On surfaces with small variation of Gaussian curvature.

Svetlana Krat

Consider a surface whose total Gaussian curvature is small, where the total curvature is defined to be the integral of the absolute value of Gaussian curvature. Of course, such a surface can arise as a small perturbation of a developing surface (that is, of a surface of zero curvature). One now asks if every surface with small total curvature is actually a small perturbation of a developing surface (as opposed to the situation where there is no developing surface close to the original one).

Establishing relations between intrinsic geometry of a surface and properties of this surface as a subset of the ambient space is one of the most classic areas of differential geometry. Most of the deep results in this area involve a condition that the curvature of the surface in question is non-positive or non-negative.

On the other hand, there are much fewer results about the immersion of 2-dimensional smooth surfaces, whose Gaussian curvature changes sign. One such result concerns the immersion of tubes. It is known due to D. Burago, that if a complete boundary-free surface diffeomorphic to a cylinder with bounded total curvature is isometrically immersed in R^3, then the image is not contained in a compact set.

D. Burago suggested that the tools he developed to prove this result can also be applied to other problems related to surfaces with small total curvature. The argument we used is indeed based on these tools.

Consider a simply connected, compact two dimensional surface with small total curvature in R^3. The surface has a boundary, which is convex from the outside with respect to the interior metric. We proved that such a surface always lies near a smooth developing surface with almost the same boundary. >