Counterexamples to Calabi conjectures on minimal hypersurfaces ...

<strong>Counterexamples</strong> <strong>to</strong> <strong>Calabi</strong> <strong>conjectures</strong> on minimalhypersurfaces cannot be properL. J. Alías ∗ G. Pacelli Bessa † M. Dajczer ‡AbstractWe give an estimate of the mean curvature of a complete submanifold lyinginside a closed cylinder B(r)×R l in a product Riemannian manifold N n−l ×R l .It follows that a complete hypersurface of given constant mean curvaturelying inside a closed circular cylinder in Euclidean space cannot be properif the circular base is of sufficiently small radius. In particular, any possiblecounterexample <strong>to</strong> <strong>Calabi</strong>’s conjecture on complete minimal hypersurfacescannot be proper. As another application of our method, we derive a resultabout the s<strong>to</strong>chastic incompleteness of submanifolds with sufficiently smallmean curvature.1 IntroductionThe <strong>Calabi</strong> problem in its original form, presented by <strong>Calabi</strong> [3] and promotedby Chern [4] about the same time, consisted on two <strong>conjectures</strong> about Euclideanminimal hypersurfaces. The first conjecture is that any complete minimal hypersurfaceof R n must be unbounded. The second and more ambitious conjecture assertedthat any complete non-flat minimal hypersurface in R n has unbounded projectionsin every (n − 2)-dimensional flat subspace.Both <strong>conjectures</strong> turned out <strong>to</strong> be false for immersed surfaces in R 3 . First Jorgeand Xavier [9] exhibit a non-flat complete minimal surface lying between two parallelplanes. Later on Nadirashvilli [12] constructed a complete minimal surface inside around ball in R 3 .∗ Partially supported by MEC projects MTM2007-64504 and PCI2006-A7-0532, and FundaciónSéneca project 04540/GERM/06, Spain. This research is a result of the activity developed withinthe framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain,by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science andTechnology 2007-2010).† Partially supported by CNPq (Brazil), MEC project PCI2006-A7-0532(Spain), and The AbdusSalam Int. Centre for Theoretical Physics-ICTP.‡ Partially supported by CNPq and Faperj (Brazil), and MEC project PCI2006-A7-0532 (Spain)1

It was recently shown by Colding and Minicozzi [5] that both <strong>conjectures</strong> holdfor embedded minimal surfaces. Their work involves the close relation betweenthe <strong>Calabi</strong> <strong>conjectures</strong> and properness. Recall that an immersed submanifold inEuclidean space is proper if the pre-image of any compact subset of R n is compact.It is a consequence of their general result that a complete embedded minimal diskin R 3 must be proper.The immersed counterexamples <strong>to</strong> <strong>Calabi</strong>’s <strong>conjectures</strong> discussed above are notproper. The example of Nadirashvilli cannot be proper since from the definition aproper submanifold must be unbounded. The same conclusion hold for the otherexample but now the argument is not so easy, one has <strong>to</strong> use the strong half-spacetheorem due <strong>to</strong> Hoffman and Meeks [7].The strong half-space theorem does not hold in R n for n ≥ 4. In fact, the higherdimensional catenoids are between parallel hyperplanes. Hence, it is natural <strong>to</strong> askif any possible higher dimensional counterexample <strong>to</strong> <strong>Calabi</strong>’s second conjecturemust be non-proper. In the special case of minimal immersion, it follows from thecorollary of our main result that a complete hypersurface of R n , n ≥ 3, with boundedprojection in a two dimensional flat subspace cannot be proper (see Corollary 2.2below).As an application of our method, we generalize the results by Markvorsen [10] andBessa and Montenegro [2] about s<strong>to</strong>chastic incompleteness of minimal submanifolds<strong>to</strong> submanifolds of bounded mean curvature. In this respect, let us recall thata Riemannian manifold M is said <strong>to</strong> be s<strong>to</strong>chastically complete if for some (andtherefore, for any) (x, t) ∈ M × (0, +∞) it holds that∫p(x, y, t)dy = 1,Mwhere p(x, y, t) is the heat kernel of the Laplacian opera<strong>to</strong>r. Otherwise, the manifoldM is said <strong>to</strong> be s<strong>to</strong>chastically incomplete (for further details about this see, forinstance, [6] or [14]).An interesting problem in submanifold geometry is <strong>to</strong> understand s<strong>to</strong>chasticcompleteness/incompleteness of submanifolds in terms of their extrinsic geometry.In [10] Markvorsen derived a mean time exit comparison theorem which impliesthat any bounded complete minimal submanifold of a Hadamard manifold N withsectional curvature K N ≤ b ≤ 0 is s<strong>to</strong>chastically incomplete. Recently, Bessa andMontenegro [2] considered minimal submanifolds of product spaces N × R, whereN is a Hadamard manifold with K N ≤ b ≤ 0, and proved a version of Markvorsen’sresult in this setting. In particular, they showed that complete cylindrically boundedminimal submanifolds of N ×R are s<strong>to</strong>chastically incomplete. Here we extend theseresults <strong>to</strong> complete submanifolds with sufficiently small mean curvature lying insidea closed cylinder B(r) × R l in a product Riemannian manifold N n−l × R l .2

2 The resultsPart (a) of Theorem 2.1 below extends the main results given in [1] for compacthypersurfaces. Part (b) generalizes s<strong>to</strong>chastic incompleteness results of [2] and [10]for minimal submanifolds.In the following we denote⎧ √ √ √⎨ b cot( bt) if b > 0, t < π/2 b,C b (t) = 1/t if b = 0,⎩ √ √−b coth( −b t) if b < 0.Theorem 2.1 Let ϕ: M m → N n−l × R l be an isometric immersion of a completeRiemannian manifold M of dimension m ≥ l + 1. Let B N (r) be the geodesic ballof N n−l centered at p with radius r. Given q ∈ M, assume that the radial sectionalcurvature K radNbounded as K radNalong the radial geodesics issuing from p = π N(ϕ(q)) ∈ N n−l is≤ b in B N(r). Suppose thatϕ(M) ⊂ B N (r) × R lfor r < min{inj N (p), π/2 √ b}, where we replace π/2 √ b by +∞ if b ≤ 0.(a) If ϕ: M m → N n−l × R l is proper, then(b) Ifthen M is s<strong>to</strong>chastically incomplete.(m − l)sup |H| ≥M m C b(r). (1)(m − l)sup |H|

3 The proofsLet ϕ: M m → N n be an isometric immersion between Riemannian manifolds. Givena function g ∈ C ∞ (N) we set f = g ◦ ϕ ∈ C ∞ (M). Sincefor every vec<strong>to</strong>r field X ∈ TM, we obtain〈grad M f, X〉 = 〈grad N g, X〉grad N g = grad M f + (grad N g) ⊥according <strong>to</strong> the decomposition TN = TM ⊕T ⊥ M. An easy computation using theGauss formula gives the well-known relation (see e.g. [8])Hess M f(X, Y ) = Hess N g(X, Y ) + 〈grad N g, α(X, Y )〉 (3)for all vec<strong>to</strong>r fields X, Y ∈ TM, where α stands for the second fundamental form ofϕ. In particular, taking traces with respect <strong>to</strong> an orthonormal frame {e 1 , . . .,e m }in TM yieldsm∑∆ M f = Hess N g(e i , e i ) + 〈grad N g, H〉. → (4)where → H= ∑ mi=1 α(e i, e i ).i=1The first main ingredient of our proofs is the Hessian comparison theorem.Theorem 3.1 Let M m be a Riemannian manifold and x 0 , x 1 ∈ M be such that thereis a minimizing unit speed geodesic γ joining x 0 and x 1 and let ρ(x) = dist(x 0 , x)be the distance function <strong>to</strong> x 0 . Let K γ ≤ b be the radial sectional curvatures of Malong γ. If b > 0 assume ρ(x 1 ) < π/2 √ b. Then, we have Hess ρ(x)(γ ′ , γ ′ ) = 0 andwhere X ∈ T x M is perpendicular <strong>to</strong> γ ′ (ρ(x)).Hess ρ(x)(X, X) ≥ C b (ρ(x))‖X‖ 2 (5)The second main ingredient is the version proved by Pigola-Rigoli-Setti [14,Theorem 1.9] of the Omori-Yau maximum principle.Theorem 3.2 Let M m be a Riemannian manifold and assume that there exists anon-negative C 2 -function ψ satisfying the following requirements:ψ(x) → +∞as x → ∞∃ A > 0 such that |gradψ| ≤ A √ ψ off a compact set√∃ B > 0 such that ∆ψ ≤ B ψG( √ ψ) off a compact set4

where G is a smooth function on [0, +∞) satisfying:(i) G(0) > 0,(ii) G ′ (t) ≥ 0 on [0, +∞),(iii) 1/ √ G(t) ∉ L 1 (0, +∞), (iv) lim sup t→+∞tG( √ t)G(t)< +∞.(6)Then, given a function u ∈ C 2 (M) with u ∗ = sup M u < +∞ there exists a sequence{x k } k∈N ⊂ M m such thatu(x k ) > u ∗ − 1/k; |gradu|(x k ) < 1/k; ∆u(x k ) < 1/k.Observe that a function G satisfying the above conditions isNow we are ready <strong>to</strong> prove Theorem 2.1.G(t) = (t + 2) 2 (log(t + 2)) 2 . (7)Proof of Theorem 2.1: Define σ : N n−l × R l → [0, +∞) byσ(z, y) = ρ R l(y),where ρ R l(y) = ‖y‖ R l is the distance function <strong>to</strong> the origin in R l . Since ϕ is properand ϕ(M) ⊂ B N (r) × R l , then the function ψ(x) = σ ◦ ϕ(x) satisfies ψ(x) → ∞ asρ M (x) = dist M (q, x) → +∞. Off a compact set, we now have|grad M ψ(x)| ≤ |grad N×Rl σ(ϕ(x))| = |grad Rl ρ R l| = 1 ≤ √ ψ(x).To compute ∆ M ψ we start with bases {∂/∂ρ N , ∂/∂θ 2 , . . .,∂/∂θ n−l } of TN and{∂/∂ρ R l, ∂/∂γ 2 , . . .,∂/∂γ l } of TR l (polar coordinates) orthonormal at x ∈ M.Then, we choose an orthonormal basis {e 1 , . . ., e m } for T x M as follows∂ ∑n−l∂ ∂e i = α i + a ij + β i∂ρ N ∂θ j ∂ρ R lj=2+l∑t=2b it∂∂γ t·Hence, we haveHess N×R l σ(ϕ(x))(e i , e i ) = Hess R l ρ R l(π R le i , π R le i ) =1σ(ϕ(x))where π R l denotes the orthogonal projection on<strong>to</strong> TR l . Here, we are usingn−l|e i | = 1 = αi 2 + ∑l∑a 2 ij + β2 i +j=25t=2b 2 itl∑t=2b 2 it ≤ 1ψ(x) ,

that yields ∑ lt=2 b2 it ≤ 1.Since ψ(x) → ∞ as ρ M (x) = dist M (q, x) → +∞, off a compact set we mayassume that√| H|(x) → = m|H|(x) ≤ ψ(x)G( √ ψ(x))where G(t) is given by (7). Otherwise, sup M |H| = +∞ and there is nothing <strong>to</strong>prove. Besides, off a compact set we also have that√1ψ(x) ≤ ψ(x)G( √ ψ(x)).Hence, from (4) we have off a compact set that∆ M ψ(x) =m∑Hess N×R l σ(ϕ(x))(e i , e i ) + 〈grad N×Rl σ(ϕ(x)), H → (x)〉i=1≤mψ(x) + m|H|(x)√≤ (m + 1) ψ(x)G( √ ψ(x)).Therefore, by Theorem 3.2 the Omori-Yau maximum principle holds on M.Define ρ: N n−l × R l → R byand u: M m → R byρ(z, y) = ρ N (z) = dist N (p, z)u(x) = ρ ◦ ϕ(x).Since ϕ(M) ⊂ B N (r) × R l , we have that u ∗ = sup M u ≤ r < ∞, Therefore, by themaximum principle there is a sequence {x k } k∈N ⊂ M m such thatHence, we have1k > ∆u(x k) =u(x k ) > u ∗ − 1/k; |gradu|(x k ) < 1/k; ∆u(x k ) < 1/k.m∑Hess N×R lρ(ϕ(x k ))(e i , e i ) + 〈grad N×Rl ρ(ϕ(x k )), H → (x k )〉 (8)i=1where {e 1 , . . .,e m } is an orthonormal basis for T xk M. Start with an orthonormalbasis {∂/∂ρ N , ∂/∂θ 2 , . . .,∂/∂θ n−l } for TN and standard coordinates {y 1 , . . .y l } forR l . Then, choose an orthonormal basis for T xk M as follows∂ ∑n−l∂e i = α i + a ij +∂ρ N ∂θ jj=26l∑t=1c it∂∂y t·

Using Theorem 3.1, a straightforward computation yieldssinceHess N×R lρ(ϕ(x k ))(e i , e i ) = Hess N ρ N (z(x k ))(π TN e i , π TN e i )=≥=∑n−la 2 ijHess N ρ N (z(x k ))(∂/∂θ j , ∂/∂θ j )j=2∑n−la 2 ijC b (r) (9)j=2(1 − α 2 i −l∑t=1c 2 it)n−l|e i | = 1 = αi 2 + ∑ l∑a 2 ij + c 2 it ,j=2t=1C b (r)where π TN denotes the orthogonal projection on<strong>to</strong> TN. Therefore,m∑Hess N×R lρ(ϕ(x k ))(e i , e i ) ≥i=1At x k , we haveand hence(m − ∑ iα 2 i − ∑ i,tgrad N×Rl ρ(ϕ(x k )) = gradu(x k ) + (grad N×Rl ρ(ϕ(x k ))) ⊥|gradu| 2 (x k ) =m∑i=1〈 ∂∂ρ N, e i 〉 = ∑ ic 2 it)C b (r). (10)α 2 i < 1/k2 . (11)Taking in<strong>to</strong> account |grad N×Rl ρ| = |grad N ρ N | = 1, from (8) and (10) we obtain1(k > m − ∑ iα 2 i − ∑ i,tc 2 it)C b (r) − m sup |H|.MIt follows using (11) that1k + C b(r)k 2(+ m sup |H| ≥ m − ∑Mi,tc 2 it)C b (r). (12)Observe now that∑c 2 it =i,tl∑ m∑c 2 it =t=1 i=1l∑|grad(y t ◦ ϕ)| 2 ≤ l,t=17

since |grad(y t ◦ ϕ)| 2 ≤ |grad Rl y t | 2 = 1. Thus,( )m − ∑ i,tc 2 it≥ (m − l)and we have letting k → +∞ in (12) thatm sup |H| ≥ (m − l)C b (r).MThis concludes the proof of the first part of Theorem 2.1.For the proof of the second part, we make use of the following characterizationof s<strong>to</strong>chastic completeness given in [13] (see [14, Theorem 3.1]): A Riemannianmanifold M is s<strong>to</strong>chastically complete if and only if for every u ∈ C 2 (M) withu ∗ = sup u < ∞ there exists a sequence {x k } such that u(x k ) > u ∗ − 1/k and∆u(x k ) < 1/k for every k ≥ 1.whereSuppose that M is s<strong>to</strong>chastically complete. Define g: N n−l × R l → R byg(z, y) = ĝ(z) = φ b (ρ N (z))⎧⎨ 1 − cos( √ bt) if b > 0, t < π/2 √ b,φ b (t) = t 2 if b = 0,⎩cosh( √ −bt) if b < 0.Then f = g ◦ ϕ is a smooth bounded function on M. Thus there exists a sequenceof points {x k } in M such thatf(x k ) > f ∗ − 1/k and ∆f(x k ) < 1/kfor k ≥ 1, where f ∗ = sup M f ≤ φ b (r) < ∞. Similar as before, we haveHess N×R lg(ϕ(x k ))(e i , e i ) = Hess N ĝ(z(x k ))(π TN e i , π TN e i )∑n−l= φ ′′b(r k )αi 2 + φ ′ b(r k ) a 2 ijHess N ρ N (z(x k ))(∂/∂θ j , ∂/∂θ j )j=2∑n−l≥ φ ′′b(r k )αi 2 + φ ′ b(r k )C b (r k )= φ ′′b(r k )α 2 i + φ ′ b(r k )C b (r k )= φ ′ b (r k)C b (r k )(81 −l∑t=1c 2 itj=2()a 2 ij1 − α 2 i −l∑t=1c 2 it)

m − ∑ i,tsince φ ′′b (t) − φ′ b (t)C b(t) = 0. Here, we are writing r k = ρ N (z(x k )). Therefore,1k > ∆f(x k) =m∑Hess N×R lg(e i , e i ) + 〈grad N×Rl g, H〉→i=1( )≥ φ ′ b (r k)C b (r k )c 2 it+ φ ′ b (r k)〈grad N×Rl ρ N , → H〉≥ φ ′ b (r k) ((m − l)C b (r k ) − m sup |H|).Finally, since lim k→∞ φ ′ b (r k) > 0, letting k → ∞ we havesup |H| ≥(m − l)mC b(r).Proof of Corollary 2.2: If ϕ is proper in R n+1 , from part (a) of Theorem 2.1 wewould have |H| ≥ 1/nr, and that is a contradiction.References[1] G. P. Bessa and J. F. Montenegro, On compact H-Hypersurfaces of N × R.Geom. Dedicata. 127 (2007), 1–5.[2] G. P. Bessa J. Fabio Montenegro, Mean time exit and isoperimetric inequalitiesfor minimal submanifolds of N × R. To appear in Bull. London Math.Soc. Available at http://arxiv.org/pdf/0709.1331[3] E. <strong>Calabi</strong>, Problems in Differential Geometry (S. Kobayashi and J. Eells, Jr.,eds.) Proc. of the United States-Japan Seminar in Differential Geometry,Kyo<strong>to</strong>, Japan, 1965, Nippon Hyoronsha Co. Ltd., Tokyo (1966) 170.[4] S. S. Chern, The Geometry of G-structures. Bull. Amer. Math. Soc. 72(1966), 167–219.[5] T. Colding and W. Minicozzi II, The <strong>Calabi</strong>-Yau <strong>conjectures</strong> for embeddedsurfaces. Annals of Math. 161 (2005) 727–758.[6] A. Grigor’yan, Analytic and geometric background of recurrence and nonexplosionof the Brownian motion on Riemannian manifolds, Bull. Amer.Math. Soc. (N.S.) 36 (1999), 135–249.[7] D. Hoffman and W. Meeks, The Strong Half-space Theorem for minimalsurfaces. Invent. Math. 101 (1990) 373–377.[8] L. Jorge and D. Koutrofiotis, An estimate for the curvature of bounded submanifolds.Amer. J. Math. 103 (1980) 711–725.9

[9] L. Jorge and F. Xavier, A complete minimal surface in R 3 between two parallelplanes. Ann. of Math. 112 (1980) 203–206.[10] S. Markvorsen, On the mean exit time from a submanifol. J. DifferentialGeom. 29 (1989), 1-8.[11] F. Martín and S. Morales, A complete bounded minimal cylinder in R 3 .Michigan Math. J. 47 (2000), 499–514.[12] N. Nadirashvili, Hadamard’s and <strong>Calabi</strong>-Yau’s <strong>conjectures</strong> on negativelycurved and minimal surfaces. Invent. Math. 126 (1996), 457–465.[13] S. Pigola, M. Rigoli and A. Setti, A remark on the maximum principle ands<strong>to</strong>chastic completeness. Proc. Amer. Math. Soc. 131 (2003), 1283–1288.[14] S. Pigola, M. Rigoli and A. Setti, Maximum Principle on Riemannian Manifoldsans Applications. Memoirs Amer. Math. Soc. 822 (2005).Luis J. AliasDepartamen<strong>to</strong> de MatematicasUniversidad de MurciaCampus de Espinardo E-30100 – Spainljalias@um.esGregorio Pacelli BessaUFC - Departamen<strong>to</strong> de MatematicaBloco 914 – Campus do Pici60455-760 – Fortaleza – Ceara – Brazilbessa@mat.ufc.brMarcos DajczerIMPAEstrada Dona Cas<strong>to</strong>rina, 11022460-320 – Rio de Janeiro – Brazilmarcos@impa.br10