Intrinsic Diameter and Curvature Integrals of Surfaces Immersed in ℝn

We prove that if Σ is a complete connected surface without boundary smoothly immersed in ℝn, then its intrinsic diameter satisfies the bound diamΣ < Cn volN(Σ), where N(Σ) ⊂ ℝn × Sn−1 is the manifold of unit normals to Σ and Cn is a universal constant depending only on n. In fact we establish a stronger bound, linear in the sum of the area of Σ and the L1 norms of its second fundamental form and Gauss curvature (in dimension n = 3 the two bounds are equivalent).