Curvature-blowup criterion for inextendibility of a surface
= Curvature-blowup criterion for inextendibility of a surface
Let $S\subset\mathbb R^3$ be a smooth surface. Suppose its <Gaussian curvature> tends to infinity in absolute value along every <maximal geodesic> with a finite affine-parameter endpoint. Then $S$ is an <inextendible embedded surface>: a smooth extension would let some geodesic reach the extension boundary in finite time while its curvature remained locally bounded.