Complete Open Surfaces Research Articles

On Huber-type theorems in general dimensions

In this paper we present some extensions of the celebrated finite point conformal compactification theorem of Huber [20] for complete open surfaces to general dimensions based on the n-Laplace equation in conformal geometry. We are able to conclude a domain in the round sphere has to be the sphere deleted finitely many points if it can be endowed with a complete conformal metric with the negative part of the smallest Ricci curvature satisfying some integrable conditions. Our proof is based on the improvements of the Arsove-Huber-type theorem on n-superharmonic functions in our earlier work [26]. Moreover, using p-parabolicity, we push the injectivity theorem of Schoen-Yau to allow some negative curvature and therefore establish the finite point conformal compactification theorem for manifolds which have a conformal immersion into the round sphere. As a side product we establish the injectivity of conformal immersions from n-parabolicity alone, which is interesting by itself in conformal geometry.

Total curvature of complete submanifolds of Euclidean space

The classical Cohn-Vossen inequality states that for any complete 2-dimensional Riemannian manifold the difference between the Euler characteristic and the normalized total Gaussian curvature is always nonnegative. For complete open surfaces in Euclidean 3-space this curvature defect can be interpreted in terms of the length of the curve at infinity''. The goal of this paper is investigate higher dimensional analogues for open submanifolds of Euclidean space with cone-like ends. This is based on the extrinsic Gauss-Bonnet formula for compact submanifolds with boundary and its extension to infinity''. It turns out that the curvature defect can be positive, zero, or negative, depending on the shape of the ends at infinity''. We give an explicit example of a 4-dimensional hypersurface in Euclidean 5-space where the curvature defect is negative, so that the direct analogue of the Cohn-Vossen inequality does not hold. Furthermore we study the variational problem for the total curvature of hypersurfaces where the ends are not fixed. It turns out that for open hypersurfaces with cone-like ends the total curvature is stationary if and only if each end has vanishing Gauss-Kronecker curvature in the sphere at infinity''. For this case of stationary total curvature we prove a result on the quantization of the total curvature.

Convexity of the Ideal Boundary for Complete Open Surfaces

Convexity of the Ideal Boundary for Complete Open Surfaces

Convexity of the ideal boundary for complete open surfaces

For complete open surfaces admitting total curvature, we define several kinds of convexity for the ideal boundary, and provide examples of each of them. We also prove that a surface with most strongly convex ideal boundary is in fact a generalization of a Hadamard manifold in the sense that the ideal boundary consists entirely of Busemann functions.

The extension of pointed Hausdorff approximation maps to the ideal boundaries of complete open surfaces

The extension of pointed Hausdorff approximation maps to the ideal boundaries of complete open surfaces

The ideal boundaries of complete open surfaces

0. Introduction. It is an interesting problem to investigate compactiίications of complete noncompact Riemannian manifolds. The ideal boundary of an Hadamard manifold Zis defined to be the set of equivalence classes of rays in X. Here the equivalence relation between two rays in X is obtained by an asymptotic relation between them. Busemann first defined (see [Bu, Chap. 3, §22]) an asymptotic relation between two rays (which he called co-ray relation and used to define parallelism on a straight G-space). This asymptotic relation is not symmetric in general and hence the equivalence classes of rays are not defined by it. If X is an Hadamard manifold, then this asymptotic relation becomes symmetric, and makes it possible to define the ideal boundary X(co) of X (see [EO] and [BGS]). Gromov defined in [BGS] the Tits metric on X{ao). Recently, Kause constructed an ideal boundary of an asymptotically nonnegatively curved manifold. The purpose of the present paper is, first of all, to define the ideal boundary M(oo) with the metric d^ as the set of natural equivalence classes of rays in a finitely connected, oriented, complete and noncompact surface M admitting total curvature. Then we investigate the geometry on the ideal boundary in terms of the total curvature. Here the total curvature c(M) of such a surface M is defined by an improper integral over M of the Gaussian curvature G:

The ideal boundaries and global geometric properties of complete open surfaces

In this paper we study the ideal boundaries of surfaces admitting total curvature as a continuation of [Sy2] and [Sy3]. The ideal boundary of an Hadamard manifold is defined to be the equivalence classes of rays. This equivalence relation is the asymptotic relation of rays, defined by Busemann [Bu]. The asymptotic relation is not symmetric in general. However in Hadamard manifolds this becomes symmetric. Here it is essential that the manifolds are focal point free.

Mass of rays on complete open surfaces

The total curvature of a complete open surface describes certain properties of the Riemannian structure which defines it. We study relationships between the total curvature and the mass of rays on a finitely connected complete open surface and obtain some integral formulas. 0. Introduction. Throughout this paper let M be a connected, finitely connected, oriented, complete and noncompact Riemannian 2-manifold without boundary. The total curvature c(M) of M is defined to be an improper integral over M of Gaussian curvature G with respect to the area element dM of M. A well-known theorem due to Cohn-Vossen [1] states that if M admits total curvature, then 2πχ(M) — c(M) > 0, where χ(M) is the Euler characteristic of M. Clearly c(M) depends on the choice of Riemannian metric. This phenomenon gives rise to the idea that the value 2πχ(M) - c{M) should describe certain properties of Riemannian metric which defines it. A ray (respectively, a straight line) on M is by definition a unit speed geodesic parametrized on [0, oo) (respectively, on R) every subarc of which realizes distance between its terminal points. For a point p G M let Sp(l) be the unit circle centered at the origin of the tangent space Mp to M at p. Let A(p) be the set of all unit vectors tangent to rays emanating from p. A(p) is closed in Sp(l). Let ΐDl be the natural measure on Sp(l) induced from the Riemannian metric. A relation between the mass of rays and the total curvature was first investigated by Maeda in [6], [7]. He proved that if M is homeomorphic to R2 and if G > 0, then ΐOloA > 2π — c(M), and in particular inf^ SDT ° A = 2π - c(M) . These results were extended by Shiga in [10], [11] to Riemannian planes whose Gaussian curvatures change sign, and later by Oguchi [9] to finitely connected M with one endpoint. In connection with an isoperimetric problem discussed by Fiala [3] and Hartman [4], the first-named author proved in [14] that if M has one end and if 2πχ(M) - c(M) < 2π , then for every monotone increasing sequence {Kj} of compact sets with \JKj = M,

An integral formula for the measure of rays on complete open surfaces

An integral formula for the measure of rays on complete open surfaces

Total Curvatures and Minimal Areas of Complete Open Surfaces

Total Curvatures and Minimal Areas of Complete Open Surfaces

Total curvatures and minimal areas of complete surfaces

Minimal areas for certain classes of finitely connected complete open surfaces are obtained by using a Bonnesen-style isoperimetric inequality for large balls on the surfaces. In particular, the minimal area of Riemannian planes whose Gaussian curvatures are bounded above by 1 is 4 π 4\pi .