On non-parametric surfaces in three dimensional spheres
Ryosuke Ichida · Kodai Mathematical Journal · 1976
Introduction.Let D be a bounded domain with boundary dD in the Euclidean 2-plane E 2 .We denote by C\D) the set of real-valued functions of class C 2 on D. For a function u<=C 2 (D) we consider the non-parametric surface M in the Euclidean 3-sρace E 3 defined by (0.1) β(*)=(*ι, *z, wNow we take the unit normal vector field η on M as follows :where p=du/dx 1 , q-du/dx 2 and |Fw| 2 =ί 2 +g 2 .Then the mean curvature # of M with respect to -η is expressed aswhere ^-/rrrrr-r 2 (P, #)• It can be rewritten as follows:where r=d 2 u/dx 1 2 , s=d 2 u/dx ί dx 2 , t=d 2 u/dx 2 2 .Conversely, let H be a given continuous real-valued function on D. If we C\D} is a solution of the equation (0.2), then for this u the mean curvature of the surface in E 3 defined by (0.1) is equal to H. Now, we assume that the boundary dD of D is smooth.Let Jl and X be the area of D and the length of dD respectively.The following theorem was proved by R. Finn [3].THEOREM.For a function u<=C\D) and a positive constant H Q suppose that the mean curvature H of the non-parametric surface in E B defined by (0.1) satisfies the inequality \H(x)\^H Q for all x^D.Then we have JL/-£^1/2H 0 .In particular, if D is the disk of radius R, then