Regularity of averages over hypersurfaces

Averages over smooth measures on smooth compact hypersurfaces inR ⁿ are studied. With assumptions on the decay of the Fourier transform of the measure we obtain mixed norm estimates for these means, for exampleL ^p estimates of multiparameter maximal functions over compact hypersurfaces.     

Closed Characteristics on Asymmetric Convex Hypersurfaces in R2n and the Corresponding Pinching Conditions

Abstract In this paper, we construct first a new concrete example of asymmetric convex compact C ^1,1-hypersurfaces in R ²ⁿ possessing precisely n closed characteristics. Then we prove multiplicity results on the closed characteristics on convex compact hypersurfaces in R ²ⁿ pinched by not necessarily symmetric convex compact hypersurfaces. *Partially supported by the 973 Program of STM, Funds of EC of Jiangsu, the Natural Science Funds of Jiangsu (BK 2002023), the Post-doctorate Funds of China, and the NNSF of China (10251001) **Partially supported by the 973 Program of STM, NNSF, MCME, RFDP, PMC Key Lab of EM of China, S. S. Chern Foundation, and Nankai University     

Spherical-type hypersurfaces in a Riemannian manifold

We extend some rigidity results of Aleksandrov and Ros on compact hypersurfaces inR ⁿ to more general ambient spaces with the aid of the notion of almost conformal vector fields. These latter, at least locally, always exist and allow us to find interesting integral formulas fitting our purposes.     

Weak type estimates for maximal operators over convex hypersurfaces

We prove sharp weak type (p,p) estimates on H ^p spaces for the maximal operators with a rough distance function over convex hypersurfaces.     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

An Integral Formula for Lipschitz-Killing Curvature and the Critical Points of Height Functions

We have established (see Shiohama and Xu in J. Geom. Anal. 7:377–386, 1997; Lemma) an integral formula on the absolute Lipschitz-Killing curvature and critical points of height functions of an isometrically immersed compact Riemannian n-manifold into R ^n+q . Making use of this formula, we prove a topological sphere theorem and a differentiable sphere theorem for hypersurfaces with bounded L ^n/2 Ricci curvature norm in R ⁿ⁺¹. We show that the theorems of Gauss-Bonnet-Chern, Chern-Lashof and the Willmore inequality are all its consequences.     

Singular homologically area minimizing surfaces of codimension one in Riemannian manifolds

For n≥7, it is shown how to construct examples of smooth, compact Riemannian manifolds (N ^{n +1},g), with non-trivial n dimensional integer homology, such that for some Γ∈H _n (N,Z), the hypersurface (n-current) M, which minimizes area among all hypersurfaces representing Γ, has singularities. The singular set of M consists of two isolated points, and the tangent cone at these points can be prescribed as any strictly stable, strictly minimizing, regular cone. To my knowledge these are the first examples of codimension one homological minimizers with singularities. Oblatum: 3-I-1997 & 13-II-1998 / Published online: 18 September 1998     

Harnack estimate for curvature flows depending on mean curvature

The maximum principle is applied to prove the Harnack estimate of curvature flows of hypersurfaces in Rn＋1,where the normal velocity is given by a smooth function f depending only on the mean curvature.By use of the estimate,some corollaries are obtained including the integral Harnack inequality.In particular,the conditions are given with which the solution to the flows is a translation soliton or an expanding soliton.     

On Complete Hypersurfaces with Constant Mean Curvature and Finite L^P-norm Curvature in R^n＋1

By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1＋1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n＋1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1＋1 are considered.     

Existence of codimension one foliations with minimal leaves

Every smooth closed manifold of dimension 4 or greater that has a smooth codimension one foliation, has such aC ¹ foliation whose leaves are minimal hypersurfaces for someC ¹ Riemannian metric.     

The Degree of Symmetry of Certain Compact Smooth Manifolds Ⅱ

We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2×V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.     

Prescribed Mean Curvature Hypersurfaces in Hn+1(-1) with Convex Planar Boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hⁿ⁺¹(-1). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.     

On hypersurfaces inR n+1 with integral bounds on curvature

We show that the L ^p norm of the second fundamental form of hypersurfaces in R ⁿ⁺¹ is very coercive in the GMT setting of Gauss graphs currents, when p exceeds the dimension n. A compactness result for immersed hypersurfaces and its application to a variational problem are provided.     

Interpolation and sampling hypersurfaces for the Bargmann-Fock space in higher dimensions

We study those smooth complex hypersurfaces W in having the property that all holomorphic functions of finite weighted L^p norm on W extend to entire functions with finite weighted L^p norm. Such hypersurfaces are called interpolation hypersurfaces. We also examine the dual problem of finding all sampling hypersurfaces, i.e., smooth hypersurfaces W in such that any entire function with finite weighted L^p norm is stably determined by its restriction to W. We provide sufficient geometric conditions on the hypersurface to be an interpolation or sampling hypersurface. The geometric conditions that imply the extension property and the restriction property are given in terms of some directional densities. The first author is supported by projects MTM 2005-08984-Co2-O2 and 2001SGR00611 The third author is partially supported by NSF grant DMS0400909     

A finiteness theorem for isoparametric hypersurfaces

In this note we prove that for eachn there are only finitely many diffeomorphism classes of compact isoparametric hypersurfaces ofS ⁿ⁺¹ with four distinct principal curvatures.     

Estimates for stable hypersurfaces of prescribed F-mean curvature

We consider immersed hypersurfaces :Mⁿ→ℝⁿ⁺¹ with prescribed anisotropic mean curvature . Such hypersurfaces can be characterized as critical points of parametric functionals of the type with an elliptic Lagrangian F depending on normal directions and a smooth vectorfield Q satisfying . We establish curvature estimates for stable hypersurfaces of dimension n≤5, provided F is C³-close to the area integrand.     

（L^p,L^q） Estimates for Multilinear Oscillatory Singular Integralswith Smooth Phases

In this paper,the authors establish the weighted (L^p,L^q) estimates for a class of multilinear oscillatory singular integrals with smooth phases.Certain endpoint estimates are also considered.     

An Optimal Inequality and Extremal Classes of Affine Spheres in Centroaffine Geometry

A hypersurface f : M → Rⁿ⁺¹ in an affine (n+1)-space is called centroaffine if its position vector is always transversal to f_*(TM) in Rⁿ⁺¹. In this paper, we establish a general optimal inequality for definite centroaffine hypersurfaces in Rⁿ⁺¹ involving the Tchebychev vector field. We also completely classify the hypersurfaces which verify the equality case of the inequality.     

Convex mean curvature flow with a forcing term in direction of the position vector

A smooth, compact and strictly convex hypersurface evolving in ℝ ⁿ⁺¹ along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.