Regularity of Harmonic Functions in Cheeger-Type Sobolev Spaces

We give a geometric approach to the study of the regularity of harmonic functions in Cheeger-type Sobolev spaces, and prove the Hölder continuity of such functions. In the proof, we give a definition of an upper curvature bound of the unit sphere of a Banach space, which seems to be of independent interest.     

A note on 4-dimensional locally conformally flat Walker manifolds

A 4-dimensional Walker metric on a semi Riemanian manifold M, for the canonical metric with c = 0, have been investigated by M. Chaichi, E. García—Río and Y. Matsushita. The paper generalizes these notions to the case of constant c ≠ 0. Specially the form of defining functions of this metric in locally conformally flat 4-dimensional Walker manifolds is found.     

Nonnegative curvature and cobordism type

We show that in each dimension n = 4k, k≥ 2, there exist infinite sequences of closed simply connected Riemannian n-manifolds with nonnegative sectional curvature and mutually distinct oriented cobordism type. W. Tuschmann’s research was supported in part by a DFG Heisenberg Fellowship.     

The equations of a holomorphic subspace in a complex Finsler space

In [Mu1] we underlined the motifs of holomorphic subspaces in a complex Finsler space: induced nonlinear connection, coupling connections, and the induced tangent and normal connections. In the present paper we investigate the equations of Gauss, H-and A-Codazzi, and Ricci equations of a holomorphic subspace. We deduce the link between the holomorphic curvatures of the Chern-Finsler connection and its induced tangent connection. Conditions for totally geodesic holomorphic subspaces are obtained. Communicated by János Szenthe     

Geometrical study of paraxial light beam transmission in free space

By introducing an imaginary space transform curvature ρ_s, a complex space called Riemannian space is constructed, in which the light propagating in free space has the trajectory of straight line while propagating. Moreover, this curvature couples with that of the wave front of the paraxial beam ρ_w, and therefore a complex curvature ρ_c is constructed, which can be employed to investigate the behavior of the light transmission and to generalize the ABCD law. Project supported by the National Hi-Tech Inertial Confinement Fusion Committee, the Guangdong Natural Science Foundation the Postdoctoral Foundation of Guangdong and National Postdoctoral Foundation of China.     

Hypersurface with Constant Main Curvature Symmetric Functions

ＨｙｐｅｒｓｕｒｆａｃｅｗｉｔｈＣｏｎｓｔａｎｔＭａｉｎＣｕｒｖａｔｕｒｅＳｙｍｍｅｔｒｉｃＦｕｎｃｔｉｏｎｓ￥ＷｕＢａｏｑｉａｎｇ；ＳｏｎｇＨｏｎｇｚａｏ（ＸｕｚｈｏｕＴｅａｃｈｅｒｓＣｏｌｌｅｇｅ，２２１００９）（ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ，...     

Stability of translating solutions to mean curvature flow

We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates. The authors are members of SFB 647/B3 “Raum – Zeit – Materie: Singularity Structure, Long-time Behaviour and Dynamics of Solutions of Non-linear Evolution Equations”.     

Orbifold fibrations of Eschenburg spaces

Most of the few known examples of compact Riemannian manifolds with positive sectional curvature are the total space of a Riemannian submersion. In this article we show that this is true for all known examples, if we enlarge the category to orbifold fibrations. For this purpose we study all almost free isometric circle actions on positively curved Eschenburg spaces, which give rise to principle orbifold bundle structures, and we examine in detail their geometric properties. In particular, we obtain a new family of 6-dimensional orbifolds with positive sectional curvature whose singular locus consists of just two points.      

When is a Riemannian submersion homogeneous?

We study the structure of the most common type of Riemannian submersions, namely those whose fibers are given by the orbits of an isometric group action on a Riemannian manifold. Special emphasis is given to the case where the ambient space has nonnegative curvature. The first author was supported by research grant MTM2004-04794-MEC. Most of this work was done during a visit of the second author to Madrid, financed in part by funds of the aforementioned grant.     

Entire radial solutions of elliptic systems and inequalities of the mean curvature type

In this paper we study first nonexistence of radial entire solutions of elliptic systems of the mean curvature type with a singular or degenerate diffusion depending on the solution u. In particular we extend a previous result given in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389]. Moreover, in the scalar case we obtain nonexistence of all entire solutions, radial or not, of differential inequalities involving again operators of the mean curvature type and a diffusion term. We prove that in the scalar case, nonexistence of entire solutions is due to the explosion of the derivative of every nonglobal radial solution in the right extremum of the maximal interval of existence, while in that point the solution is bounded. This behavior is qualitatively different with respect to what happens for the m-Laplacian operator, studied in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389], where nonexistence of entire solutions is due, even in the vectorial case, to the explosion in norm of the solution at a finite point. Our nonexistence theorems for inequalities extend previous results given by Naito and Usami in [Y. Naito, H. Usami, Entire solutions of the inequality div(A(|Du|)Du)?f(u), Math. Z. 225 (1997) 167-175] and Ghergu and Radulescu in [M. Ghergu, V. Radulescu, Existence and nonexistence of entire solutions to the logistic differential equation, Abstr. Appl. Anal. 17 (2003) 995-1003].