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Cobordisms in problems of algebraic topology by Bukhshtaber

By Bukhshtaber

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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 . 6 . = 6 . 6 . + X ( v . 6 . v -vi6jvh)6h 1 1 3 1 x. fiivj A 6 . v - = j i ,J E l l h 6ivj6h6h6ivj 6ivj6hvi6hvk6kvj h , k , i tj = h,i,j n,i,j , i so 6iv j. 6 i hv j + DIFFERENTIAL PROPERTIES OF SURFACES 38 a t t h e f i x e d p o i n t . From Schwarz i n e q u a l i t y , we have 2 2 " i=l Thus, where e x i s t s , which i s t h e c a s e almost everywhere, we g e t 6c n i ,h=l L e t us observe now t h a t n 1 2 (6,6,Vi) " z = i,h=l 2 (ai"Vi' " + II " 1 (GhSiVi) 2 , i = l h=l i=l hCi moreover, a t t h e f i x e d p o i n t , f o r E6 2 o=-c v .

Nitsche's Review Article on Minimal Surfaces, E 7 4 1 ) . SLOPE OF MINIMAL GRAPHS 27 What w e f o l l o w h e r e i s e s s e n t i a l l y T r u d i n g e r ' s argument. VdH n ' ' T h i s i n e q u a l i t y , which h o l d s f o r a l l compact s u b s e t s KCAX R for which t h e d i v e r g e n c e theorem makes s e n s e , i m p l i e s t h a t g r a p h s w i t h bounded mean c u r v a t u r e h a v e l o c a l l y bounded area. i n our further considerations. 2 SLOPE ESTIMATE FOR GRAPHS.

I n f a c t , also h e r e w e have, i f relations a(X) <+m, t h e f o l l o w i n g sequence of 47 SETS O F FINITE PERIMETER t j > h a(XnM. d. m We have so proved t h a t m sets, then smallest aIB must c o n t a i n a l l B o r e l s e t s , t h a t i s t h e e l e m e n t s o f t h e f u n c t i o n with val u es i n + ") The r e s t r i c t i o n u - a l g e b r a o f s e t s , t o which open s e t s b e l o n g . a t o the family of 0 - a l g e b r a of s e t s c o n t a i n i n g a l l open is a % KO, o f Borel s e t s i s a c o m p l e t e l y a d d i t i v e a(%,) C E O , + + m 1 and l o c a l l y f i n i t e , t h a t i s * % A function l i k e t h a t , defined over f i n i t e over %o , LO, with values i n +a] and c o m p l e t e l y a d d i t i v e i s what one u s u a l l y means f o r a non n e g a t i v e Radon measure.

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