By William M. Goldman

This is often the 1st finished remedy of the geometry of advanced hyperbolic area, a wealthy quarter of study with quite a few connections to different branches of arithmetic, together with Riemannian geometry, advanced research, symplectic and make contact with geometry, Lie teams, and harmonic research.

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JS To conclude we summarize what w e proved i n t h i s s e c t i o n : i f i s a s o l u t i o n of t h e minimal s u r f a c e e q u a t i o n , t h e n i t s g r a p h i . e . t h e r e must e x i s t a € R5 f(x)=a*x+b , and VxER bE R 5 . such t h a t S 5 f :R +R is flat, CHAPTER TWO SETS OF FINITE PERIMETER AND M I N I M A L BOUNDARIES En+l W e s h a l l d e f i n e f o r a l l Lebesgue measurable s e t s o f a general c o n c e p t o f boundary measure, c a l l e d p e r i m e t e r . W e w i l l prove t h e c l a s s i c a l i s o p e r i m e t r i c i n e q u a l i t y and o t h e r g l o b a l p r o p e r t i e s o f t h i s new n-dimensional m e a s u r e , we pass t h e n t o t h e a n a l y s i s of l o c a l p r o p e r t i e s of sets wi t h f i n i t e p erim e te r.

2dHn S xE B 2P P (x,) , $ = - B (x,) P 0 , in Q - 2P (x ) we obtain, i f and 0 - B 2 D ( ~ O ) C f, i DIFFERENTIAL PROPERTIES OF SURFACES 36 and a l s o , r e c a l l i n g 1 . 6 . 2 BERNSTEIN THEOREM Let f : R n be a s o l u t i o n , n e c e s s a r i l y a n a l y t i c , of t h e minimal R -f s u r f a c e e q u a t i o n , and l e t J1+(Dfrz V = (-Df,l)/ En+l be a u n i t v e c t o r f i e l d d e f i n e d on variable. )' w e i Lac2 = 1. r 2 1 1 1 6 . 6 , =~ c~L 6 . 6 . v. i , . + h,i,j ~ Thus, w e g e t I n t h i s i d e n t i t y we s u b s t i t u t e 6iA with and what e l s e w i l l be necessary, r e c a l l i n g t h a t 6 .

2 AN 1SOPERIMETRIC INEQUALITY. If X C S is a H -measurable set and w i t h compact s u p p o r t i n xES-X Letting , , with t h e n f r o m Theorem 1 . 4 . 3 P(x) X X in S , if we obtain X", X coincides with the is s u f f i c i e n t l y r e g u l a r . THE MONOTONIC BEHAVIOR OF AREA OF MINIMAL SURFACES. Going b a c k t o t h e i n e q u a l i t y (l), w h i c h w e w r i t e for p