The Levy-Gromov Isoperimetric Inequality in Convex Manifolds with Boundary

1 Oct ober 10, 2007 T he Le v y-G ro m o v Is o p e rim et ri c Inequa lity in C onv ex M anif o lds w ith B o un dar y Frank Mo rg a n De par tmen t of Ma t he matics and Sta tis tics W il li a ms Co lle g e W il li a mst o wn, Ma s sa chuse t t s 01267 Frank. Mo rg a n@wi lli a ms.edu A b s trac t W e observ e a f ter B ayl e and Rosa les t hat t he L ev y- Gromov iso perim e tric in equ al it y ge ner al izes to conv e x m an ifolds wi th boundar y. 1. In tro ductio n. The Le v y- Gromov iso perime tr i c ine qua lit y ([ G, §2.2], [BZ, 34.3. 2 ]) p r ovid es a shar p lower bound on the perimete r re qu ired to enc lose g iv en vo lu me in a c losed man ifol d M in terms of a p osi tive l ower bo und on the Ri cc i curva tur e, b y co m paris on wi th t he s pher e. Our Theore m 1 show s tha t t he res ul t and its p r oof g e n era l i ze to co m pact, conv e x m an ifol ds with boun dar y. T he h eart of t he p roof is the observation that t he re g ion o f p rescr ibed vol um e is cov ered by ra y s norma l to t he en cl osin g, ar ea -m in imiz i n g surfa c e, yie ldin g a n ineq ual it y f or the vol um e in terms of the surf a ce and it s me an c urvature a f ter He int ze and Karc her [HK]. In a conve x m an ifold with bounda r y, t he surfac e should m eet t he bounda r y of M or t ho gon al l y, t he norm a l r a y s should still cov er the re g ion , and the sam e re sul t sh ou ld hold. T he case of smoo th boundar y wa s pr oved b y Ba yle and R osal es [BR, Thm. 4. 8] , usin g the se cond v ar iation for m u la . So l et M be a c ompac t, conve x n- d im ension al Ri em ann ian m an ifold with bounda r y B, smoo th on the in ter ior. The ( parti a lly fre e bound ar y) is o p eri metric p roble m seeks a re g ion of pre scr ibed vo lum e f r action 0 < V < 1 of le ast perim e ter, not countin g pe rim eter insid e B. W e r evi ew some re sul ts f rom g eom etric me asure theor y (see [ M1 ], [ M 3]). E x is tenc e fo llows from standard compact ness ar gu me nt s. Le t S be the c losure of t he part of t he bounda r y of the re g io n i n the interio r of M . A t an interior p oint p , S h as a tan ge n t cone , ar ea -mi ni m izin g withou t volu me con s t raint; i f such a c one is a plan e, then S is a smooth, constan t -m ean -curva tur e h y p ersurfac e a t p . ( Actual l y it is known t hat t he interior sin g ul ar se t has Hau sdorff dim ens ion at mo s t n − 8.) A t boundary p oint s of S , tan ge n t “cones” s till e x is t, and one e xpe c ts S to be norma l to B at mos t re gu la r poin ts of B, a lthoug h we do not need s uch d el ic ate re gu la rit y . For our c onve x it y h y p o t hesi s, a ll w e ne ed t o know is t hat M ha s a conve x tan g e n t con e at al l bound ar y p oint s, t hat some shor tes t pa t h bet ween int er ior poin ts li es on the interior , a nd that some shor tes t pa t h be tween an interior poin t and a bound ar y p oint q is not tan g ent t o the boundar y a t q. 2 1. Th eor em (L ev y- Gromov f o r conve x m an ifo l ds wit h boundar y). Let M b e an n- dime n siona l, c ompact, con vex , conn ec te d Riemann ian mani fol d w it h boundary, sm ooth on th e i nt erior, wit h R ic c i cur v ature bound ed be low by n − 1 and volume λ tim es the volume of th e uni t sphere . Then the isoperime tri c profi le P( V), the leas t per imet er to e nc lose given volu me fr act ion 0< V<1, satis fie s (9) P ≥ λ P 0 , where P 0 is the isoperime tric profi le o f th e un it spher e. Equa li ty ho lds only if M is the round sus pension of a space for m wi th boundary . W e provide a sketch of the whol e proof. The m ai n new i n g re di ent f or the g e ne ra lization ap p ears in para g ra ph t hree, the observation that s hor tes t p a th s from an interior p oin t t o an iso p eri metric surfac e me et t he surfac e on the in ter ior, s o t hat norm al ray s from the interior of the surf a ce cov er the in ter ior of the re g io n. Proof s ke tch . For g i ve n vol um e fra ction 0 < V < 1, le t P be the perime ter o f a m in im izin g h y p ers urfa ce S i n M and let P 0 be the perim eter o f the h y p ers p here S 0 in the unit s p here. B y re pla ci n g V b y 1 − V (whi ch ch an g es the s ig n o f t he m ea n curv a tures) if ne c es s ar y, we may a ssume that t he me an curvature of S is g r eater than or equ al to t hat of S 0 . T he i d ea is t o es timate V b y t he vo lu me of the uni on of r a y s norm al to S at re g u la r p oin t s of S. Consid er a shorte s t p at h γ f ro m a p oint p t o t he surfac e S , m eetin g S at a p oin t q. We c la i m that if q is an interior poin t, then q mus t be a re gu la r poin t of S. S i nc e γ is a shorte s t p a th, an (ar ea- mi ni m i zin g ) tan ge nt cone C to S a t q li es in the far h al fs p ace bounded by t he hy p er plane no rma l to γ . I t fol low s that C mus t be a hy p er plane ( or mov in g the v erte x i n the dire ction of the continuation of γ would de c re ase ar ea to first order) . Mor eover , as a shorte s t pa t h, γ m us t meet S nor ma l l y. W e c la i m that q cannot be a boundary p oint ; su p p o se i t were. S inc e γ is a shor tes t pa t h, C must li e in the fa r ha lf-s pace i n tersec t t he c onve x tan g ent cone C ' to t he bound ar y of M at q. I t f ol lows t hat C must be the por tion of a hy p er plane inside C ' and tha t the li ne tan g e n t to γ at q must l ie in C' (or movi n g the v er te x i n the di re c tion of the continuation of γ would de cre ase are a to f i rst order; if the l ine le a ves C' k eep onl y t he par t of C ins ide C' and do bett er). Thu s γ is tan g ent t o t he bound ar y of M a t q, a contradi ction of conv e x it y. Therefo re the union of ray s norma l to S at re gu l ar p oin t s of S c over the re g i on of volu me frac ti on V . B y t he ca lc ulus estimate of He in t ze-Karche r [HK], t he vo lum e λ V en cl osed b y S sati sfi es ! " V P ≤ ! V P 0 , as desir ed. If e qua lit y holds for som e V , t hen a penci l of g e ode sics norm als to S m us t be isom etric to a pencil of norm als to t he equ ator of a r ound s phere, M is t he round su s p ension of a conve x s pac e form wi th boundar y , and equ al it y holds for a ll V. 3 Techn ic al no te . Wh il e the e x istenc e of a t an g e n t cone depend s on so-ca ll ed monotonic i t y, we jus t need a non- zero t an g e nt obj ect ( weak l im it of ho mo thet i c e xpansions), whi ch fol lows from lower de nsit y bo unds ( which c an b e p roved for e x ample as in [ M2, Le mm a 3. 6 ]) and trivi al up p er densit y b ounds. T his no t an issue when M i s smoot h u p t o the boundar y. 2. Coro l l ary. Le t R b e a c on ve x s ub region o f the n D unit h emisphere wi th volume λ tim es th e volume o f th e h emisphere. Then t he isoperimetri c prof ile P (V ) (l east p erimet er to en cl ose given volume fr act ion 0< V<1) sat isfi es (9) P ≥ λ P 1 , where P 1 is the isoperime tric profi le o f th e un it hemisphere . I f e quali ty ho lds for s ome V, th en M is the suspension of a convex subset of an e quator ial hyp ersphe r e (and h enc e equa li ty hol ds for all V). Proof. Coro ll ary 2 f o llows im me di ate from T heor em 1 be ca use P 1 = . 5P 0 and the λ of the co r o ll ary is t wic e the λ of the theore m. As a fur ther c or o ll a r y we hav e an iso perim e tric resu l t sta t ed in Mo r ga n [ M2 , Rmk . 3.11] , a pa per which d ea lt p ri ma rily wi th t he surfac e of p ol y to pe s. Corol la r y 3 wa s p r oved e a rli er wi thou t uniquen ess b y Lions and Pac el l a [ L P , Thm. 1 . 1] . Se e a lso [ M 4, Rmk . a f ter Thm. 10 . 6] and for t he smoot h c ase [RR,T hm. 4.11]. 3. Coro l l ary . Le t P n be a sol id (c onvex) pol yt ope i n R n . For s mal l prescribed volume, isoperime tri c regions are bal ls about a ve rt ex . Proof. The proof for s olid pol y t o pe s f ol lows t he p roof for s ur fa ces of pol y t o pe s [ M2 ]. T he proof a p plies an is o p erim etric comparison theore m [Ros, T hm. 3.7] for p roduc ts and cone s t o t he con e C ov er a v ert e x of P. Rou gh l y , since b al ls about the ori g in a re iso perim etric in a ha lf plane , t he y are is o perim etric i n C. There is a hy p o the sis on the l ink, which r equi res Coroll ary 2 abov e, wi thou t t he ne e d for smoothin g or the cons equent loss of uniqu en es s w hi ch o cc ur i n the ori g i na l p roof f or surfac es of pol y t o p es. R e f er e nc e s [BR] Vincent Bayle and César Rosale s, Some isoperimetric compari s on theorems for convex bodies in Riema nni an manifolds, India na Univ. Math. J. 54 (2005), 1371-1394. [BZ ] Y u. D. Bura g o and V . A. Zal g a l le r, Geo metric Ineq ual ities, Spri n g e r- V er la g , 1988. 4 [G] M . G romov, Iso p erim etric in equa litie s in Ri em ann ia n m ani fol ds, A p p endix I t o Vital i D. M il man and Gide on Sc he chtman, As y m p to t ic Theor y of F in ite Dim ension al Nor m ed S pac es, L ec . Note s M a th. 1200, S prin g e r-V e rl a g , 1986. 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