GENERALIZED ISOPERIMETRIC INEQUALITIES FOR EXTRINSIC BALLS IN MINIMAL SUBMANIFOLDS. Steen Markvorsen and Vicente Palmer*

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1 GENEALIZED ISOPEIMETIC INEQUALITIES FO EXTINSIC BALLS IN MINIMAL SUBMANIFOLDS Steen Markvorsen and Vicente Palmer* Abstract. The volume of an extrinsic ball in a minimal submanifold has a well defined lower bound when the ambient manifold has an upper bound on its sectional curvatures, see e.g. [3] and [10]. When this upper bound is non-positive, the second named author has shown an isoperimetric inequality for such domains, see [11]. This result again gives the comparison result for volumes alluded to above together with a characterization of the totally geodesic submanifolds of hyperbolic space forms. In the present paper we find a corresponding sharp isoperimetric inequality for minimal submanifolds in spaces with sectional curvatures bounded from above by any constant. As a corollary we find again a characterization of the totally geodesic submanifolds of spherical space forms. 1. Introduction Let P m be an immersed submanifold of a complete riemannian manifold N n. The distance function on the ambient space N n is denoted by d. If p is a point in P, we define r(q := d(p, q for every q N. We shall also denote by r the restriction r P : P I. Let us define the extrinsic ball of radius and center p P, D (p P, to be the smooth connected component of B n(p P = {q P r p(q } which contains p. Here, B n (p denotes the geodesic -ball around p in the ambient space N, subject to π the restriction that min{i N (p, 2 }, where b is the supremum of the sectional b curvatures of N, and i N (p is the injectivity radius of N from p. We shall denote by S n 1 (p the geodesic -sphere in N. We observe that, when considering the n-dimensional simply connected space forms of constant curvature b I, (denoted from now on as IK n (b and its totally geodesic submanifolds IK m (b IK n (b, then the corresponding extrinsic -ball centered at p IK m (b is just the geodesic -ball B b,m and its boundary is the geodesic sphere S b,m 1. centered at p in this submanifold, 1991 Mathematics Subject Classification. 53C21; 58J65. Key words and phrases. isoperimetric inequality, Brownian motion, mean exit time, minimal submanifold, extrinsic ball. *Supported by a Grant of the Spanish Ministerio de Educación y Cultura 1 Typeset by AMS-TEX

2 2 S. Markvorsen and V. Palmer One way to study the relationship between the inner and the outer geometry of an immersed m-dimensional submanifold P m in an ambient riemannian manifold N n is to find estimates for suitable inner invariants of extrinsic balls of the submanifold. The volume of an extrinsic ball and the volume of its boundary are but two such invariants. The estimates are typically established via comparison with the corresponding values for the totally geodesic balls in the constant curvature setting. We shall refer to this as the standard comparison situation. The actual comparison may be obtained by transplantation. We use the distance function to transplant suitable radially (distance-symmetric PDE-solutions from the constant curvature totally geodesic balls into the extrinsic balls of P m. The transplanted solution is then pointwise compared with the true solution of the corresponding PDE problem for the extrinsic ball in question. Using this technique we get our main result: Theorem 1. Let P m be a minimally immersed submanifold of N n and let D (p be an extrinsic -ball in P m. If the sectional curvatures K N of N satisfy K N b, (b > 0, then we have for all 0 : (1.1 Vol(D Vol(B b,m Vol( D Vol(S b,m 1 mh b (, and as a consecuence (1.2 Vol(D Vol( D 1 mh b (, where h b (r is the constant curvature of any geodesic sphere of radius r in a space form of constant curvature b. Equality for some in (1.1 implies that D is a minimal cone in N n. Hence, if N n = S n (b, the sphere of constant curvature b > 0, then P m is a totally geodesic submanifold of N n. emarks We recall that b cot br, if b > 0 1 h b (r = r, if b = 0 b coth br, if b < 0 We also note that unless the dimension of P is m = 1, the inequality (1.2 is always a strict inequality for all > 0. In fact (1.1 implies that Vol(D Vol( D Vol(S b,m 1 mh b ( + mh b ( Vol(B b,m

3 Generalized Isoperimetric Inequalities for extrinsic balls... 3 and when b > 0 and m > 1 we have the strict inequality: mh b ( Vol(B b,m <, (see Lemma 2.4 and emark 2.5 below. Vol(S b,m 1 If the submanifold P m is minimally immersed in the real space form IK n (b, S.-Y. Cheng, P. Li and S.-T. Yau obtained sharp lower bounds for the volume of the corresponding extrinsic balls via comparison theorems for their Dirichlet and Neumann heat kernels, see [3]. In [10], the first named author of the present paper extended these results to minimal submanifolds of ambient spaces with sectional curvatures bounded from above by a real constant. In [1] J. Choe has established similar inequalities of isoperimetric type for minimal surfaces. In [3] and [10] the idea is to compare the Dirichlet and Neumann heat kernels defined on the extrinsic -balls D in P m, with the corresponding heat kernels defined on the geodesic -ball B b,m of dimension m in IKn (b. To be precise this and the volumes in IK m (b, respectively. gives the following comparison between the volumes of D and B b,m of the boundaries D and the geodesic -sphere S b,m 1 Theorem A. ([10] Let P m be a minimally immersed submanifold of N n and let D (p be an extrinsic -ball in P m. If the sectional curvatures K N of N satisfy K N b, (b I, then (1.3 Vol(D Vol(B b,m, (1.4 Vol( D Vol(S b,m 1. In [11], the second named author obtained an isoperimetric inequality for such extrinsic balls in a minimal submanifold P m of a riemannian manifold N n whose sectional curvatures K N are bounded from above by a nonpositive constant. This inequality is based on the comparison, established in [9], between the mean exit time function defined on the extrinsic ball in P m and the corresponding function defined in IKm (b. If the sectional curvatures of the ambient manifold are bounded from above by a nonpositive constant this isoperimetric inequality together with the co-area formula gives an alternative proof of the volume comparison theorems. The results in [11] can be stated explicitly as follows: on the geodesic ball B b,m Theorem B. ([11] Let P m be a minimally immersed submanifold of N n and let D (p be an extrinsic -ball in P m. If the sectional curvatures K N of N satisfy K N b 0, then (1.5 Vol(D Vol( D Vol(Bb,m Vol(S b,m 1. As a consecuence, we have inequality (1.3.

4 4 S. Markvorsen and V. Palmer When b 0, equality in (1.4 for some implies that D is a minimal cone in N n. Hence, if N n = IK n (b, the hyperbolic space of constant curvature b < 0 or the euclidean space I n, then P m is a totally geodesic submanifold of N n. It is natural to ask for a characterization of the totally geodesic submanifolds of spheres S n (b from equality in (1.3, in the same way that it was obtained in Theorem B for hyperbolic or euclidean ambient spaces. We present an alternative proof of this inequality which then gives the following characterization of totally geodesic submanifolds of spheres: Theorem 2. Suppose the assumptions of Theorem A are satisfied. If equality is attained in (1.3 for some, then D is a minimal cone in N n. Hence, if N n = IK n (b, b I, then P m is a totally geodesic submanifold of N n. Finally, and still in the general framework of Theorem B, we conjecture the characterization of totally geodesic submanifolds of euclidean spaces from assuming that the isoperimetric equality (1.6 Vol(D Vol( D = Vol(B0,m Vol(S 0,m 1 is satisfied by an extrinsic ball with given center p P and given radius > 0. We know (see [9] and [11], that for b 0 the isoperimetric equality Vol(D Vol( D = Vol(Bb,m Vol(S b,m 1 leads to the equality between the mean exit time function E and the accurately transplanted radially symmetric standard mean exit time function E b,m (which are both defined on the extrinsic ball D. As was pointed out in [9], when the ambient space is the hyperbolic space IK n (b, this equality together with the minimality of the submanifold P m and the analytic prolongation principle implies that P m must be totally geodesic in IK n (b. However, when the ambient space is I n, the equality E = E 0,m on D is always satisfied on extrinsic balls defined in minimal submanifolds of the euclidean space, where it even characterizes minimal hypersurfaces (Theorem 3 in [9]. We show the following partial results towards the conjecture stated above. Theorem 3. Let P m be a minimally immersed submanifold of I n. (a Let us suppose that we have equality (1.6, for all p P, and for some fixed radius. Then P m is a totally geodesic submanifold of I n. (b Let us suppose that we have equality (1.6, for all > 0, and for some fixed point p P. Then P m is a totally geodesic submanifold of I n. Our general method of attack in this paper is via the mean exit time function, which can be characterized as follows. Let E (x denote the mean time of first exit

5 Generalized Isoperimetric Inequalities for extrinsic balls... 5 from D (p for a Brownian particle starting at x D (p. A remark due to Dynkin in [4] then states that E is the continuous solution to the following Poisson equation with Dirichlet boundary data: (1.7 E = 1 E D = 0. We shall denote by E b,m the mean exit time function on the geodesic ball Bb,m in IK m (b. The intrinsic ball B b,m has maximal isotropy at the center p, so we have that E b,m only depends on the extrinsic distance r (of x from p. Therefore, we will write (r. E b,m = Eb,m We shall also consider the general radially symmetric solutions of the Poisson equation (1.7 in the constant curvature totally geodesic balls, but without imposing continuity at r = 0. We call these functions the generalized mean exit time functions, and denote them by E b,m C, integration. It is easy to check that E b,m 0, = Eb,m for E b,m C, in Lemma The constant C denotes the arbitrary constant of and we show an explicit expression Using these solutions in combination with the Laplacian comparison technique descibed below we get upper as well as lower bounds for the generalized isoperimetric quotient Vol(D C Vol( D, where C is a suitable (best possible choice of the integration constant in E b,m C,. The constant depends on m,, and b and is subject to the conditions under which the Laplacian comparison techniques can be applied. From the upper bounds we obtain the results of Theorems 1 and B and from the lower bounds we get Theorem 2. The geometric isoperimetric type inequalities are based on a comparison involving the laplacian P E b,m C, (x, the first and the second derivative of Eb,m C,, where Eb,m C, denotes here the radial solution E b,m C, (r, accurately transplanted to D P m, and the gradient in P of the extrinsic distance r, grad P r. The comparison follows from Lemma 2.1, (see [6], see also [9]: When the sectional curvatures K N are bounded from below and P is minimal, we have: If the first derivative E b,m C, (r 0 r, (respectively, if Eb,m C, (r 0 r, then (1.9 P E b,m C, ( {Eb,m C, Eb,m C, h b} grad P r 2 +mh b E b,m C,.

6 6 S. Markvorsen and V. Palmer In order to extract the geometric conclusions from these estimates using grad P r 2 1 we thus need the quantity E b,m C, Eb,m C, h b to be nonnegative when (r 0 and to be nonpositive when Eb,m C, (r 0. E b,m C, The outline of the paper is the following: We prove Theorem 1 in Section 3, Theorem 2 in Section 4 and Theorem 3 in Section 5. Section 2 is devoted to the describe some previous results about the standard situations and to discuss the general solutions of the Poisson equation as alluded to above. This work was done during a stay of the second author at the Department of Mathematics, Technical University of Denmark, while he was supported by a Grant from the Spanish Ministerio de Educación y Cultura, (Programa Sectorial de Formación de Profesorado y Perfeccionamiento del Personal Investigador. He would like to thank the staff at the Department of Mathematics for the cordial hospitality during this period. 2. Preliminaries Let f : P I be a smooth function that depends only on the extrinsic distance r along P in N. Thus f(r denotes the composition f r : P I. We denote by grad N r and grad P r the gradients of r in N and P respectively. Let us first remark that grad P r(q is just the tangential component in P of grad N r(q, for all q P. Then we have the following basic relation on D (p, for all : (see [6], eq. (2.1 (2.1 grad N r = grad P r + (grad P r where (grad P r (q is perpendicular to T q P for all q D (p. In a standard situation we have that grad IKm (b r 2 = 1 because as IK m (b IK n (b is totally geodesic, the restriction of the extrinsic distance from p in IK m (b is just the intrinsic distance from p, and hence grad IKm (b r = grad IKn (b r. Concerning the functions f(r we then have: (2.2 < grad P f, grad P r >= grad P r(f = df(grad P r = f (rdr(grad P r = f (r grad P r 2. We may now recall the following inequality, which was proved in [6], see also [9]: 2.1 Lemma. Let q D (p \ {p}. Suppose that f (r 0, for all r [0, ], and that the sectional curvatures of the ambient manifold N satisfy K N b, (respectively, K N b, b I. Then (2.3 P f q ( (f (r f (rh b (r grad P r 2 + mf (r(h b (r+ < H(q, grad N r >,

7 Generalized Isoperimetric Inequalities for extrinsic balls... 7 where P denotes the Laplacian on P m, and H(q is the mean curvature vector of P m at q in N n. If f (r 0, for all r [0, ], and the sectional curvatures of the ambient manifold satisfy K N b, (respectively, K N b, b I. Then the inequalities in (2.3 are reversed: (2.4 P f q ( (f (r f (rh b (r grad P r 2 + mf (r(h b (r+ < H(q, grad N r > 2.2 emark When P is a minimal submanifold in a space form IK n (b, the inequalities in (2.3 and (2.4 are equalities. In particular we have for all q D (p \ {p}, (2.5 P r q = h b (r grad P r 2 + m(h b (r+ < H(q, grad N r >. 2.3 Lemma. ([5] Let Br b,m and Sr b,m 1 be a geodesic r-ball and a geodesic r-sphere, respectively, in a real space form IK m (b. Then (2.6 (2.7 where d dr Vol(Bb,m r = Vol(Sr b,m 1 r > 0 Vol(Br b,m = Vol(S 0,m 1 1 F b (t = r 0 (F b(t m 1 dt r > 0 1 b (1 cos bt, if b > 0 t 2 2, if b = 0 1 b (1 cosh bt, if b < 0 Using this Lemma, we obtain the following inequalities 2.4 Lemma. In a real space form IK m (b, we have (2.8 (2.9 (2.10 If b > 0 Vol(Sr b,m 1 > mh b (r Vol(Br b,m r > 0. If b = 0 Vol(Sr 0,m 1 = mh 0 (r Vol(Br 0,m r > 0. If b < 0 Vol(Sr b,m 1 < mh b (r Vol(Br b,m r > 0. Proof. Inequality (2.9 is immediate from Lemma 2.3, taking into account that F 0 (r = r2 2 and h 0 (r = 1 r. We are going to prove inequality (2.8. Before that we must remark that, for all b I, (2.10 F b (r = h b(rf b (r r > 0.

8 8 S. Markvorsen and V. Palmer It is straightforward, using Lemma 2.3, that inequality (2.8 holds for all r if and only if (2.12 d (Vol(Bb,m r > 0 r > 0. dr (rm (F b This last inequality is true, using (2.11, if and only if we have, for all r > 0, (2.13 (F b(r m > mf b (r r 0 (F b(t m 1 dt. In order to prove (2.13 when b > 0, and taking into account that, in this case, F b (r = sin br b, we must check that ( ( b m tan br(sin br m 1 > m ( b m 1 r 0 (sin bt m 1 dt r > 0. and We may show this last inequality using the functions f(r = 1 b tan br(sin br m 1 g(r = m r 0 (sin bt m 1 dt satisfying f(0 = g(0 = 0. It is then easy to check that (f g (r > 0 r > 0. The proof of inequality (2.10 follows in a similar way, changing the sign of inequality (2.12 and using that, in this case, F b (r = sinh br b. 2.5 emark By virtue of Lemma 2.4 we have the relation between the bounds for the isoperimetric quotient in inequality (1.2, (positive curvature case and in inequality (1.5, (negative curvature case. The maximum between the isoperimetric quotient in the standard situations and the inverse of mean curvature of geodesic spheres in the real space forms depends of the curvature of the ambient space, i.e. max{ Vol(Bb,m r Vol(Sr b,m 1, 1 mh b (r } = Vol(Bb,m r Vol(Sr b,m 1 Vol(Br b,m Vol(Sr b,m 1 = 1 mh b (r max{ Vol(Bb,m r Vol(Sr b,m 1, 1 mh b (r } = 1 mh b (r r if b < 0 r if b = 0 r if b > 0 The following technical results will also be useful in the proof of Theorem 1.

9 Vol(Br b,m is a strictly increasing func- 2.6 Lemma. When b > 0, Ψ(r = 1 tion, Ψ(r > 0 for r > 0. Generalized Isoperimetric Inequalities for extrinsic balls... 9 mh b (r Vol(Sb,m 1 r Proof. We only have to check that Ψ (r > 0, and this is straightforward using Lemma 2.3, equality (2.11 and taking into account that, since b > 0, h b (r = b tan br. 2.7 Lemma. The generalized radially symmetric solutions of the Poisson equation with Dirichlet boundary data (2.15 IKm (b E = 1, on B b,m ( p p E S b,m 1 = 0 are given explicitly by (2.16 E b,m C, (r = r Vol(Bu b,m C Vol(S 0,m 1 Vol(S b,m 1 u where C is an arbitrary constant of (first integration. 1 du, Proof. Using Lemma 2.1 and emark 2.2, equation (2.15 can be written as (2.17 Ë(r + (m 1h b (rė(r = 1. The first integration of this ordinary differential equation gives us, using equation (2.11, (2.18 E b,m C, (r = 1 {C (rm 1 (F b r 0 (F b(t m 1 dt} Integration of (2.18 now gives the result via Lemma 2.3 and the Dirichlet boundary condition. 2.8 Definition We are going to transplant the radial functions E b,m C, to the extrinsic domain D (p via the extrinsic distance function along P in N. For every x D (p we will write as follows (note that the radial and the transplanted functions are - by sligth abuse of notation - denoted by the same symbol: C, : D (p I; E b,m C, (x := Eb,m C, (r p(x E b,m

10 10 S. Markvorsen and V. Palmer The new transplanted function only depends, in fact, of the extrinsic distance r to p in P. Let us also define the following quantities which will help us keep track of the assumptions under which we are going to apply the Laplacian comparison technique: K 1 (b, m, = Vol(Bb,m Vol(S 0,m 1 K 2 (b, m, = 1 1 Vol(S 0,m 1 1 {Vol(Bb,m 1 mh b ( Vol(Sb,m 1 } Note that, for all b I, and for all > 0, K 1 (m, b, > 0. However, by virtue of Lemma 2.4, when b > 0 we have K 2 (m, b, < 0, for all > 0, whereas K 2 (m, b, 0, for all > 0 when b 0. The respective signs of the derivative E b,m C, (r and the quantity Eb,m C, (r Eb,m C, (rh b(r play a fundamental role in the comparison between the laplacians mentioned above. In the following result we describe the behaviour of these signs in terms of b, the constant C and the quantities K 1 (m, b, and K 2 (m, b,. 2.9 Proposition. The functions E b,m C, (r, defined on the extrinsic -ball D (p, satisfy: (a For all r we have E b,m C, ( 0 if and only if C K 1(m, b,. In particular, if C 0, then E b,m C, (r 0. (b The inequality E b,m C, (r Eb,m C, (rh b(r > 0 holds when b > 0, C K 2 (m, b,, and r < when b = 0, C < 0, and r when b < 0, C 0, and r. (c When C K 1 (m, b, we have as well: E b,m C, (r Eb,m C, (rh b(r < 0 r. Proof. Assertion (a follows from equation (2.16 and taking into account that, by Lemma 2.7 and Lemma 2.3, we have (2.19 C Vol(S 0,m 1 1 Vol(B b,m = 0 if and only if C = K 1( = Vol(Bb,m Vol(S b,m 1.

11 Generalized Isoperimetric Inequalities for extrinsic balls On the other hand, when b > 0, then K 2 (m, b, < 0, so that E b,m C, (r 0 r. This follows from equation (2.16 and the fact that in both cases (b > 0 and b 0 C is a nonpositive constant. The proof of (b is based on the following considerations: When b > 0 and C K 2 (m, b,, (respectively, when b 0 and C 0, we know from (a that the sign of E b,m C, (r does not change in the interval [0, ]. Then, using Lemma 2.1, remark 2.2, equation (2.15 and the fact that grad IKm (b r 2 = 1, we have: (2.20 IKm (b E b,m C, = Eb,m C, and, therefore, using (2.16, (2.21 E b,m C, (r Eb,m C, (rh b(r = Vol(S b,m 1 r (r Eb,m C, (rh b(r + mh b (re b,m C, (r = 1, + mh b (r Vol(B b,m mh b (rc Vol(S 0,m 1 r Vol(S b,m 1 r Then, by (2.21, when b > 0 and C K 2 (m, b,, (2.22 E b,m C, (r Eb,m C, (rh b(r 1 r. mh b(r Vol(Sr b,m 1 {Ψ( Ψ(r} 0 r, and these expressions are strictly positive if r <, because Ψ(r = 1 mh b (r Vol(Sb,m 1 r Vol(Br b,m is a strictly increasing function, when b > 0, by Lemma 2.6. On the other hand, when b = 0 and C < 0, we have, using (2.9 and (2.21, 1 Vol(Sr b,m 1 (2.23 E b,m C, (r Eb,m C, (rh b(r = mh b(rc Vol(S 0,m 1 If C 0, then E b,m C, (r Eb,m C, (rh b(r is nonnegative on [0, ]. When b < 0 and C 0, using inequality (2.10 and (2.21, we have 1 Vol(Sr b,m 1 (2.24 E b,m C, (r Eb,m C, (rh b(r > mh b(rc Vol(S 0,m 1 Finally, if C K 1 (m, b, > 0, then > 0 r. 0 r. (2.25 E b,m C, (r Eb,m C, (rh b(r Vol(S b,m 1 r + mh b (r{vol(br b,m Vol(B b,m Vol(S b,m 1 r } < 0 r. We shall need two results concerning small extrinsic balls and their volumes.

12 12 S. Markvorsen and V. Palmer 2.10 Proposition. Let P m be an immersed submanifold of a riemannian manifold N n. Let D r (p be an extrinsic ball of radius r in P m. Then grad P r dσ D lim r = 1 Vol( D r As a consequence, lim Eb,m C, (r D r grad P r dσ = lim E b,m C, (r Vol( D r Proof. We recall the decomposition for the gradient of the extrinsic distance grad N r = grad P r + (grad P r On the other hand we can consider, on each q D r the following orthonormal basis of the orthogonal complement in T q N of the tangent space T q D r : { gradp r grad P r, ξ i} n i=m+1 where {ξ i } n i=m+1 is an orthonormal frame of the normal bundle of P m in N n. Since T q D r T q Sr n 1, we have (T q Sr n 1 (T q D r, so, expressing grad N r in this basis and using that grad N r = 1, we have (2.26 grad P r 2 = 1 n i=m+1 < grad N r, ξ i > 2 1. Integrating (2.26 along D r and taking into account that we obtain 1 n i=m+1 Therefore, if we just prove that lim 0 grad P r 2 grad P r 1, D r < grad N r, ξ i > 2 Vol( D r D r < grad N r, ξ i > 2 Vol( D r D r grad P r Vol( D r 1 = 0 i = m + 1,..., n,

13 Generalized Isoperimetric Inequalities for extrinsic balls then the result follows. For each i = m + 1,..., n and for a given fixed radius r I, we have that < grad N r, ξ i > 2 : D r I is a continuous function defined on the compact set D r. Then, we know that there exists one point q r D r where the maximum of this function is reached, so, for each radius r I, < grad N r, ξ i > 2 D r grad N r, ξ i 2 (q r. Vol( D r When r tends to zero, lim q r = p, where p is the center of the extrinsic ball, so < grad N r, ξ i > 2 D 0 lim r lim < grad N r, ξ i > 2 (q r = 0, Vol( D r since grad N r(p T p P Proposition. Let P m be an immersed minimal submanifold of a iemannian manifold N n. Let D r (p be an extrinsic ball of radius r in P m. Let us suppose that the sectional curvatures K N of N satisfy K N b for some b I. Let E b,m C, ball B b,m (r denote the generalized mean exit time function, defined on the geodesic. Then (2.27 lim E b,m C, (r Vol( D r = C Vol(S 0,m 1 1. Proof. The volume of the boundary of an extrinsic ball of any radius r, in a minimal submanifold P m, is bounded in the following way (2.28 Vol(S b,m 1 r Vol( D r d dr Vol(D r. See Corollary A in [10] for the first inequality and the proof of Corollary 1.2 in [11] for the second one. From Proposition 2.9, if C 0, then E b,m C, (r 0 r 0, and, from (2.16 in Lemma 2.7 if C > 0, then E b,m C, (r 0 for sufficiently small radius r. We are going to divide this proof in two cases: when C > 0 and when C 0. If C > 0, we have (2.29 E b,m C, (r Vol(Sb,m 1 r E b,m C, (r Vol( D r E b,m C, (r d dr Vol(D r

14 14 S. Markvorsen and V. Palmer for those small values of r where the first derivative E b,m C, (r is nonnegative. Using the asymptotic expansion for the volume of an extrinsic ball in a submanifold of an arbitrary iemannian manifold obtained in [8], (see the main theorem and the final remark there, we can express the derivative of the volume as (2.30 d dr Vol(D r = Vol(S 0,m 1 1 r m 1 1 Ar m+1 + O(r m+2 8m where the coefficient A = A( B P, H P involves the curvature of N, the norm of the second fundamental form of P, B P, and the norm of its mean curvature normal, H P. Since + Vol(S0,m 1 (2.31 Vol(S 0,m 1 1 r m 1 = Vol(S 0,m 1 r, we get from (2.16 and the expansion (2.30 that (2.32 E b,m C, (r d dr Vol(D r = {C Vol(S 0,m 1 1 Vol(Br b,m + O(r m+2 With some computations, it is easy to see that (2.33 lim and then, using (2.32, Vol(Sr 0,m 1 Vol(Sr b,m 1 } Vol(S0,m 1 r Vol(Sr b,m 1 = 1 b I (2.34 lim E b,m C, (r d dr Vol(D r = C Vol(S 0,m 1 1 On the other hand we have that (2.35 lim Eb,m C, (r Vol(Sb,m 1 r = lim{c Vol(S 0,m 1 1 Vol(Br b,m = C Vol(S 0,m 1 1, A( B, H (1 + r 2 8m } Vol(S0,m 1 r Vol(Sr b,m 1 so we have the result using (2.34, (2.35 and inequalities (2.29. When C 0, the proof is the same, only now C Vol(S 0,m 1 1 = lim E b,m C, (r d dr Vol(D r lim E b,m C, (r Vol( D r lim E b,m C, (r Vol(Sb,m 1 r = C Vol(S 0,m 1 1.

15 Generalized Isoperimetric Inequalities for extrinsic balls Proof of Theorem 1 Let be a fixed radius. To show inequality (1.1, we are going to prove, in first place, that for this radius and for all C K 2 (m, b, we have the following inequality (3.1 Vol(D C Vol(S 0,m 1 1 Vol( D Vol(Bb,m C Vol(S0,m 1 1 Vol(S b,m 1 Then (3.1 implies (1.1 when C = K 2 (m, b,. Inequality (1.2 follows from (1.1 in this way: From (1.1 we have, (if Vol( D > Vol(S b,m 1, that Vol(D Vol( D Vol(S b,m 1 b,m 1 + mh b ( Vol(B b,m Vol( D Vol(S mh b ({Vol( D Vol(S b,m 1 Then, having into account that, when b > 0 and > 0, we can conclude inequality (1.2. We are going to prove that Vol(S b,m 1 + mh b ( Vol(B b,m < 0 (3.2 1 P E b,m C, When b > 0 and C K 2 (m, b,, we know, by Proposition 2.9, that E b,m C, (r 0 r. Then, under the hypothesis of Theorem 1, using Lemma 2.1, equation (2.20, assertion (b in Proposition 2.9 and that grad P r 2 1, (3.3 P E b,m C, {Eb,m C, (r Eb,m C, (rh b(r} grad P r 2 + mh b (re b,m C, (r IKm (b E b,m C, = 1 Once we have (3.2, the proof of (3.1 runs as follows: Let D r (p be an extrinsic ball with center p and radius r. Then, using inequality (3.2, we have (3.4 Vol(D (p D r (p P E b,m D (p D r (p C, dw div P (grad P E b,m C, dw D (p D r (p

16 16 S. Markvorsen and V. Palmer Now, we are going to apply the divergence theorem. We have that div P (grad P E b,m C, C 1 (D (p D r (p and, since the boundary of the extrinsic balls agrees with its boundary in the relative topology on P, we have (D (p D r (p = D (p D r (p. We must also note that the outward unit normal vector field on (D (p D r (p, grad denoted as ν, is the same as P r grad P r on D and equal to gradp r grad P r on D r. With these considerations, using equation (2.2, that grad P r 1, that E b,m C, ( 0 and applying divergence theorem, (3.5 Vol(D (p D r (p = E b,m C, ( D (p D r (p div P (grad P E b,m C, dw = < grad P E b,m C,, ν > dσ (D (p D r (p D (p grad P r dσ + E b,m C, (r D r (p E b,m C, ( Vol( D + E b,m C, (r grad P r dσ D r (p grad P r dσ. Taking limits when r tends to zero, and applying Propositions 2.10 and 2.11, (3.6 Vol(D = lim Vol(D D r E b,m C, ( Vol( D + C Vol(S 0,m 1 Therefore Vol(D C Vol(S 0,m 1 1 Vol( D Vol(Bb,m C Vol(S0,m 1 1 Vol(S b,m 1. 1 If we have equality in (1.1 for some fixed radius 0 and some fixed center p P, then we have equality in (3.1 for this extrinsic ball D 0 (p, when C = K 2 (m, b, 0. We are going to see that equality in (3.1 for some radius 0 and some constant C K 2 (m, b, 0 implies that, given any radius < 0, D (p is a minimal cone in N n. When equality in (3.1 is satisfied by D 0 (p, then inequality (3.6 becomes an equality. Hence, taking limits in (3.4 and (3.5 when r tends to zero, inequalities become equalities, so we have, using (3.4, (3.7 Vol(D 0 (p = lim D P E b,m C, dw. 0 (p D r (p Then, taking into account that IKm (b E b,m C, = 1, (3.8 { IKm (b E b,m C, P E b,m C, }dw = 0. D 0 (p {p}

17 Generalized Isoperimetric Inequalities for extrinsic balls However, since P E b,m C, 1 = IKm (b E b,m C, on D 0 (p, we conclude that on D 0 (p. Using (3.3 and the fact that P E b,m C, = IKm (b E b,m C, E b,m C, (r Eb,m C, (rh b(r > 0 r < 0 we have that, given any < 0, grad P r 2 = 1 on D (p. Hence, the extrinsic ball D (p is a minimal cone in N n. When the ambient space N n is the sphere S n (b, we have that D is a minimal cone in a real space form, and hence, by analytic continuation from D = B b,m we finally get that P m is a totally geodesic submanifold of N n. This concludes the proof of Theorem 1. emarks (i It is possible, with totally analogous arguments, to prove inequality (3.1 for any radius, when b 0 and C 0, (in fact, inequality (1.5 in Theorem B follows from inequality (3.1, when b 0 and C = 0. Under these assumptions, we have, by Proposition 2.9, that, for all r, (3.9 E b,m C, (r 0 E b,m C, (r Eb,m C, (rh b(r 0. In fact, if b < 0, then E b,m C, (r Eb,m C, (rh b(r > 0 r and for all C 0. Then, we can conclude inequality (3.2 in the same way as before, and, applying the divergence theorem and taking limits when r tends to zero, we obtain inequalities (3.4, (3.5 and (3.6 for this case. (ii When b < 0, if we have the equality in (3.1 (for one fixed constant C 0 attained by one extrinsic ball D (p, with fixed center and radius, then it is possible to show the equality P E b,m C, = (b E b,m IKm C, on D (p, so, using Lemma 2.1 and the fact that, for this fixed constant C 0, we have, r, E b,m C, (r Eb,m C, (rh b(r > 0, we can conclude that grad P r 2 = 1 on D (p. Hence, the extrinsic ball D (p is a minimal cone in N n. Moreover, if the ambient space is the hyperbolic space, then P is totally geodesic by analytic prolongation from D (p = B b,m (p.

18 18 S. Markvorsen and V. Palmer (iii When b = 0, if equality in (3.1 is attained, for one fixed constant C < 0, by one extrinsic ball D (p, with fixed center and radius, then D (p is a minimal cone in N, with the same arguments as in (ii, because in this case, as C < 0 Ë 0,m C, (r Ė0,m C, (rh 0(r > 0 r. Therefore, if the ambient space is the euclidean space, we have that P is a totally geodesic submanifold by analytic prolongation from D (p = B 0,m (p. However, equality in (3.1 with C = 0, attained by one extrinsic ball D (p does not by itself characterize the ball as a minimal cone in N, and hence, when the ambient space is I n, we are not able to conclude from this fact that P is totally geodesic. 4. Proof of Theorem 2 To show inequality (1.3 in Theorem A under the more general hypothesis of Theorem 2, we prove that for every b I and for all C K 1 (m, b,, we have (4.1 Vol(D C Vol(S 0,m 1 1 Vol( D Vol(Bb,m C Vol(S0,m 1 1 Vol(S b,m 1. Then, inequality (1.3 follows from (4.1 by taking C = K 1 (m, b,. In order to prove inequality (4.1, we shall proceed in a similar way as in Section 3, showing, (now, for all b I, the inequality: (4.2 1 P E b,m C,. When C K 1 (m, b, > 0, it can be proved, using (2.16, that (4.3 E b,m C, (r Eb,m C, ( 0 r. Then, using Lemma 2.1, assertion (c of Proposition 2.9, equation (2.20 and the fact that grad P r 2 1, we have (4.4 P E b,m C, {Eb,m C, (r Eb,m C, (rh b(r} grad P r 2 + mh b (re b,m C, (r IKm (b E b,m C, = 1. With (4.2 in hand, we can argue as in the preceding proof, taking the domains D D r, (r, applying the divergence theorem, and using equation (4.2: grad P r 1 and E b,m C, ( 0, we get, computing as in (3.4 (4.5 Vol(D (p D r (p E b,m C, ( Vol( D + E b,m C, (r grad P r dσ D r (p

19 Generalized Isoperimetric Inequalities for extrinsic balls Taking limits when r tends to zero, and applying Propositions 2.10 and 2.11, (4.6 Vol(D = lim Vol(D D r E b,m C, ( Vol( D + C Vol(S 0,m 1 1, and hence we have inequality (4.1. The equality discussion, for all b I, is as follows: the equality in (1.3, satisfied by one fixed extrinsic ball D (p is a particular case of the equality in (4.1, taking C = K 1 (m, b,. When we have equality in the inequality (4.1 for some and some C K 1 (m, b,, then inequality (4.6 becomes an equality for this radius. This implies, as in Section 3, that by taking limits when r tends to zero in (4.5, we get P E b,m C, = IKm (b E b,m C,. Then, from (c in Proposition 2.9, and (4.4, we have that grad P r 2 = 1 on D so D is a minimal cone in N. If N is a real space form IK n (b, (b I, then D = B b,m, so P is totally geodesic from analytic continuation from this geodesic ball, as in preceding proofs. emark When b 0, there is an alternative characterization of equality in (1.3 in Theorem A, based on inequality (1.5 in Theorem B and the co-area formula, see the proof of Corollary 2 in [11]. 5. Submanifolds of I n As we have remarked before, when the ambient manifold is the euclidean space I n, we can get that P m is a totally geodesic submanifold of I n from the equality in (3.1, realized for some radius and some fixed constant C < 0. However, it is not possible to conclude the same from the isoperimetric equality (5.1 Vol(D Vol( D = Vol(B0,m Vol(S 0,m 1 attained for some fixed extrinsic ball D (p, and corresponding to equality in inequality (3.1 for the value C = 0. The explanation of this fact, in view of the proofs above, is centered at the following point: when b = 0 and C = 0, we have that Ë0,m 0, we cannot get that grad P r 2 = 1 on D (p. On the other hand, since (r Ė0,m 0, (rh 0(r = 0 for all r, so lim Ė 0,m 0, (r grad P r dσ = C Vol(S 0,m 1 1 = 0, D r (p

20 20 S. Markvorsen and V. Palmer equality (5.1 implies the equality in inequalities (3.4, (3.5 and (3.6 for the case b = 0 and C = 0, so we get (5.2 Vol(D = lim Vol(D D r = Ė0,m 0, ( grad P r dσ D (p By equation (2.15, Ė 0,m 0, = Ė0,m 0, ( Vol( D (p. ( = Vol(B0,m Vol(S 0,m 1 0, so (5.2 implies that gradp r = 1 on D, and therefore, grad P r = grad In r on D. Proof of Theorem 3 We first prove assertion (a. Let P m be a minimal submanifold of I n and let us suppose that there exists one fixed radius such that we have the equality (5.1 for all p P. Then we know that, for all p P, and given the fixed radius, grad P r = grad In r on D. Hence, given p P and given X T p P unitary, there exists a point q P such that X = grad P r q, (here, r q denotes the extrinsic distance from q P. As the extrinsic ball D (q satisfies (5.1, we can conclude that Then, so P is totally geodesic in I n. X = grad P r q = grad In r q < L P ξ X, X > = < In X ξ, X >=< ξ, In X X > =< ξ, In grad In r gradin r >= 0 Now show assertion (b. Given a fixed radius 0. If equality (5.1 holds for all the extrinsic spheres D, with fixed center p P, then (5.1 will be satisfied on the extrinsic ball D 0 (p, so grad P r = grad In r on D 0 (p and hence, D 0 (p is a minimal cone in I n. Therefore, by analytic prolongation from D 0 (p = B b,m 0 (p, we finally get that P m is a totally geodesic submanifold of I n. eferences [1] J. Choe, The isoperimetric inequality for minimal surfaces in a iemannian manifold, J. reine angew. Math. 506 (1999, [2] I. Chavel, iemannian geometry. A modern introduction, Cambridge University Press, New York, 1993.

21 Generalized Isoperimetric Inequalities for extrinsic balls [3] S.Y. Cheng, P. Li, and S.T. Yau, Heat equations on minimal submanifolds and their applications, Amer. J. Math. 106 (1984, [4] E.B. Dynkin, Markov processes, Springer Verlag, [5] A. Gray, Tubes, Addison-Wesley, eading, [6] L.P. Jorge, and D. Koutroufiotis, An estimate for the curvature of bounded submanifolds, Amer. J. Math. 103 (1981, [7] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, New York, [8] L. Karp, and M. Pinsky, Volume of a small extrinsic ball in a submanifold, Bull. London Math. Soc. 21 (1989, [9] S. Markvorsen, On the mean exit time from a minimal submanifold, J. Diff. Geom. 29 (1989, 1 8. [10] S. Markvorsen, On the heat kernel comparison theorems for minimal submanifolds, Proc. Amer. Math. Soc. 97 (1986, [11] V. Palmer, Isoperimetric Inequalities for extrinsic balls in minimal submanifolds and their applications, J. London Math. Soc. (2 60 (1999, Steen Markvorsen Department of Mathematics Technical University of Denmark DK-2800 Lyngby, Denmark address: S.Markvorsen@mat.dtu.dk Vicente Palmer Departament de Matemàtiques Universitat Jaume I Castelló, Spain palmer@mat.uji.es