Quasilinearization and curvature of Aleksandrov spaces

We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine—a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space is an Aleksandrov domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in . We also observe that a geodesically connected metric space is an domain if and only if, for every quadruple of points in , the quadrilateral inequality (known as Euler’s inequality in ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.      

The mean curvature of cylindrically bounded submanifolds

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder in a product Riemannian manifold . It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabi on complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature. Dedicated to Professor Manfredo P. do Carmo on the occasion of his 80th birthday.     

Highly degenerate harmonic mean curvature flow

We study the evolution of a weakly convex surface in with flat sides by the Harmonic Mean Curvature flow. We establish the short time existence as well as the optimal regularity of the surface and we show that the boundaries of the flat sides evolve by the curve shortening flow. It follows from our results that a weakly convex surface with flat sides of class C ^k,γ , for some and 0  <  γ ≤ 1, remains in the same class under the flow. This distinguishes this flow from other, previously studied, degenerate parabolic equations, including the porous medium equation and the Gauss curvature flow with flat sides, where the regularity of the solution for t  >  0 does not depend on the regularity of the initial data. M. C. Caputo partially supported by the NSF grant DMS-03-54639. P. Daskalopoulos partially supported by the NSF grants DMS-01-02252, DMS-03-54639 and the EPSRC in the UK.     

The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

We study constant mean curvature graphs in the Riemannian three- dimensional Heisenberg spaces . Each such is the total space of a Riemannian submersion onto the Euclidean plane with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in with respect to the Riemannian submersion over certain domains taking on prescribed boundary values. L. J. Alías was partially supported by MEC/FEDER project MTM2004-04934-C04-02 and Fundación Séneca project 00625/PI/04, Spain.     

Stable constant mean curvature hypersurfaces in the real projective space

In this paper, we prove that the only compact two-sided hypersurfaces with constant mean curvature H which are weakly stable in and have constant scalar curvature are (i) the twofold covering of a totally geodesic projective space; (ii) the geodesic spheres in ; and (iii) the quotient to of the hypersurface obtained as the product of two spheres of dimensions k and n − k, with k = 1,..., n − 1, and radii r and , respectively, with .     

On static manifolds and related critical spaces with zero radial Weyl curvature

Monatshefte für Mathematik - The aim of this paper is to study compact Riemannian manifolds $$(M,\,g)$$ that admit a non-constant solution to the system of equations $$\begin{aligned} -\Delta...     

Mean curvature rigidity of horospheres,hyperspheres, and hyperplanes

Archiv der Mathematik - We prove that horospheres, hyperspheres, and hyperplanes in a hyperbolic space $${\mathbb {H}}^n,\,n\ge 3$$ , admit no perturbations with compact support which increase...     

Free boundary constant mean curvature surfaces in a strictly convex three-manifold

Annals of Global Analysis and Geometry - Let C be a strictly convex domain in a three-dimensional Riemannian manifold with sectional curvature bounded above by a constant, and let $$\Sigma $$ be a...     

Conformal deformations of prescribing scalar curvature on Riemannian manifolds with negative curvature

We study the conformal deformation for prescribing scalar curvature function on Cartan-Hadamard manifoldM ⁿ (n≥3) with strongly negative curvature. By employing the supersubsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for near infinity so that the problem of complete conformal deformation is solvable. In more general cases, we prove an asymptotic estimation on the solutions of the conformal scalar curvature equation. Project partially supported by the NNSF of China     

Constant mean curvature graphs in a class of warped product spaces

We introduce a new existence result for compact normal geodesic graphs with constant mean curvature and boundary in a class of warped product spaces. In particular, our result includes that of normal geodesic graphs with constant mean curvature in hyperbolic space over a bounded domain in a totally geodesic .      

On constant mean curvature hypersurfaces with finite index

We discuss the non-existence of complete noncompact constant mean curvature hypersurfaces with finite index in a 4- or 5-dimensional manifold. As a consequence, we obtain that a complete noncompact constant mean curvature hypersurface in with finite index must be minimal. Received: 30 May 2005     

On complete noncompact submanifolds with constant mean curvature and finite total curvature in Euclidean spaces

In this paper, we consider a complete noncompact n-submanifold M with parallel mean curvature vector h in an Euclidean space. If M has finite total curvature, we prove that M must be minimal, so that M is an affine n-plane if it is strongly stable. This is a generalization of the result on strongly stable complete hypersurfaces with constant mean curvature in Received: 30 June 2005     

On stable surfaces of prescribed mean curvature with partially free boundaries

We study surfaces of prescribed bounded mean curvature H in a partially free boundary configuration . If is projectable onto and is a cylinder surface over the x¹,x²-plane, we show that also the spanned H-surface is projectable onto this plane. Besides certain conditions on and H, we have to suppose that is stationary and freely stable w.r.t. the generalized area functional , and the vector field is assumed to be tangential on . Consequences are uniqueness of freely stable H-surfaces and solvability of a mixed boundary problem for the nonparametric prescribed mean curvature equation. Mathematics Subject Classification (2000): 53 A 10, 35 J 65, 35 R 35, 49 Q 05     

Asymptotic symmetries for conformal scalar curvature equations with singularity

We give conditions on a positive Hölder continuous function C²such that every C ² positive solution u((x)) of the conformal scalar curvature equation in a punctured neighborhood of the origin in R ⁿ either has a removable singularity at the origin or satisfies for some positive singular solution u ₀(|x|) of where is the Hölder exponent of K.Mathematics Subject Classification (2000) Primary 35J60, 53C21     

Submanifolds of positive Ricci curvature in a Euclidean space

In this paper we study the role of constant vector fields on a Euclidean space R ^n+p in shaping the geometry of its compact submanifolds. For an n-dimensional compact submanifold M of the Euclidean space R ^n+p with mean curvature vector field H and a constant vector field on R ^n+p , the smooth function is used to obtain a characterization of sphere among compact submanifolds of positive Ricci curvature (cf. main Theorem).      

Affine normal curvature of hypersurfaces from the point of view of singularity theory

We consider smoothly embedded hypersurfaces under the action of the special affine group . We construct a differential invariant, called affine normal curvature, which assigns to a point and a tangent direction a number. We prove some of its nice properties which connect it with affine principal directions, affine umbilics, and affine mean curvature.      

Constant mean curvature hypersurfaces with two principal curvatures in a sphere

In this paper we consider a compact oriented hypersurface M ⁿ with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S ⁿ⁺¹. Denote by the trace free part of the second fundamental form of M ⁿ , and Φ be the square of the length of . We obtain two integral formulas by using Φ and the polynomial . Assume that B _H,m is the square of the positive root of P _H,m (x) = 0. We show that if M ⁿ is a compact oriented hypersurface immersed in the sphere S ⁿ⁺¹ with constant mean curvatures H having two distinct principal curvatures λ and μ then either or . In particular, M ⁿ is the hypersurface .      

Equi-Gaussian curvature folding

In this paper we introduce a new type of folding called equi-Gaussian curvature folding of connected Riemannian 2-manifolds. We prove that the composition and the cartesian product of such foldings is again an equi-Gaussian curvature folding. In case of equi-Gaussian curvature foldings, f: M → P _n , of an orientable surface M onto a polygon P _n we prove that

The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle

Geometriae Dedicata - Let $$\pi :\mathcal {X}\rightarrow M$$ be a holomorphic fibration with compact fibers and L a relatively ample line bundle over $$\mathcal {X}$$ . We obtain the asymptotic of...     

On the generalized Chern conjecture for hypersurfaces with constant mean curvature in a sphere

Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constant γ(n) depending only on n such that if |H| ≤γ(n) and β(n,H)≤S ≤β(n,H)+n/18,then S≡β(n,H) and M is a Clifford torus.Here,β(n,H)=n+n~3/2(n-1)H~2+n(n-2)/2(n-1)(1/2)n~2H~4+4(n-1)H~2.