A note on constant geodesic curvature curves on surfaces

In this paper we are concerned with the structure of curves on surfaces whose geodesic curvature is a large constant. We first discuss the relation between closed curves with large constant geodesic curvature and the critical points of Gauss curvature. Then, we consider the case where a curve with large constant geodesic curvature is immersed in a domain which does not contain any critical point of the Gauss curvature.     

Timelike surfaces of constant mean curvature ±1 in anti-de Sitter 3-space ℍ3 1(−1)

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ?³ ₁(?1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in ?SL₂? via Bryant type representation formulae. These Bryant type representation formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson–Guichard correspondence, between timelike surfaces of constant mean curvature ± 1 and timelike minimal surfaces in Minkowski 3-space E ³ ₁. The hyperbolic Gauß map of timelike surfaces in ?³ ₁(?1), which is a close analogue of the classical Gauß map is considered. It is discussed that the hyperbolic Gauß map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ?³ ₁(?1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauß map and timelike surface of constant mean curvature ± 1 in ?³ ₁(?1) is studied.     

Existence and uniqueness of CMC parabolic graphs in with boundary data satisfying the bounded slope condition

In this work, we investigate the existence of parabolic graphs of constant mean curvature H in whose boundary is given a priori, under hypothesis relating H with the geometry of the domain and a condition on the boundary data that, by analogy with a similar problem for vertical graphs in , we denominated it by bounded slope condition.     

Forward and inverse scattering on manifolds with asymptotically cylindrical ends

We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form ²(dy)+h(x,dx), h(x,dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to ²(dy)+h(x,dx). Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energies, we show that these two manifolds are isometric.     

Uniqueness and Sharp Estimates on Solutions to Hyperbolic Systems with Dissipative Source

Global weak solutions of a strictly hyperbolic system of balance laws in one-space dimension were constructed (cf. Christoforou, 2006 Christoforou , C. ( 2006 ). Hyperbolic systems of balance laws via vanishing viscosity . J. Diff. Eq. 221 ( 2 ): 470 – 541 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) via the vanishing viscosity method under the assumption that the source term g is dissipative. In this article, we establish sharp estimates on the uniformly Lipschitz semigroup 𝒫 generated by the vanishing viscosity limit in the general case which includes also nonconservative systems. Furthermore, we prove uniqueness of solutions by means of local integral estimates and show that every “viscosity solution” can be constructed as a limit of vanishing viscosity approximations.     

The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities

We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in L_p-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.     

Canonical Euler-Lagrange equations and Jacobi's theorem on regular surfaces

In this article we establish conditions under which canonical variables can be defined for a variational problem defined on a geometric (compact) surface. Also, we show the form the corresponding Euler-Lagrange equations assume once we rewrite them in terms of such canonical variables. Furthermore, we prove a version of Jacobi's theorem generalizing the univariate standard version of this theorem. The main results are applied to the conformal Gauss curvature functional.     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

Fibrations and stability for compact group actions on manifolds with local bounded Ricci covering geometry

In the study of the collapsed manifolds with bounded sectional curvature,the following two results provide basic tools:a(singular)fibration theorem by K.Fukaya[J.Differential Gcom.,1987.25(1):139156]and J.Cheeger,K.Fukaya,and M.Gromov[J.Airier.Math.Soc.,1992.5(2):327372],and the stability for isometric compact Lie group actions on manifolds by R.S.Palais[Bull.Amer.Math.Soc.,1961,67(4):362364]and K.Grove and H.Karcher[Math.Z..1973,132:1120].The main results in this paper(partially)generalize the two results to manifolds with local bounded Ricci covering geometry.     

Groupoid C *-Algebras and Index Theory on Manifolds with Singularities

The simplest case of a manifold with singularities is a manifold M with boundary, together with an identification M M × P, where P is a fixed manifold. The associated singular space is obtained by collapsing P to a point. When P = Z/k or S ¹, we show how to attach to such a space a noncommutative C ^*-algebra that captures the extra structure. We then use this C ^*-algebra to give a new proof of the Freed–Melrose Z/k-index theorem and a proof of an index theorem for manifolds with S ¹ singularities. Our proofs apply to the real as well as to the complex case. Applications are given to the study of metrics of positive scalar curvature.