Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux

The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood–Paley theory, in a noncommutative setting, defined via heat flow on surfaces. Received: April 2004 Revision: June 2005 Accepted: July 2005 The first author is partially supported by NSF grant DMS-0070696. The second author is a Clay Prize Fellow and is partially supported by NSF grant DMS-01007791.     

On space-like hypersurfaces in the de Sitter space

This paper gives the intrinsic conditions for a compact space-like hypersurface in a de Sitter space to be isometric to a sphere.     

ON A CONJECTURE OF K. OGIUE FOR KAEHLER HYPERSURFACES

ＯＮ Ａ ＣＯＮＪＥＣＴＵＲＥ ＯＦ Ｋ． ＯＧＩＵＥ ＦＯＲ ＫＡＥＨＬＥＲ ＨＹＰＥＲＳＵＲＦＡＣＥＳ￥ＳＨＡＮＧＷＥＩＭＩＮＧＡｂｓｔｒａｃｔ：ＡｎａｆｆｉｒｍａｔｉｖｅａｎｓｗｅｒｔｏａｃｏｎｊｅｃｔｕｒｅｏｆＫ．Ｏｇｉｕｅｆｏｒｍｕｌａｔｅ...     

Some connections between symmetry results for semilinear PDE in real and hyperbolic spaces

Taking advantage of the “invariance” under conformal transformations of certain elliptic operators and combining it with symmetry results obtained by moving totally geodesic hypersurfaces in Hn, we are able to prove the symmetry of positive solutions of −Δu=f(r,u), in balls in Rn, for a class of nonlinearities that do not satisfy the classical hypothesis of f being decreasing in r.     

S4中等参超曲面的谱刻画

本文证明了如果Ｓ⁴中的具常平均曲率ｈ的超曲面Ｍ与其具平均曲率ｈ的等参超曲面Ｍ_０（强）等谱，则Ｍ＝Ｍ_０．     

Spacelike hypersurfaces with constant higher order mean curvature in Minkowski space–time

In this paper, we develop a series of general integral formulae for compact spacelike hypersurfaces with hyperplanar boundary in the (n+1)-dimensional Minkowski space–time . As an application of them, we prove that the only compact spacelike hypersurfaces in having constant higher order mean curvature and spherical boundary are the hyperplanar balls (with zero higher order mean curvature) and the hyperbolic caps (with nonzero constant higher order mean curvature). This extends previous results obtained by the first author, jointly with Pastor, for the case of constant mean curvature [J. Geom. Phys. 28 (1998) 85] and the case of constant scalar curvature [Ann. Global Anal. Geom. 18 (2000) 75].     

On submanifolds whose tubular hypersurfaces have constant higher order mean curvatures

Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k~v(k ≥ 1) of a submanifold M~n-is defined as the k-th power sum of the principal curvatures,or equivalently,of the shape operator with respect to the unit normal vector v.We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures,then the submanifold M itself has some constant higher order mean curvatures Q_k~v independent of the choice of v.Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained.In particular,we generalize several classical results in isoparametric theory given by E.Cartan,K.Nomizu,H.F.Miinzner,Q.M.Wang,et al.As an application,we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.     

Polar actions on compact Euclidean hypersurfaces

Given an isometric immersion of a compact Riemannian manifold of dimension n ≥ 3 into Euclidean space of dimension n + 1, we prove that the identity component Iso ⁰(M ⁿ ) of the isometry group Iso(M ⁿ ) of M ⁿ admits an orthogonal representation such that for every . If G is a closed connected subgroup of Iso(M ⁿ ) acting polarly on M ⁿ , we prove that Φ(G) acts polarly on , and we obtain that f(M ⁿ ) is given as Φ(G)(L), where L is a hypersurface of a section which is invariant under the Weyl group of the Φ(G)-action. We also find several sufficient conditions for such an f to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their underlying warped product structure.