It was recently shown by Colding and Minicozzi [5] that both <str<strong>on</strong>g>c<strong>on</strong>jectures</str<strong>on</strong>g> holdfor embedded <strong>minimal</strong> surfaces. Their work involves the close relati<strong>on</strong> betweenthe <str<strong>on</strong>g>Calabi</str<strong>on</strong>g> <str<strong>on</strong>g>c<strong>on</strong>jectures</str<strong>on</strong>g> 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 c<strong>on</strong>sequence of their general result that a complete embedded <strong>minimal</strong> diskin R 3 must be proper.The immersed counterexamples <str<strong>on</strong>g>to</str<strong>on</strong>g> <str<strong>on</strong>g>Calabi</str<strong>on</strong>g>’s <str<strong>on</strong>g>c<strong>on</strong>jectures</str<strong>on</strong>g> discussed above are notproper. The example of Nadirashvilli cannot be proper since from the definiti<strong>on</strong> aproper submanifold must be unbounded. The same c<strong>on</strong>clusi<strong>on</strong> hold for the otherexample but now the argument is not so easy, <strong>on</strong>e has <str<strong>on</strong>g>to</str<strong>on</strong>g> use the str<strong>on</strong>g half-spacetheorem due <str<strong>on</strong>g>to</str<strong>on</strong>g> Hoffman and Meeks [7].The str<strong>on</strong>g half-space theorem does not hold in R n for n ≥ 4. In fact, the higherdimensi<strong>on</strong>al catenoids are between parallel hyperplanes. Hence, it is natural <str<strong>on</strong>g>to</str<strong>on</strong>g> askif any possible higher dimensi<strong>on</strong>al counterexample <str<strong>on</strong>g>to</str<strong>on</strong>g> <str<strong>on</strong>g>Calabi</str<strong>on</strong>g>’s sec<strong>on</strong>d c<strong>on</strong>jecturemust be n<strong>on</strong>-proper. In the special case of <strong>minimal</strong> immersi<strong>on</strong>, it follows from thecorollary of our main result that a complete hypersurface of R n , n ≥ 3, with boundedprojecti<strong>on</strong> in a two dimensi<strong>on</strong>al flat subspace cannot be proper (see Corollary 2.2below).As an applicati<strong>on</strong> of our method, we generalize the results by Markvorsen [10] andBessa and M<strong>on</strong>tenegro [2] about s<str<strong>on</strong>g>to</str<strong>on</strong>g>chastic incompleteness of <strong>minimal</strong> submanifolds<str<strong>on</strong>g>to</str<strong>on</strong>g> submanifolds of bounded mean curvature. In this respect, let us recall thata Riemannian manifold M is said <str<strong>on</strong>g>to</str<strong>on</strong>g> be s<str<strong>on</strong>g>to</str<strong>on</strong>g>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<str<strong>on</strong>g>to</str<strong>on</strong>g>r. Otherwise, the manifoldM is said <str<strong>on</strong>g>to</str<strong>on</strong>g> be s<str<strong>on</strong>g>to</str<strong>on</strong>g>chastically incomplete (for further details about this see, forinstance, [6] or [14]).An interesting problem in submanifold geometry is <str<strong>on</strong>g>to</str<strong>on</strong>g> understand s<str<strong>on</strong>g>to</str<strong>on</strong>g>chasticcompleteness/incompleteness of submanifolds in terms of their extrinsic geometry.In [10] Markvorsen derived a mean time exit comparis<strong>on</strong> theorem which impliesthat any bounded complete <strong>minimal</strong> submanifold of a Hadamard manifold N withsecti<strong>on</strong>al curvature K N ≤ b ≤ 0 is s<str<strong>on</strong>g>to</str<strong>on</strong>g>chastically incomplete. Recently, Bessa andM<strong>on</strong>tenegro [2] c<strong>on</strong>sidered <strong>minimal</strong> submanifolds of product spaces N × R, whereN is a Hadamard manifold with K N ≤ b ≤ 0, and proved a versi<strong>on</strong> of Markvorsen’sresult in this setting. In particular, they showed that complete cylindrically bounded<strong>minimal</strong> submanifolds of N ×R are s<str<strong>on</strong>g>to</str<strong>on</strong>g>chastically incomplete. Here we extend theseresults <str<strong>on</strong>g>to</str<strong>on</strong>g> 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 compact<strong>hypersurfaces</strong>. Part (b) generalizes s<str<strong>on</strong>g>to</str<strong>on</strong>g>chastic incompleteness results of [2] and [10]for <strong>minimal</strong> 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 immersi<strong>on</strong> of a completeRiemannian manifold M of dimensi<strong>on</strong> 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 secti<strong>on</strong>alcurvature K radNbounded as K radNal<strong>on</strong>g 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<str<strong>on</strong>g>to</str<strong>on</strong>g>chastically incomplete.(m − l)sup |H| ≥M m C b(r). (1)(m − l)sup |H|
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