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Ends, Fundamental Tones, and Capacities of Minimal Submanifolds Via Extrinsic Comparison Theory

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Ends, Fundamental Tones, and Capacities of Minimal Submanifolds Via Extrinsic Comparison Theory ENDS, FUNDAMENTAL TONES, AND CAPACITIES OF MINIMAL SUBMANIFOLDS VIA EXTRINSIC COMPARISON THEORY VICENT GIMENO AND S. MARKVORSEN ABSTRACT. We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotation- ally symmetric model manifold. Using the asymptotic behavior of the volumes and ca- pacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question. 1. INTRODUCTION Let M be a complete non-compact Riemannian manifold. Let K ⊂ M be a compact set with non-empty interior and smooth boundary. We denote by EK (M) the number of connected components E1; ··· ;EEK (M) of M n K with non-compact closure. Then M EK (M) has EK (M) ends fEigi=1 with respect to K (see e.g. [GSC09]), and the global number of ends E(M) is given by (1.1) E(M) = sup EK (M) ; K⊂M where K ranges on the compact sets of M with non-empty interior and smooth boundary. The number of ends of a manifold can be bounded by geometric restrictions. For ex- ample, in the particular setting of an m−dimensional minimal submanifold P which is properly immersed into Euclidean space Rn, the number of ends E(P ) is known to be re- lated to the extrinsic properties of the immersion. Indeed, V. G. Tkachev proved in [Tka94, Theorem 2] (see also [Che95]) that for any properly immersed m−dimensional minimal n submanifold P in R with finite volume growth Vw0 (P ) < 1 the number of ends is bounded from above by (1.2) E(P ) ≤ CmVw0 (P ) ; m where Cm = 1 (Cm = 2 in the original [Tka94]) and the volume growth Vw0 (P ) is n R Vol P \ BR (o) (1.3) Vw (P ) = lim m : 0 R!1 R Vol BR (o) m m R R arXiv:1401.1329v1 [math.DG] 7 Jan 2014 Here Vol BR (o) is the volume of a geodesic ball BR (o) of radius R centered at o in Rm. The inequality (1.2) thus shows a significant relation between the number of ends (i.e. a topological property) and the behavior of a quotient of volumes (i.e. a metric property). Motivated by Tkachev’s application of the volume quotient appearing in equation (1.3), we will consider the corresponding flux quotient and capacity quotient of the minimal sub- manifolds. These quotients are constructed in the same way as indicated by the volume quotient but here we generalize the setting as well as Tkachev’s result to minimal sub- manifolds in more general ambient spaces as alluded to in the abstract. Specifically we assume that the minimal immersion goes into an ambient manifold N with a pole and with 2010 Mathematics Subject Classification. Primary 53A, 53C. Key words and phrases. First Dirichlet eigenvalue, Capacity, Effective resistance, Minimal submanifolds. Work partially supported by DGI grant MTM2010-21206-C02-02. 1 2 V. GIMENO AND S. MARKVORSEN sectional curvatures KN bounded from above by the radial curvatures Kw of a rotationally n + n−1 M = × g n symmetric model space w R S1 , with warped metric tensor Mw constructed us- + + 2 2 ing a positive warping function w : ! in such a way that gM n = dr +w(r) g n−1 R R w S1 is also balanced from below (see [MP06] and x3 for a precise definitions). Our generalization of inequality (1.2) is thence the following: Theorem 1.1. Let ' : P m ! N n be a proper minimal and complete immersion into a n−dimensional ambient manifold N n which possesses a pole o 2 N n and its sectional curvatures KN at any point p 2 N are bounded by above by the radial curvatures Kw of n a balanced from below model space Mw w00 (1.4) KN (p) ≤ KM n (r (p)) = − (r (p)) : w w 0 Suppose moreover, that w > 0 and there exist R such that K n (R) ≤ 0 for any R > R . 0 Mw 0 N Then, the number of ends EDR (P ) with respect to the extrinsic ball DR = P \ BR (o) for R > R0 is bounded from above by !m t ! 2 R w(s)m−1ds Vol(D ) (1.5) E (P ) ≤ 0 t ; DR R tm=m Vol(Bw) 1 − t t for any t > R. Using the above theorem we can estimate the global number of ends as follows: Corollary 1.2. Under the assumptions of theorem 1.1, suppose moreover R t m−1 ! 0 w(s) ds (1.6) lim sup m = Cw < 1 ; t!1 t =m and suppose also that the submanifold has finite volume growth, namely Vol(Dt) (1.7) Volw(P ) = lim w < 1 : t!1 Vol(Bt ) Then m (1.8) E(P ) ≤ 2 Cw Volw(P ) : m Remark a. If we choose w(t) = w0(t) = t, the model space becomes R , which is bal- anced from below, and the hypothesis of theorem 1.1 are therefore automatically fulfilled for any complete minimal submanifold properly immersed in a Cartan-Hadamard ambient manifold. Inequality (1.5) becomes !m 2 Vol(D ) (1.9) E (P ) ≤ t ; DR R V tm 1 − t m m For any R > 0 and any t > R, being Vm the volume of a geodesic ball of radius 1 in R . From inequality (1.6) we get (1.10) Cw0 = 1 : Thus inequality (1.8) becomes m Vol(Dt) (1.11) E(P ) ≤ 2 lim m ; t!1 Vmt which is the original inequality obtained by Tkachev (inequality (1.2)), but now inequality (1.11) is valid for any minimal submanifold properly immersed in a Cartan-Hadamard ambient manifold with finite volume growth. ENDS, FUNDAMENTAL TONES, AND CAPACITIES 3 FIGURE 1. Two examples of extrinsic annuli in R3: A catenoid on the left and the singly periodic Scherk surface on the right. The extrinsic annuli are constructed by cutting the surfaces with two spheres (with the same center but of different radii) in the ambient manifold (R3). The catenoid has two ends and finite total curvature. Hence, by theorem 1.4, the capacity of the extrinsic annulus of the catenoid is greater than the capacity of the corresponding annulus of the Euclidean 2-plane but is smaller than two times that capacity. The same is true for the extrinsic annulus of the singly periodic Scherk surface (we refer the reader to the introduction of [MW07] for the area growth of the singly periodic Scherk surface). In [GP12, Che95] are also obtained lower bounds for the number of ends, but we note that those lower bounds seem to need stronger assumptions: Dimension greater than 2, or embeddedness of the ends and codimension 1, decay on the second fundamental form, and a rotationally symmetric ambient manifold. As a counterpart, those lower bounds are associated to the so-called gap type theorems. Combining the results of [GP12, Theorem 3.5] and corollary 1.2, and taking into ac- count the role of sectional curvatures of the model space (see [GP12, Proposition 2.6]) we have m n Corollary 1.3. Let ' : P ! Mw be a minimal and proper immersion into a balanced n from below model space Mw with increasing warping function w satisfying the following conditions: R t m−1 ! 0 w(s) ds lim sup m = Cw < 1 ; t!1 t =m (1.12) there exist R0 such that for any R > R0 8 w00(R) <− w(R) ≤ 0 2 1−(w0(R)) : w(R)2 ≤ 0 n −1 Suppose moreover that m > 2, that the center ow of Mw satisfies ' (ow) 6= ; and that the norm of the second fundamental form kBP k of the immersion is bounded for large r by (r) (1.13) kBP k ≤ ; w0(r)w(r) where is a positive function such that (r) ! 0 when r ! 1. Then the number of ends is bounded from below and from above by m (1.14) Volw(P ) ≤ E(P ) ≤ 2 Cw Volw(P ) : By using our results about the behavior of the comparison quotients we can also estimate N N the capacity of an extrinsic annulus Aρ,R = P \ BR (o) n Bρ (o) (see figure 1 and, x2 and x3 for a precise definition of capacity and extrinsic annulus): 4 V. GIMENO AND S. MARKVORSEN Theorem 1.4. Let ' : P m ! Rn denote a complete and proper minimal immersion into the Euclidean space Rn. Then, for any R > ρ > 0, the capacity of the extrinsic annulus Aρ,R is bounded by Vol(Dρ) Cap(Aρ,R) Vol(DR) (1.15) ≤ m ≤ ; m R m Vmρ Cap(Aρ,R) VmR m m R R m where Cap(Aρ,R) is the capacity of the geodesic annulus Aρ,R in R of inner radius ρ and outer radius R. Vol(Ds) Remark b. Since, from Theorem 2.1, the quotient m is a non-decreasing function on Vms s, we can state Theorem 1.4 in the limit case ( ρ ! 0 and R ! 1) and inequality (1.15) there becomes Cap(Aρ,R) Vol(DR) (1.16) 1 ≤ m ≤ lim = Vw (P ) : R R!1 m 0 Cap(Aρ,R) VmR When we deal with a minimal surface Σ ⊂ R3 which is properly embedded into the Euclidean space R3 the limit 3 R Vol(Σ \ BR (o)) (1.17) Vw (Σ) = lim 0 R!1 πR2 is well understood.
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