### On the mean curvature estimates for bounded submanifolds

Leslie Coghlan, Yoe Itokawa, Roman Kosecki
1992 Proceedings of the American Mathematical Society
A Liouville-type theorem is proved for strongly subharmonic functions on complete riemannian manifolds of bounded curvature. We use this to give a simple proof of a theorem of Jorge. Koutroufiotis and Xavier, which gives an estimate for the exterior size of a submanifold in terms of the sup of the length of its mean curvature. We give a short proof of the following theorem of Jorge and Xavier . Theorem 1. Let M and M be riemannian manifolds and let f ' : M -> M be an isometric immersion.
more » ... tric immersion. Suppose that M is complete with inf scalar curvature > -oo. Let S := sup of the sectional curvature of M, H := sup of the length of mean curvature vector of f, and k be such that there exists some closed normal ball Bx~ of radius X in M containing f(M). Then, ' (l/y/^ô)tanh-x(Vzô/H) ifS<0, A>¡ l/H ifô = 0, , Min{(l/y/ô)tan-x(VS/H),n/(2Va)} ifô > 0. Theorem 1 follows from Theorem 2. Let M be a complete riemannian manifold with bounded sectional curvature and 6 a positive constant. Then, every C2 solution to the inequality Au > 6 is unbounded. Proof. First, we claim that given any r > 0, there exists a > 0 such that for any p e M, we can construct a C2 function vPjr := M -► 3?+ such that vp,r(p) = l>Vp,r vanishes outside Br(p), and |Hess^7)Pir| < a for all x € M. That such a vp ; r exists independent of the injectivity radius of M follows by smoothing a bump function by convolution in the tangent space using the techniques of Theorem 1.8 in  . Set vp,r := dvPtr/(a\fd), where d := dimAf. Then, |Hess vp 0 fixed, there exists a point q e M such that n -u(q) < 6/(aVd) where a is as in the