By Robert Osserman

ISBN-10: 0486649989

ISBN-13: 9780486649986

Divided into 12 sections, this article explores parametric and nonparametric surfaces, surfaces that reduce zone, isothermal parameters on surfaces, Bernstein's theorem and masses extra. Revised variation contains fabric on minimum surfaces in relativity and topology, and up to date paintings on Plateau's challenge and on isoperimetric inequalities. 1969 version.

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Extra resources for A survey of minimal surfaces

Example text

Whi ch minim izes th e distance to Ç(O) . '! whi ch tend to Ç, and let x( k ) be the cor­ x(k) cannot have any point of accu­ r esponding points in D. '!. Thus 1 x(k ) l ... R as k and sin ce ... oc , 1 Ç ( k ) - Ç{O) I > 1 x ( k ) l by (5 . 5), it follows that 1 Ç- Ç{O)I 2: R, which proves the lemma. + /{ x 1 , x 2) be a solution of the minimal sur­ face equation (3 . 1 0) for x + x < R 2 . 8), (4. 12), th e map {x ' x 2 ) ... (Ç1 , Ç2 ) delined by (4. 13) is a dif­ l LEM M A 5 . 4 . '! which includes a disk of radius R about the point Ç(O).

N- 1, m a y be extended by setting xk(u 1 , 0) 0, xk (u 1, u 2) = - xk (u 1, - u 2). 3. {Reflection principle). = = . . = = in the full disk, by the r eflection principle for harmonie functions. T hus the functio n s k= l, . . ,n - 1 are analytic in the full disk and are pure i maginary on the real axis. MINIMAL SURFACES WITH BOUNDARY 55 By the equation 4>; we see that 4>; n- 1 = - I k= 1 4>; extends continuously to the real axis and has non- negative real values th ere. lt follows that 4> n extends continuous­ 1 xn a xn ;au2 0 ly to the real axis and has real values there.

9) may be used to defi ne harmonie fun etions . 7) into equations (5. 8) valid in the whole x l ' x2 -plan e. Proof of Coro11ary 1: This is an i mmediate consequence of Theorem S . l and Le mma S . S . t 39 BERNSTEIN'$ THEOREM Proof of Corollary 2. B y Lemma 4 . 2, each xk , for k 3, . . , n, will be a bounded harmonie function of (u 1 , u2) in the whole u 1 , u2J pl ane, hence constant. + Proof of Corollary 3: This is an immediate consequence of L emm a 4. 3 , using the fact th at in view of (5 .

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A survey of minimal surfaces by Robert Osserman


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