The isoperimetric problem in complete annuli of revolution with increasing Gauss curvature

The isoperimetric problem in complete annuli of revolution with increasing Gauss curvature Antonio Ca˜nete Departamento de Matem´aticas, Universidad de Le´on, 24071 Le´on, Spain ([email protected]) Manuel Ritore´ Departamento de Geometr´ıa y Topolog´ıa, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain ([email protected]) In this work we describe the isoperimetric regions in complete symmetric annuli of revolution with Gauss curvature non-decreasing from the shortest parallel. This description allows us to complete the classification of isoperimetric regions in quadrics of revolution. 1. Introduction It is well known that least-perimeter sets of given area in the plane, in hyperbolic planes and in round spheres are geodesic discs [3]. However, even in very simple surfaces, this isoperimetric problem has remained open. In 1996, Benjamini and Cao [2] proved that the least-perimeter way to enclose a given area in a paraboloid of revolution is by means of a circle of revolution. This result was recovered by different methods by Pansu [9], Topping [12], Morgan et al. [8] and Ritor´e [10]. In these works, the isoperimetric regions were classified for some new types of surfaces. Benjamini and Cao [2] solved the problem for complete planes of revolution with non-increasing curvature from the origin which are convex at infinity. Morgan et al. [8, §4.3] removed this convexity assumption, and characterized the isoperimetric regions in real projective planes of revolution with non-increasing Gauss curvature from the origin. In [10], amongst other results, the isoperimetric problem was solved for spheres of revolution with an equatorial symmetry and Gauss curvature either non-increasing or non-decreasing from the equator to the poles. An approach to the classification of isoperimetric regions in tori of revolution has been given by Ca˜nete [4], who has classified the stable regions in such surfaces. Even in this simple class of examples, the geometry of the perimeter-minimizing regions of given area can be quite complex. In some planes [8] and spheres [10] of revolution, these regions can be either discs or annuli, and in annuli of revolution with decreasing curvature from one end of finite area the isoperimetric regions are bounded by a single circle of revolution [8, 10]. On the other hand, in tori of revolution, the boundary of a stable region can be composed of curves of constant geodesic curvature which are not circles of revolution [4]. Moreover, in non-compact surfaces, isoperimetric regions may not exist. In general, a minimizing sequence of sets of a given area whose perimeters converge to the infimum of perimeters for this area may have a convergent part of smaller area and a diverging part of positive area, so that in the limit we obtain an isoperimetric region of smaller area, and possibly some minimizing object at infinity. This paper is devoted to the classification of isoperimetric regions in a complete annulus of revolution with an equatorial symmetry and Gauss curvature which is non-decreasing from this equator. Examples of such annuli are minimal catenoids and one-sheeted hyperboloids. We shall use techniques from the calculus of variations to treat this problem by classifying the embedded curves with constant geodesic curvature which can be part of the boundary of an isoperimetric region. Our results allow us to complete the classification of isoperimetric regions in quadrics of revolution. In the resolution of this problem we shall find all the difficulties mentioned before: non-existence of isoperimetric regions, the break of a minimizing sequence into two parts and the existence of isoperimetric regions of several different types. We prove in our main result, theorem 3.9, that in a complete symmetric annulus of revolution with non-decreasing Gauss curvature from the shortest parallel, the isoperimetric regions may be (i) a ‘disc at infinity’, (ii) a symmetric annulus, (iii) an asymmetric annulus, or (iv) an annulus bounded by an unduloid-type curve and a circle of revolution. All possibilities occur in different annuli, as shown in example 3.8, where we exhibit isoperimetric regions of the last type, and in §4. We have organized the remainder of the paper into three sections. In §2we establish notation and give preliminary results. In §3, we shall prove our main results, mainly that a minimizing sequence cannot be broken into two pieces, and that there exist isoperimetric regions in these annuli which are not of revolution. Finally, in §4, we apply the previous results to the classification of isoperimetric regions in the one-sheeted hyperboloid. This allows us to conclude, in corollary 4.4, the classification for quadrics of revolution (invariant by a one-parameter group of rotations around a line). 2. Preliminaries 2.1. Annuli of revolution with non-decreasing curvature We shall denote by Mthe product S1×Rendowed with a complete warped metric ds2:= f(t)2dθ2+dt2, for t∈R,θ∈S1and f:R→Ra smooth positive function. The Gauss curvature depends only on the t-coordinate, and is given by K(t):=−f(t) f(t).(2.1) Moreover, the length and the geodesic curvature of the parallels S1×{t}, are given by L(t):=2πf(t),h(t):=f(t) f(t).(2.2) We shall suppose that the annulus Mis symmetric with respect to the parallel S1×{0}, which is equivalent to the symmetry f(t)=f(−t), and that the Gauss curvature Kis a non-decreasing function of the distance tfrom S1×{0}. We note that the monotonicity of K(t) is equivalent to that of the function (4π2)((f)2−ff)(t)=L2(K+h2)(t). From the monotonicity of K, since there are no complete ends with positive curvature, it follows that K⩽0. Moreover, if Kvanishes at some point t0, then K≡0 in [t0,+∞), and so our annulus is flat near infinity. Since Kis non-decreasing and non-positive, we shall define K∞:= lim t→∞ K(t)⩽0. As K⩽0, it follows that f ⩾0, and so fis non-decreasing. Since f(0)=0by the symmetry of f, we deduce that f⩾0 for t⩾0 (and non-positive for t⩽0). Then fis non-decreasing for t⩾0 and, consequently, Mis complete and S1×{0}is a shortest parallel of M. We will refer to it as the shortest geodesic loop (although it is not necessarily unique). Given some Ω⊂M, we shall denote the Riemannian area of Mby A(M). If Cis a rectifiable curve, the length of Cwill be denoted by L(C). If Ωis a finite perimeter set in M, then its perimeter will be denoted by P(Ω). We shall consider the isoperimetric problem of minimizing perimeter under an area constraint in these symmetric annuli of revolution with non-decreasing Gauss curvature from the shortest geodesic loop. 2.2. Isoperimetric regions For a surface M, given a∈(0,A(M)), we consider the isoperimetric profile of M, defined by I(a) = inf{L(∂B):B⊂M, smooth with A(B)=a}. An isoperimetric region Ω⊂Mis a finite perimeter set such that P(Ω)=I(A(Ω)). The regularity results by Morgan [7] imply that an isoperimetric region has smooth boundary. The existence of an isoperimetric region for a given area a>0isnot guaranteed in a non-compact surface, since a minimizing sequence {Ωn}n∈Nof sets of area a, and satisfying lim n→∞ L(∂Ωn)=I(a), may lose all or part of its area at infinity. However, we have the following result. Lemma 2.1 (Ritor´e [10, lemma 1.8]).Let Mbe a Riemannian surface, A>0, and let {Ωn}nbe a minimizing sequence for area A. Then Ωncan be decomposed as Ωn = Ωcn ∪ Ωnd, where (i) Ωc nconverges to a set Ω⊂M, with A(Ω)∈[0,A], (ii) Ωd ndiverges, (iii) if Lc= lim L(∂Ωc n)and Ld= lim L(∂Ωd n), then Lc+Ld=I(A), and (iv) Ωis an isoperimetric region of area A(Ω). From this result, we may conclude that if the loss of area at infinity Ad= limn→∞ area(Ωd n) is zero, then Ωis an isoperimetric region for area A. 2.3. Constant geodesic curvature curves Classical variational formulae for length and area [11], together with the regularity results for isoperimetric regions in surfaces, imply that the boundary of an isoperimetric region has constant geodesic curvature with respect to the inner normal. Constant geodesic curvature curves in rotationally symmetric surfaces can be classified due to the existence of a first integral coming from the one-parameter group of isometries. From [4,10] we have the following result. Theorem 2.2.Let Mbe a symmetric annulus of revolution with non-decreasing Gauss curvature from the shortest geodesic loop. Let Cbe a curve with constant geodesic curvature and finite length. Then Cis a parallel, a nodoid-type curve or an unduloid-type curve. In nodoid-type curves, the tangent vector turns monotonically, presenting points with vertical tangent vector. When these curves are closed, they bound discs in the surface. On the other hand, unduloid-type curves are periodic graphs over θ, symmetric with respect to every critical point of its t-coordinate. We define the period of the latter as the θ-distance between two consecutive maxima (or minima) points of the t-coordinate (see figure 1). The following lemma gives necessary and sufficient conditions for these curves to be closed and embedded. Lemma 2.3 (Ritor´e [10, proposition 1.3]).Let Cbe a curve with constant geodesic curvature in a warped product S1×I. (i) If Cis a nodoid-type curve, it yields a closed embedded curve if and only if the maximum and the minimum of t|Care in the same vertical line. (ii) If Cis an unduloid-type curve, it yields a closed embedded curve if and only if the period of Cis equal to 2π/k, with k∈N. 2.4. Stability We shall say that a smooth curve Cof finite length and constant geodesic curvature is stable if it is actually a local minimum of the perimeter for variations t θ Figure 1. Nodoid-type and unduloid-type curves in the product S1×I. preserving the area. For such variations, the second derivative of length is given [1] by I(u)=−C ud2u ds2+(K+h2)uds, (2.3) where u:C→Ris the normal component of the vector field associated to the variation and sis the arc-length parameter in C. We find that Cis stable if and only if I(u)⩾0 for any function usuch that C uds=0. We will say that a Ω⊂Mis a stable region if ∂Ω is an embedded stable curve with constant geodesic curvature with respect to the inner normal. We remark that any isoperimetric region is stable. Related to (2.3), we consider the Jacobi operator defined by J(u)=d2u ds2+(K+h2)u, (2.4) and, on every connected component C⊂C, the corresponding eigenvalue problem J(u)+λu =0, for C2functions u:C→R. The eigenvalues associated to the Jacobi operator will be of interest throughout this paper. We note that a stable curve cannot have more than one connected component with a negative first eigenvalue. We recall some information about these eigenvalues in the following lemma (see [4] for details). Lemma 2.4.Let C⊂Mbe a connected curve with constant geodesic curvature, and consider λ1(C)the first eigenvalue associated to the Jacobi operator (2.4) in C. (i) If Cis a nodoid-type or an unduloid-type curve, then λ1(C)<0. (ii) If Cis the parallel S1×{t}, then λ1(C)=−(K+h2)(t). The following results discuss the stability of the curves described in theorem 2.2, and of the horizontal annuli bounded by two parallels. Lemma 2.5 (Ritor´e [10, lemma 1.6]).A parallel S1×{t}in Mis stable if and only if L2(K+h2)(t)⩽4π2,(2.5) or, equivalently, if ((f)2−ff)(t)⩽1. Lemma 2.6 (Ritor´e [10, lemma 1.7]).A horizontal annulus S1×[t1,t 2]⊂Mis stable if and only if the parallels S1×{ti}are stable (i=1,2) and L−1(K+h2)(t1)+L−1(K+h2)(t2)⩽0.(2.6) Moreover, in the case of a symmetric annulus S1×[−t, t], condition (2.6) reduces to (K+h2)(t)⩽0.(2.7) Lemma 2.7.Let C⊂Mbe a closed embedded nodoid-type curve, not contained in a region with constant Gauss curvature. Then Cis unstable. Proof. By [10, lemma 2.3], the only closed embedded nodoids will necessarily intersect the parallel S1×{0}. But, from [10, lemma 3.4], we conclude that such nodoids are unstable, since S1×{0}is the parallel where Kachieves its minimum. Remark 2.8.We recall that closed embedded nodoids contained in regions with constant Gauss curvature are stable, by the classical isoperimetric inequalities. Applying the same reasoning as in [4, §2], it can be proved that there exist closed and embedded unduloid-type curves (and even stable ones) in some of our annuli. Moreover, we shall see in §3 that these curves actually appear as the boundaries of isoperimetric regions. Let us recall a necessary condition for the stability of unduloid-type curves. Lemma 2.9 (Ca˜nete [4, lemma 2.2]).Let C⊂Mbe a closed embedded stable unduloid-type curve, not contained in the region where (f)2−ff =1. Then the curve Ctouches the regions where (f)2−ff <1and (f)2−ff >1. Stable symmetric annuli of small area satisfy dh/dA>0. These annuli grow up to reach the parallels where K+h2= 0. At this point we obtain stable asymmetric annuli by letting one of these boundary curves approach and the other move away from the shortest parallel with the same geodesic curvature. In this process we obtain asymmetric annuli with dh/dA<0. The deformation continues until the largest parallel in the boundary of the asymmetric annulus reaches a parallel with L2(K+h2)=4π2, where unduloids appear. This latter phenomenon has been studied in detail in [4]. 3. Main results As in [4], from theorem 2.2 we can classify the stable regions in our surfaces. Theorem 3.1.Let Mbe a symmetric annulus of revolution with non-decreasing Gauss curvature from the shortest parallel. Then the stable regions in Mmay be (i) discs bounded by a nodoid-type curve, contained in a region with constant Gauss curvature, (ii) horizontal annuli symmetric with respect to the shortest parallel, and bounded by two parallels contained in the region K+h2⩽0, (iii) non-symmetric horizontal annuli bounded by two parallels satisfying the stability condition (2.6), and contained in the region L2(K+h2)⩽4π2, (iv) annuli bounded by a stable unduloid-type curve, and a parallel contained in K+h2<0,or (v) unions of a disc of constant Gauss curvature, and a symmetric annulus contained in K+h2<0, with the same geodesic curvature. Proof. Let Ωbe a stable region in M. Since the first eigenvalue of the Jacobi operator (2.4) associated to a nodoid-type or an unduloid-type curve is negative, it follows that ∂Ω will contain at most one of these curves. On the other hand, the geodesic curvature h(t) of parallels has a different sign in each half of the annulus and, consequently, ∂Ω will contain at most one parallel in each half. If ∂Ω does not contain either a nodoid-type curve or an unduloid-type curve, then Ωis a horizontal annulus of type (ii) or (iii) satisfying the conditions described in lemma 2.6. If ∂Ω has a nodoid-type curve, by lemma 2.7 it turns out that Ωis of type (i) or (v). If ∂Ω contains an unduloid-type curve, then Ωmust be of type (iv), satisfying the condition of lemma 2.9. From now on we shall denote by M(K∞) the complete plane with constant Gauss curvature K∞. Theorem 3.2.Let Mbe a symmetric annulus of revolution with non-decreasing Gauss curvature from the shortest geodesic loop. Consider A>0, and a minimizing sequence {Ωn}n∈Nfor area A. If the area Acof the convergent part and the area Adof the divergent part are both positive, then the value of the isoperimetric profile I(A)is given by the sum of the perimeter of a stable symmetric annulus in Mand the perimeter of a disc in M(K∞), both with the same geodesic curvature. Proof. Since Ldis finite, the boundary curves of the sets of the divergent part of the minimizing sequence are homotopically trivial for nlarge enough. Since K⩽K∞, applying the classical isoperimetric inequality to Ωd nand passing to the limit we get L2 d⩾4πAd−K∞A2 d. Consider a disc D⊂M(K∞) of area Ad>0. As K∞⩽0, the injectivity radius of the complement of any compact set in Mis infinite. From the isoperimetric inequality it follows that L(∂D)2=4πAd−K∞A2 d⩽L2 d, where Lddenotes the limit length of the divergent part of the minimizing sequence {Ωn}n.IfL(∂D)<L d, it is easy to get a contradiction from the minimizing character of {Ωn}n, simply by considering a family of geodesic discs in Mof area Ad whose centres diverge. Then L(∂D)=Ld. The above reasoning shows that I(A) is given by the perimeter of the union D∪Ω, where Ωis the limit set of the convergent part of {Ωn}n∈N. The configuration D∪Ω in M(K∞)∪Mcannot be unstable, since otherwise it could be deformed to a least perimeter configuration with the same area, and the deformation of Dcould be approximated by a set in M, thus giving a contradiction. Since the first eigenvalue associated to the Jacobi operator (2.4) in Dis negative, we conclude that Ωmust be a symmetric annulus. We shall see in the following results that the possibility of the previous theorem 3.2 cannot hold. Lemma 3.3.Let Mbe a symmetric annulus of revolution with non-decreasing Gauss curvature from the shortest geodesic loop. If Ris a stable symmetric annulus of area A, perimeter Land geodesic curvature h, then L > hA. Proof. We shall prove the equivalent inequality L2−LhA > 0. If R=S1×[−t, t], t>0, it suffices to check that f(t)2−f(t)t 0 f(s)ds>0.(3.1) Let gbe the derivative with respect to tof the left-hand term of (3.1). We have g(t)=f(t)f(t)+K(t)f(t)t 0 f(s)ds and g(t)=h(t)g(t)+m(t), with m(t)=K(t)f(t)t 0f(s)ds. As g(0) = 0, it follows (see [6, corollary 2.1, p. 48]) that g(t) = exp t 0 h(s)dst 0 exp −s 0 h(u)dum(s)ds⩾0. Therefore, the left-hand term in (3.1) is an increasing function, and the desired inequality holds. Proposition 3.4.Let Mbe a symmetric annulus of revolution with non-decreasing Gauss curvature from the shortest geodesic loop. Let h>0. Then the union of a stable symmetric annulus Rhand a disc Dhin M(K∞)with the same geodesic curvature hhas a larger perimeter than the disc Din M(K∞)of area A(Rh)+ A(Dh). Proof. Let −b2:= K∞and A:= A(Rh)+A(Dh). From the isoperimetric inequality in M(−b2)wehave L(∂D)2=4πA +b2A2. On the other hand, it is easy to check that, for the geodesic disc Dhof geodesic curvature hin M(−b2), we have hL(∂Dh)=2π+b2A(Dh),(3.2) h>b. (3.3) Let us prove that (L(∂Dh)+L(∂Rh))2>4πA +b2A2.(3.4) By applying the isoperimetric inequality in M(−b2)toDh, and lemma 3.3 to Rh, we have (L(∂Dh)+L(∂Rh))2=L(∂Dh)2+L(∂Rh)2+2L(∂Dh)L(∂Rh) >4πA(Dh)+b2A(Dh)2+h2A(Rh)2+2hL(∂Dh)A(Rh). (3.5) From (3.3) it is clear that h2A(Rh)2>b 2A(Rh)2,(3.6) Finally, by using (3.2) and (3.6), we obtain from (3.5) that (L(∂Dh)+L(∂Rh))2>4π(A(Dh)+A(Rh)) + b2(A(Dh)+A(Rh))2, which proves the statement. From proposition 3.4 we obtain two interesting consequences. Corollary 3.5.Consider a minimizing sequence for area A. Then the areas Ad, Acof the divergent and convergent parts cannot be positive simultaneously. Corollary 3.6.The value of the isoperimetric profile for some given area A>0 cannot be achieved by the sum of the perimeters of a stable symmetric annulus and a disc in M(K∞). In other words, the union of a stable symmetric annulus and a disc in M(K∞) with the same geodesic curvature ‘cannot be an isoperimetric region’ in M. Example 3.7.Let Mbe a minimal catenoid defined by x2+y2=λ2cosh2z λ,λ>0. This surface is included in our family of annuli, and the corresponding warped function is f(t)=(t2+λ2)1/2,t∈R. As (f)2−ff <1, there are no stable closed unduloid-type curves in M.Bya comparison argument, it follows that the discs in M(K∞)=M(0) have a smaller perimeter than horizontal annuli, and so the isoperimetric profile of the catenoids is given by the planar isoperimetric inequality I(a)=(4πa)1/2. The validity of the planar isoperimetric inequality in minimal surfaces is an extremely interesting subject (see [5]). We now give an example of a symmetric annulus of revolution with non-decreasing Gauss curvature from the shortest parallel, where the largest area stable asymmetric annulus has a smaller perimeter than a disc in M(K∞). Then, by the continuity of the isoperimetric profile, it turns out that annuli bounded by an unduloid-type curve and a parallel are also isoperimetric regions.