The relative isoperimetric inequality in Cartan-Hadamard 3 ... - KIAS

184 Choe and Ritoré, Relative isoperimetric inequalityHS 2 areaðSÞ f Ð H 2 dA f Ð Ðð3:7ÞdA b nðu b Þ ds:SS qSLet us see thatÐÐð3:8Þlim dA b ¼ lim e 2u bdA ¼ 2p:b!yS b!ySLet rð pÞ :¼ dð p; p 0 Þ be the extrinsic distance to p 0 . Fix r 0 > 0 small. Observe that e 2u b convergesuniformly to 0 in S out of the ball Bð p 0 ; r 0 Þ when b ! y. For r 0 small enough, themodulus of the gradient of rj S over S X Bð p 0 ; r 0 Þ is approximately 1, and the length ofS X qBð p 0 ; rÞ, for r A ð0; r 0 Þ, is approximately pr. By applying the coarea formula to rj Swe getÐSXBð p 0 ; r 0 Þ dA b ¼ p Ðr 02b 2r1 þ b 2 r 2 dr þ oðr 0 Þ¼2p þ oðr 0 Þ;where oðr 0 Þ converges to 0 when r 0 ! 0. Letting r 0 ! 0 we get (3.8).0On the other hand, qS is a curve (possibly disconnected) that bounds a regionR H qC. Recall that n, the inner normal to qS on S, is also the outer normal to qC. Asp 0 A qC we see that a geodesic g p : ½0; t p Š!M from p 0 to any other point p A qS H qCmust satisfy g g 0 p ðt pÞ; nð pÞ f 0 by the convexity of C. Hence gð‘d; nÞ f 0 for all p A qSand we deduce thatnðu b Þ¼ 2b2 dgð‘d; nÞ1 þ b 2 d 2is a nonnegative function. Discarding the last summand in (3.7) and taking limits we obtain(3.1).Let us consider now the equality caseð3:9ÞH 2 SareaðSÞ ¼2p:If (3.9) holds then there is also equality in (3.6), which implies thatð3:10aÞ ‘ 2 d 2 ðv; vÞ ¼2gðv; vÞ; for any p A S; v A T p S:We also have that limb!yÐqSnðu b Þ ds ¼ 0 since, after taking limits, there is equality in (3.7).The sequence f nðu b Þg b is pointwise increasing and converges to 2nðdÞ=d. By the MonotoneConvergence Theorem0 e Ð qS2nðdÞ Ðds ¼ limd b!yqSnðu b Þ ds ¼ 0;and soð3:10bÞgð‘d; nÞ 1 0along qS:

186 Choe and Ritoré, Relative isoperimetric inequalitywhere N is the unit normal to S in the direction of the mean curvature vector of S. Integrating(3.11) on S, we obtain the classical Minkowski formula3 volðWÞ ¼ 1 H areaðSÞ:This concludes the proof in the Euclidean case.Let us consider now the hyperbolic case. By scaling the metric g of M we may assumethat k ¼ 1, so that the inequality we intend to prove isð3:12Þð 1 þ HS 2 Þ areaðSÞ f 2p;1Þ of constant sec-with equality if and only if W is a half-ball in the hyperbolic space M 3 ðtional curvature equal to 1.Consider again a point p 0 A qS and let d be the distance to p 0 . We define the family ofconformal metrics g b ¼ e 2u b g, given by the functionsð3:13Þu b ¼ logð12bb 2 Þþð1 þ b 2 ; b > 1:Þ coshðdÞWhen M is the three-dimensional hyperbolic space, this family of metrics is obtainedby homothetically expanding the spherical metric! 21 tanh 2 ðd=2Þ1 þ tanh 2 g:ðd=2ÞFrom (3.2) we obtain the following relation for the sectional curvatures of the tangentplane to S.e 2u bðK s Þ b¼ K s þ 1 þ e 2u b1 þ b 2 ð3:14Þþð1 b 2 Þþð1 þ b 2 Þ coshðdÞ P 2 ‘ 2 coshðdÞðe i ; e i Þ 2 coshðdÞ ;i¼1where fe 1 ; e 2 g is a g-orthonormal basis of the tangent plane to S. By the Hessian ComparisonTheorem we know that ‘ 2 coshðdÞ f coshðdÞg. So we obtainð3:15Þe 2u bðK s Þ bf K s þ e 2u bþ 1:Equality holds in (3.15) if and only if ‘ 2 coshðdÞðe i ; e i Þ¼coshðdÞ for i ¼ 1; 2. From equation(3.3) and inequality (3.15) we obtainÐ1 þ HSð 2 Þ dA f Ð ÐdA b nðu b Þ ds:S qSAs in the previous case one shows that lim dA b ! 2p and, by the convexity of C, thatb!yÐnðu b Þ f 0, which yields the desired estimate (3.12).S

Choe and Ritoré, Relative isoperimetric inequality187If equality holds in (3.12) then we conclude, as in the Euclidean case, that‘ 2 coshðdÞðv; vÞ ¼coshðdÞ for any unit tangent vector v to S at any point of S, that H isconstant, and that gð‘d; nÞ ¼0 at any point of qS.Condition ‘ 2 coshðdÞðv; vÞ ¼coshðdÞ for any unit tangent vector v to S at any pointof S implies, by the standard comparison theorems, that the metric g of W (i.e., on the coneover S with vertex p 0 ), has constant sectional curvatures equal to 1. Moreover, conditiongð‘d; nÞ ¼0 at any point of qS implies that R is a totally geodesic surface. Finally, as S hasconstant mean curvature H > 1, we see as in [Mo], Theorem 9, by taking inner parallels,thatÐcoshðdÞþH sinhðdÞh‘d; Ni dA f 0; N ? SSwith equality if and only if W is a half ball in hyperbolic space. But since the metric of W ishyperbolic, we have ‘ 2 coshðdÞ ¼coshðdÞg in W so that integratingon S we getD S coshðdÞ ¼2 coshðdÞþ2H sinhðdÞh‘d; NiÐScoshðdÞþH sinhðdÞh‘d; Ni dA ¼ 0:This completes the proof in the hyperbolic case k ¼ 1.Finally, if S is merely C 1; 1 , the arguments in the proof apply without changes sincethe principal curvatures of S, and hence the mean and the Gauss curvature, are defined almosteverywhere. The divergence theorem still holds under our weak hypothesis. rBy using the mean comparison result in Proposition 3.1 we can now prove the isoperimetriccomparison theorem.Theorem 3.2. Let M be a three-dimensional Cartan-Hadamard manifold with sectionalcurvatures bounded above by a nonpositive constant k, and let MðkÞ be the threedimensionalCartan-Hadamard manifold with constant sectional curvatures equal to k. Assumethat C H M is a proper convex domain with smooth boundary and that H H MðkÞ isa half space. Then we haveð3:16ÞI C f I H ;which in turn implies that for any bounded finite perimeter set D H M C we haveð3:17ÞareaðqDÞ Cf I H volðDÞ :Moreover, equality holds in (3.17) if and only if D is isometric to a half ball in MðkÞ.Proof. Since the existence of isoperimetric regions is a crucial point of our arguments,but it is not guaranteed in the noncompact M C , we first construct an exhaustionof M C by relatively compact sets fE k g k A N . We take p 0 A qC, and define E k ¼ Bð p 0 ; r k Þ C,where r k is an increasing diverging sequence of positive numbers, and Bð p 0 ; r k Þ is the closedball centered at p 0 of radius r k .

188 Choe and Ritoré, Relative isoperimetric inequalitySince E k is bounded, isoperimetric regions exist on E k for any given volumev A ð0; vol E k Þ. The boundary S of any isoperimetric region W in E k satisfies the regularityproperties of Lemma 2.1 provided there is no point in S X qC X ðqE k Þ C. Assume there isq 0 A S X qC X ðqE k Þ C. The set qE k is contained in the geodesic sphere qBð p 0 ; r k Þ, whichmeets qC at q 0 at an angle less than or equal to p=2. Reflecting locally W with respect toqC and blowing up W and the metric from q 0 , [Gr1], we obtain a cone in R 3 which is areaminimizing in a wedge of R 3 . This implies regularity if the angle is p=2 and it is not possibleif the angle is less than p=2; see [G], Thm. 15.5.Fix k A N and assume that W k is an isoperimetric region in E k . The set W k may haveseveral components. Let W k ¼ W 1 k W W2 k , where W1 k consists of the components of W k touchingqC (in an orthogonal way), and W 2 k consists of the components of W k disjoint from qC.Then by (3.1)ðk þ H 2 ðqW 1 k Þ areaðqWCÞ 1 k Þ C f 2p½Kfcomponents of W1 kgŠ f 2p:On the other hand, by [Kl] or [R],ðk þ H 2 ðqW 2 k Þ areaðqWCÞ 2 k Þ C f 4p½Kfcomponents of W2 kgŠ f 0:Adding both inequalities we getand henceðk þ H 2 ðqW k Þ CÞ areaðqW k Þ Cf 2p;ð3:18ÞH ðqWk Þ Cf H k areaðqW k Þ C;where H k ðaÞ denotes the mean curvature of a geodesic sphere of area 2a in MðkÞ. By Proposition3.1, equality holds in (3.18) if and only if W k is isometric to a geodesic half ball inMðkÞ.Denote by I k the isoperimetric profile of E k . By standard arguments, [Hs], pp. 170–172, we have I k is continuous and increasing, when I k is smooth at v 0 then I 0 k ðv 0Þ¼2H, where H is the constant mean curvaturein the interior of E k of any isoperimetric region of volume v 0 , and left and right derivatives of I k exist everywhere.Since I k is a continuous monotone function with left and right derivatives at every point, itis absolutely continuous.The continuity of I k follows from the convergence of isoperimetric regions. To provethe monotonicity of I k we just need to show that the constant mean curvature H of theboundary of an isoperimetric region W in the interior of E k is positive. Let S ¼ qW C .IfSdoes not touch ðqE k Þ Cthen there is an outer parallel qC t to qC, which is tangent to S at

Choe and Ritoré, Relative isoperimetric inequality189some point and leaves S on one side. Standard comparison theorems show that the principalcurvatures of qC t are nonnegative. By the maximum principle, H f 0. But in caseH ¼ 0, we obtain from the maximum principle that S and qC t locally coincide and, by aconnectedness argument, that a connected component of S is contained in qC t , which givesus a contradiction. If S X ðqE k Þ Cis not empty then the maximum principle shows that H islarger than or equal to the mean curvature of ðqE k Þ Cat some point, which is strictly positivesince geodesic spheres in a Cartan-Hadamard manifold are strictly convex.The computation of the derivative I 0 k ðv 0Þ is standard by making a variation supportedaround a point of S X intðE k Þ. The fact that left and right derivatives always exist followsfrom the convergence of isoperimetric regions of volumes v k ! v 0 to an isoperimetric regionof volume v 0 , see [Hs], p. 171.Let J k be the restriction of the isoperimetric profile of MðkÞ to the intervalð0; vol E k Þ. Let f ðaÞ, gðaÞ be the inverse functions of I k , J k , respectively. We know thatg 0 ðaÞ ¼Jk 0ðaÞ 1 ¼ 2H k ðaÞ 1, and that, when f0exists, f 0 ðaÞ ¼Ik 0ðaÞ 1 ¼ð2HÞ 1 , whereH is the mean curvature in the interior of E k of any isoperimetric region of volume f ðaÞ.By Proposition 3.1, we obtain g 0 ðaÞ f f 0 ðaÞ a.e. Since f is absolutely continuous (and g issmooth), we have gðaÞ f f ðaÞ. It follows that I k f J k .If equality holds for some v 0 , then for a 0 ¼ J k ðv 0 Þ¼I k ðv 0 Þ we have gða 0 Þ¼ f ða 0 Þ.Since g 0 f f 0 we obtain that f 1 g in the interval ð0; a 0 Þ and so H k ða 0 Þ 1 ¼ Hða 0 Þ 1 .IfW 0is any isoperimetric region of volume v 0 then Proposition 3.1 implies that W 0 is isometric toa half ball in MðkÞ of volume v 0 .Finally let W H M C be relatively compact with smooth boundary. Then W H E k forsome k, andPðWÞ f I k volðWÞ f I H volðWÞ :If equality holds then W is an isoperimetric region in E k and I k volðWÞ ¼ I H volðWÞ .Bythe discussion in the above paragraph we have that W is isometric to a half ball in MðkÞ ofvolume volðWÞ. rCorollary 3.3. Suppose that M is a three-dimensional Cartan-Hadamard manifold,C H M a proper convex domain with smooth boundary, and D bounded and of finite perimeterin M @ C. ThenareaðqD @ qCÞ 3 f 18p volðDÞ 2 ;and equality holds if and only if D is a flat half ball.4. The relative isoperimetric inequality for a general convex setIn this section, C will denote a bounded strictly convex body in R 3 . No assumptionon the regularity of its boundary is made. We say that a convex body is strictly convex if itsboundary does not contain a nontrivial segment [Sch], p. 77. Recall that p is an extremepoint of C if it cannot be written in the form p ¼ lx þð1 lÞy, with x; y A C, x 3 y, and

190 Choe and Ritoré, Relative isoperimetric inequalityl A ð0; 1Þ. Any point in the boundary of a strictly convex set is an extreme point. A cap of Caround p is a set of the form C X H þ , where H þ is a closed half space with p A int H þ ,[Sch], pp. 18–19.For strictly convex sets we have the following result:Theorem 4.1. Let C H R 3 be a proper convex domain which is strictly convex, andH H R 3 a half space. Then, for any v > 0,I C ðvÞ > I H ðvÞ:That is, equality never holds in the above inequality for these convex bodies.Proof. Using standard results on the Hausdor¤ metric, we can find a sequenceof convex bodies with smooth boundary C k H R 3 , with C H C k for all n A N, convergingin the Hausdor¤ distance to C. Let W H ðR 3 Þ Cbe a relatively compact set and defineW k ¼ W X ðR 3 Þ Ck. Then volðW k Þ!volðWÞ and P C ðWÞ f P Ck ðW k Þ. Since the relative isoperimetricinequality (2.1) is satisfied in ðR 3 Þ Ck, we have P C ðWÞ f P Ck ðW k Þ f I H volðW k Þ .Taking limits, we get P C ðWÞ f I H volðWÞ , and the isoperimetric inequality I C ðvÞ f I H ðvÞholds in ðR 3 Þ C.To see that this inequality is always strict, consider a region W H ðR 3 Þ Csuchthat equality P C ðWÞ ¼I H volðWÞ holds. Let p be a point in the interior, relative to qC,of qW X qC. The strict convexity implies that p is an extreme point of C. By [Sch], Lemma1.4.6, there is a cap P þ of C around p contained in the interior of qW X qC. Let P bethe half space obtained as the closure of the complement of the half space determiningP þ . Consider the convex set C 0 ¼ C X P , and W 0 ¼ W W ðC X P þ Þ. We haveP C 0ðW 0 Þ¼P C ðWÞ, and volðWÞ < volðW 0 Þ. HenceP C 0ðW 0 Þ¼P C ðWÞ ¼I H volðWÞ < I H volðW 0 Þ ;and so I C volðW 0 Þ < I H volðW 0 Þ . This is a contradiction, since we have already provedthat in ðR 3 Þ C0 the isoperimetric inequality I C ðvÞ f I H ðvÞ holds. rRemark 4.2. The first paragraph in the proof of Theorem 4.1 shows that the isoperimetricinequality areaðqD @ qCÞ 3 f 18p volðDÞ 2 holds for any region D outside a convexset C with nonsmooth boundary in R 3 . The authors have not characterized what happensin the equality case, and they believe that techniques di¤erent from the ones used in thispaper should be employed.References[A] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Di¤. Geom. 11 (1976), 573–598.[BG] E. Bombieri and E. Giusti, Harnack’s inequality for elliptic di¤erential equations on minimal surfaces,Invent. Math. 15 (1972), 24–46.[BZ] Y. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer, Berlin 1988.[C1] J. Choe, Relative isoperimetric inequality for domains outside a convex set, Arch. Inequ. Appl. 1 (2003),241–250.

Choe and Ritoré, Relative isoperimetric inequality191[C2] J. Choe, The double cover relative to a convex set and the relative isoperimetric inequality, J. Aust. Math.Soc. 80 (2006), 375–382.[Cr] C. Croke, A sharp four dimensional isoperimetric inequality, Comment. Math. Helv. 59 (1984), 187–192.[G] E. Giusti, Minimal surfaces and functions of bounded variation, Monogr. Math. 80, Birkhäuser Verlag,Basel-Boston 1984.[GMT] E. Gonzalez, U. Massari and I. Tamanini, On the regularity of boundaries of sets minimizing perimeterwith a volume constraint, Indiana Univ. Math. J. 32 (1983), 25–37.[GLP] M. Gromov, J. Lafontaine and P. Pansu, Structures métriques pour les variétés Riemanniennes, Cedic/Fernand Nathan, Paris 1981.[Gr1] M. Grüter, Optimal regularity for codimension one minimal surfaces with a free boundary, Manuscr.Math. 58 (1987), 295–343.[Gr2] M. Grüter, Boundary regularity for solutions of a partitioning problem, Arch. Rat. Mech. Anal. 97(1987), 261–270.[Hs] Wu-Yi Hsiang, On soap bubbles and isoperimetric regions in noncompact symmetric spaces, I, TohokuMath. J. (2) 44 (1992), no. 2, 151–175.[K] I. Kim, An optimal relative isoperimetric inequality in concave cylindrical domains in R n , J. Inequ. Appl.1 (2000), 97–102.[Kl] B. Kleiner, An isoperimetric comparison theorem, Invent. Math. 108 (1992), 37–47.[LY] P. Li and S. T. Yau, A new conformal invariant and its applications to the Willmore conjecture and thefirst eigenvalue of compact surfaces, Invent. Math. 69 (1982), 269–291.[Mo] S. Montiel, Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, IndianaUniv. Math. J. 48 (1999), 711–748.[MoR] S. Montiel and A. Ros, Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures,Di¤erential geometry, Longman Sci. Tech., Harlow (1991), 279–296.[R] M. Ritoré, Optimal isoperimetric inequalities for three-dimensional Cartan-Hadamard manifolds, Globaltheory of minimal surfaces, Clay Math. Proc. 2 (2005), 395–404.[Sch] R. Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge Univ. Press, Cambridge 1993.[StZ] E. Stredulinsky and W. P. Ziemer, Area minimizing sets subject to a volume constraint in a convex set, J.Geom. Anal. 7 (1997), 653–677.[Wh] B. White, Existence of smooth embedded surfaces of prescribed genus that minimize parametric evenelliptic functionals on 3-manifolds, J. Di¤. Geom. 33 (1991), 413–443.Korea Institute for Advanced Study, 207-43 Cheongryangri 2-dong, Dongdaemun-gu,Seoul 130-722, South-Koreae-mail: choe@kias.re.krDepartamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada,18071 Granada, Españae-mail: ritore@ugr.esEingegangen 16. Juli 2004, in revidierter Fassung 16. Februar 2006