4. CHARACTERIZATIO N (1) Q AN D FLATNES S 5 B(r,Q) FIGURE 1 . Reifenber g fla t domain . Note that thi s definitio n i s significant onl y fo r S small. W e will always assum e S | an d motivat e thi s choic e later . Se t (1.15) 0(r , Q) = in f ( -D [ B(r, Q) n dQ S(r , Q) n L(r , Q) ] QeL(r,Q) [ r and (1.16) 0 K (r)= su p 0(r,Q) . QednnK We say that f 2 is vanishing Reifenberg, i f it i s 6—Reifenberg flat (S 1/8) , an d moreover, fo r ever y compac t se t K C Mn+1 w e hav e (1.17) limsup9 K (r) = 0 . In a certai n sense , suc h domain s ar e "locally flat" . T o simplif y th e notation , when ther e i s no ambiguit y w e will denote L(r, Q) b y LQ. To hel p clarif y th e definition s o f Reifenber g flat an d vanishin g Reifenber g do - mains w e present a fe w examples . We begi n wit h th e tw o mos t basi c examples , namel y th e dis c an d th e wedge , that wil l emphasize th e relevanc e of the vanishing condition (1.17) . Suc h example s serve a s the prototyp e o f the classe s o f smooth an d Lipschit z domain s respectivel y (see Definition 3.15) . Example 1^.1. Fo r th e wedg e centered a t a point w an d wit h angl e 0 ip 7r, we let (3 = (7T—^)/2 , see figure 2. Elementar y computations yield infL(r?lt ) 0(r , w) = rs'mfi. Thi s show s tha t a s th e wedg e open s up , convergin g t o a line , th e domai n becomes vanishin g Reifenberg . O n th e othe r hand , th e rati o 6(r,w)/r canno t b e made smalle r b y decreasin g th e scal e r . Example ^.2. I n th e cas e o f th e dis c o f radiu s R 0 w e le t w denot e a boundary point , se e figure 3 . Choosin g L(r,w) t o b e th e tangen t w e deriv e th e formula 6(r, w) = ma x —^ V 'R? + r 2 - R , from whic h i t immediatel y follow s tha t th e dis c i s vanishing Reifenberg .
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