## Embedding Riemann surfaces in Riemannian manifolds

### Abstract (summary)

Every smooth compact orientable surface in a Riemannian manifold is naturally a Riemann surface. We may use the extrinsic metric, that is the metric induced by the surrounding Riemannian manifold, to introduce a conformal structure. As Gauss understood, and proved existence in the case of real analytic surfaces in $\IR\sp3$, the passage from the Riemannian structure on a surface to a Riemann surface structure is precisely the introduction of isothermal coordinates.

Around 1960, A. Garsia proved that every compact Riemann surface can be conformally immersed in $\IR\sp3$. In 1970, R. Ruedy extended Garsia's result to open Riemann surfaces and later he proved that every compact Riemann surface can be conformally embedded in $\IR\sp3$.

We may ask whether we can get a conformally equivalent model in a Riemannian manifold for any given compact Riemann surface. To give the answer we will study the deformation of surfaces embedded in an orientable Riemannian manifold.

Our main result here is the extension of the Garsia-Ruedy theorem to compact Riemann surfaces in orientable Riemannian manifolds.

This study was inspired by recent developments in mathematics and particle physics. Embedded Riemann surfaces occur in string theory--the so called theory of everything--as the world sheets, that is the trajectories, of strings moving in space-time. The strings are permitted to join and separate. In general, these surfaces are non-compact and have positive genus.

We prove the following two theorems in this dissertation. Theorem 1. Assume that S is a compact ${\cal C}\sp\infty$-embedded Riemann surface in the orientable Riemannian manifold M of dim M $\geq$ 3. Let S$\sb0$ be any Riemann surface structure on S. If there exists a nowhere vanishing smooth section of the normal bundle NS of S in M, then there exists an $\epsilon$ = $\epsilon$(S) so that there is an embedded $\epsilon$-normal deformation of S. And there exists an $\epsilon$-normal deformation of S which is conformally equivalent to the given Riemann surface S$\sb0$. Theorem 2. The map of the space of embedded surfaces into the Teichmuller space is continuous in the ${\cal C}\sp1$-topology. This map is not continuous in the ${\cal C}\sp0$-topology.