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.     

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.     

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.     

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.     

Hypersurfaces with Flat Centroaffine Metric and Equations of Associativity

It is demonstrated that hypersurfaces M ⁿ A ⁿ⁺¹ with a flat centroaffine metric are governed by a system of nonlinear PDEs known as the equations of associativity of 2-dimensional topological field theory. In the case of surfaces M ²A ³ this system reduces to a single third-order PDE, f _{x x x} f _{y y y} –f _{x x y} f _{x y y} =1 where x and y are the asymptotic coordinates on M ².     

Centroaffine minimal hypersurfaces in ℝ n+1

In this paper the first and the second variation formulas for the area integral of the centroaffine metric of hypersurfaces in ⁿ⁺¹ are calculated, and some interesting examples of stable and unstable centroaffine minimal hypersurfaces are given.Partially supported by the DFG-project Affine Differential Geometry at the TU Berlin.     

δ-invariants and their applications to centroaffine geometry

We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in Rn+1. The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in , satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article.     

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Characterization off-vectors of families of convex sets inR d Part I: Necessity of Eckhoff’s conditions

LetK=K ₁,...,K_n be a family ofn convex sets inR ^d . For 0≦i<n denote byf _i the number of subfamilies ofK of sizei+1 with non-empty intersection. The vectorf(K) is called thef-vectors ofK. In 1973 Eckhoff proposed a characterization of the set off-vectors of finite families of convex sets inR ^d by a system of inequalities. Here we prove the necessity of Eckhoff's inequalities. The proof uses exterior algebra techniques. We introduce a notion of generalized homology groups for simplicial complexes. These groups play a crucial role in the proof, and may be of some independent interest.     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

Ideal einstein,conformally flat and semi-symmetric immersions

Recently, B. Y. Chen introduced a new intrinsic invariant of a manifold, and proved that everyn-dimensional submanifold of real space formsR ^m (ε) of constant sectional curvature ε satisfies a basic inequality δ(n ₁,…,n _k )≤c(n ₁,…,n _k )H ²+b(n ₁,…,n _k )ε, whereH is the mean curvature of the immersion, andc(n ₁,…,n _k ) andb(n ₁,…,n _k ) are constants depending only onn ₁,…,n _k ,n andk. The immersion is calledideal if it satisfies the equality case of the above inequality identically for somek-tuple (n ₁,…,n _k ). In this paper, we first prove that every ideal Einstein immersion satisfyingn≥n ₁+…+n _k +1 is totally geodesic, and that every ideal conformally flat immersion satisfyingn≥n ₁+…+n _k +2 andk≥2 is also totally geodesic. Secondly we completely classify all ideal semi-symmetric hypersurfaces in real space forms. The author was supported by the NSFC and RFDP.     

Turbulence Phenomena in Real Analysis

The purpose of this paper is first to show that if X is any locally compact but not compact perfect Polish space and stands for the one-point compactification of X, while E^X is the equivalence relation which is defined on the Polish group C(X,R₊*) by where f, g are in C(X,R₊*), then E^X is induced by a turbulent Polish group action. Second we show that given any if we identify the n-dimensional unit sphere Sⁿ with the one-point compactification of Rⁿ via the stereographic projection, while E_n,r is the equivalence relation which is defined on the Polish group C^r(Rⁿ,R₊*) by where f, g are in C^r(Rⁿ,R₊*), then E_n,r is also induced by a turbulent Polish group action. Dedicated to my sister Alexandra and to her daughter Marianthi.     

Higher order Willmore hypersurfaces in Euclidean space

Let x ： Mn^n→ R^n＋1 be an n（≥2）-dimensional hypersurface immersed in Euclidean space Rn＋1. Let σi（0≤ i≤ n） be the ith mean curvature and Qn = ∑i=0^n（-1）^i＋1 （n^i）σ1^n-iσi. Recently, the author showed that Wn（x） = ∫M QndM is a conformal invariant under conformal group of R^n＋1 and called it the nth Willmore functional of x. An extremal hypersurface of conformal invariant functional Wn is called an nth order Willmore hypersurface. The purpose of this paper is to construct concrete examples of the 3rd order Willmore hypersurfaces in Ra which have good geometric behaviors. The ordinary differential equation characterizing the revolutionary 3rd Willmore hypersurfaces is established and some interesting explicit examples are found in this paper.     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

Integral formula of Minkowski type and new characterization of the Wulff shape

Given a positive function F on S^n which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature Mr for hypersurfaces in R^n＋1 which is a generalization of the usual r-th mean curvature Hr. We get integral formulas of Minkowski type for compact hypersurfaces in R^n＋1. We give some new characterizations of the Wulff shape by the use of our integral formulas of Minkowski type, in case F=1 which reduces to some well-known results.     

Admissible functions with multiple discontinuities

LetT be the unit circle, α irrational andF: T → R a step function. A necessary and sufficient condition for the skew of the α-rotation byf (considered as taking values mod 1) to be minimal is given. Also, the boundedness of Σ _i=1 ⁿ f(x+iα asn → α is resolved.     

Unique continuation for parabolic operators

It is shown that if a functionu satisfies a backward parabolic inequality in an open set Ω∉R ⁿ⁺¹ and vanishes to infinite order at a point (x ₀·t ₀) in Ω, thenu(x, t ₀)=0 for allx in the connected component ofx ₀ in Ω⌢(R ⁿ ×{t ₀}).     

On double sine and cosine transforms,Lipschitz and Zygmund classes

We consider complex-valued functions f ∈ L 1 (R+2),where R +:= [0,∞),and prove sufficient conditions under which the double sine Fourier transform f ss and the double cosine Fourier transform f cc belong to one of the two-dimensional Lipschitz classes Lip(α,β) for some 0 α,β≤ 1;or to one of the Zygmund classes Zyg(α,β) for some 0 α,β≤ 2.These sufficient conditions are best possible in the sense that they are also necessary for nonnegative-valued functions f ∈ L 1 (R+2).     

A representation for the remainder of the Maclaurin quadrature formula

Summary We show that the remainder of the Maclaurin quadrature formula belonging to oddn (n+1 is the number of nodes) can be represented asR _n (f)=c _n f ⁽ⁿ⁺¹⁾ (), wheneverf ⁽ⁿ⁺¹⁾ exists and is continuous The corresponding problem for evenn has already been settled by A. Walther in 1925.