Inductive Analysis on Singular Minimal Hypersurfaces

The geometric analysis of a minimal hypersurface H within some Riemannian manifold (M, g) with second fundamental form A usually involves the scalar quantity |A|2 = sum of squared principal curvatures. A few classical examples are seen from Simons type inequalities like: ∆H |A|2 ≥ −C · (1 + |A|2)2 or the stability condition (valid in particular for area minimizers): 0 ≤ Area(f) = ∫ H |∇Hf |2 − f 2(|A|2 +RicM (ν, ν))dA for infinitesimal variations f in normal direction ν to H . Now the point is that minimality, i.e. trA = 0, also implies that the scalar curvature scalH of H satisfies scalH = −|A|2 in a flat ambient space (which is central in particular when one is interested in the case where H is singular).