Integral inequalities for ovaloids in Euclidean space

Extract

Let M be a smooth surface in Euclidean space E³ and L the Weingarten map. The fundamental forms I₁, I₂, I₃,… on M are defined in terms of L and the usual inner product 〈, 〉 of E³ as follows. If X and Y are in the tangent space T_PM of M (Pε M), then I₁(X, Y) = 〈X, Y), I₂(X, Y) = 〈LX, Y〉, I₃(X, Y) = 〈L²X, Y), etc. Moreover, if M is convex, i.e., the Gaussian curvature K = k₁k₂, where k_i, (i = l,2) are the principal curvatures of M, is everywhere positive, then one can also define on M the forms I₀(X, Y) = 〈L⁻¹X, Y), I₋₁,(X, Y) = 〈L⁻²X, Y), I₋₂(X, Y) = 〈 L⁻³X, Y) etc., where L⁻¹ is the inverse of L. Since L is self-adjoint, the forms I_m are, for any integer m, symmetric bilinear functions on T_PM × T_PM. Furthermore I_m are C^∞ in the sense that if X and Y are vector fields with domain A ⊂ M, then 〈 L^mX, Y〉_P = 〉L^mX_P, Y_P) is a C^∞ real function on A. If the convex surface M is appropriately oriented, then the forms I_m define metrics on M, which we also denote by 〈, 〉_m (〈, 〉₁)≡ 〈, 〉).