there exist at least n -k geometrically distinct P-symmetric closed ∑ characteristics on ∑, as a consequence, Z carry at least n geometrically distinct P-invariant closed characteristics.展开更多
The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula...The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface with some low entropy is diffeomorphic to a round sphere. We also prove that a smooth selfshrinker with low entropy is a hyperplane.展开更多
基金supported by China Postdoctoral Science Foundation(Grant No.2013M540512)National Natural Science Foundation of China(Grant Nos.10801078,11171341 and 11271200)Lab of Pure Mathematics and Combinatorics of Nankai University
文摘In this paper, let n ≥ 2 be an integer, P = diag（-In-k,In-k,Ik） for some integer κ∈[0, n）, and ∑∪→R^2n be a partially symmetric compact convex hypersurface, i.e., x ∈∑ implies Px∈∑. We prove that if ∑ is （r, R）-pinched with R/r〈 √2, then there exist at least n -k geometrically distinct P-symmetric closed ∑ characteristics on ∑, as a consequence, Z carry at least n geometrically distinct P-invariant closed characteristics.
基金Project supported by the NSERC Discovery Grant, the Fok Ying Tung Eduction Foundation the National Natural Science Foundation of China (No.10371041) Hundred Talents Program of Chinese Academy of Sciences.
文摘In this paper the authors discuss the existence and convexity of hypersurfaces with prescribed Weingarten curvature.
文摘The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface with some low entropy is diffeomorphic to a round sphere. We also prove that a smooth selfshrinker with low entropy is a hyperplane.
