In joint work with Blaine Lawson, I proved several results about
ends of n-manifolds with scalar curvature bounded below by
n(n-2)/2. We showed in particular that such an end cannot be a
"bad" end. For complete **minimal surfaces of finite** (Morse)
**index**
in Euclidean three-space, this implies that the surface has
quadratic area growth, finite total curvature and finite
topological type [22]. I also proved a converse, verifying a
conjecture of Doris Fischer-Colbrie: a complete immersed minimal
surface has finite index if and only if it has finite total
curvature [23]. Fischer-Colbrie proved this result independently
in *Inventiones Math.* **82,** 121-132.

[22]. The structure of a stable minimal hypersurface near a
singularity (with Blaine Lawson), *Proc. Symp. in Pure
Math.*** 44,** 213-237 (1986).

[23]. Index and total curvature of complete minimal surfaces,
*Proc. Symp. in Pure Math.* **44,** 207-211 (1986).