A Perturbation Theorem for Surfaces of С onstant Mean Curvature

261

It IS well known that the energy functional associated with surfaces of constant mean curvature H is given by

Ah =A + 2HV (23)

More precisely, a regular surface g has constant mean curvature H if and only if Afj IS stationary at g with respect to all variations of g keeping the boundary fixed We need a somewhat stronger result expressed in the following Lemma 3 The author's original proof of this Lemma was unnecessarily complicated, the present version was suggested by J С С Witsche

Lemma 3. For sufju ientl\ small ь the folUming asset turn is U ue A suiface of the

form F(f, u) where f eU,^'^''( /q) and tie У,^^"" is a suiface of constant mean curiatuie

H if and only if

D , AH ( fu ) = 0^

D^ denoting the partial dent at ne with lespect to и

Pf oof According to a variation formula of Gauss ([1] §93) we have D,A(fu)i = -2\H^^iN(f) N{F{f u))dojf,,

where H^ ,, and dcOf „ are the mean curvature and the surface element of the surface F {J, w), respectively, and denotes the scalar product of E^ A similar calculation gives

D , V ( f . u ) i=\iN { f ) N(F{f.u))dcoj,,

Therefore , we obtain

D , , AH { f . u ) i=2\ { H - H^JN ( f ) N{F(f.u))dcoj,,

By continuity, the expression N(f) N{F( /, u)) is strictly positive for any /e CZ/^'^l /o) and weK/+^ provided only that i is chosen sufficiently small Thus, О^Ан([.и) vanishes if and only \{ Hj ^ = H

Lemma 4. There exists a mapping %^еС^ (U'^-^Jo)x K'^" СЦВ)) such that the formulas D^AH(f.u)i =\ Ч^н(f^u)Nn^)dœ,(^) (24)

and

O^^4^^ ( f0 ) i = -A,i+(2K^-4HH^)i (25)

are valid, where do)f.Äf, Kj, and Hf denote the surface element, the Laplace- Beltrami operator the Gaussian curvature, and the mean curvature of the surface / respectively

ProoJ In view of the representation

AH ( f . u ) = \i{J{\\DJ(x%D'f(\lu{\lDu(x))d\

with a certain algebraic function /, we deduce from Lemma 2

D , AH { f . u ) i=\ ^{f(x\Df{x\D'f(x\ii(xlDu{x))i(x)dx du

+ \-^(f(xXDf{xlD'f(x).u(x%Du{x))Di(x)dx *' dDu