EMBEDDING OPEN RIEMANN SURFACES IN 4-DIMENSIONAL RIEMANNIAN MANIFOLDS

Title & Authors
EMBEDDING OPEN RIEMANN SURFACES IN 4-DIMENSIONAL RIEMANNIAN MANIFOLDS
Ko, Seokku;

Abstract
Any open Riemann surface has a conformal model in any orientable Riemannian manifold of dimension 4. Precisely, we will prove that, given any open Riemann surface, there is a conformally equivalent model in a prespecified orientable 4-dimensional Riemannian manifold. This result along with [5] now shows that an open Riemann surface admits conformal models in any Riemannian manifold of dimension $\small{{\geq}3}$.
Keywords
embedding open Riemann surface;4-dimensional Riemannian manifold;$\small{Teichm{\ddot{u}}ller}$ space;conformal deformation;
Language
English
Cited by
References
1.
W. Abikoff, The real analytic theory of Teichmuller space, Lecture Notes, 820, Springer-Verlag, 1980.

2.
A. M. Garsia, An imbedding of closed Riemann surfaces in Euclidean space, Comment. Math. Helv. 35 (1961), 93-110.

3.
S. Ko, Embedding Riemann surfaces in Riemannian manifolds, Ph.D. Thesis, University of Connecticut, Aug. 1989.

4.
S. Ko, Embedding Bordered Riemann surfaces in Riemannian Manifolds J. Korean Math. Soc. 30 (1993), no. 2, 465-484.

5.
S. Ko, Embedding Open Riemann Surfaces in Riemannian Manifolds, J. Geom. Anal. 9 (1999), no. 1, 119-141.

6.
S. Ko, Embedding Compact Riemann Surfaces in Riemannian Manifolds, Houston J. Math. 27 (2001), no. 3, 541-576.

7.
S. Ko, Embedding Compact Riemann Surfaces in 4-dimensional Riemannian Manifolds, Houston J. Math. 31 (2005), no. 2, 355-366.

8.
S. Ko, Embedding Bordered Riemann Surfaces in 4-dimensional Riemannian Manifolds, Houston J. Math. 33 (2007), no. 3, 649-661.

9.
O. Lehto and K. Virtanen, Quasiconformal Mappings, Springer, 1965.

10.
J. Nash, \$C^1\$-isometric imbeddings, Ann. of Math. 60 (1954), 383-396.

11.
R. Ruedy, Embeddings of open Riemann surfaces, Comment. Math. Helv. 46 (1971).

12.
R. Ruedy, Deformations of embedded Riemann surfaces, Ann. of Math. Studies 66 (1971).