Normal fields of geodesic spheres

This question is related to this one (https://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) I’ve asked at math.stackexchange. Let (M,g) be a compact Riemannian manifold with no conjugate points and (˜M,˜g) its universal covering. Let ˆg the Sasaki metric on TM{0} and dˆg its associated distance function.

Fix ˜p˜M and R=1. Let ˜H:=˜M¯B1(p). For x˜H, consider the geodesic sphere centered at p and of radius rx=dg(x,p). I will denote this as Srx(p). Let N(x) be the (inner) normal unit vector at xSrx(p) pointing towards p. As we fixed p, We have defined a function xN(x) whose domain is ˜H. My question is the following:

Is there some constant k such that
dˆg(N(x),N(y))
for all x,y\in\tilde{H}?

Roughly asking, is the normal vector field of “all large spheres” uniformly Lipschitz? I see that if we fix the radius r_x, this is true by the smoothness of geodesic spheres, but I’m trying to get some estimatives letting r_x\in(1,+\infty).

Other way of viewing this question is the following: is there some upper bound for the normal curvature of large geodesic spheres of universal coverings of compact manifolds without conjugate points?

In some sense, I see that the abscense of conjugate points could play some role: if we take the sphere \mathbb{S}^n with the usual metric and fix p, the normal vector field of each geodesic sphere centered in p is Lipschitz, but as we take the radius going to \pi (and the geodesic spheres get “closer” to -p) we see the normal curvatures growing arbitrarily. On the other hand, I see that in the hyperbolic space \mathbb{H}^n, the round spheres have constant mean curvature (a result by Alexandrov – Uniqueness theorems for surfaces in the large, V. Vestnik Leningrad
Univ., 13, No. 19, A.M.S. (Series 2), 21 (1958), 412–416.).

As I pointed out in https://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres, I’m aware that the normal curvature satisfies a Riccati equation and then, maybe, one could be able to use some results describing bounds to solutions of this equations in abscense of conjugate points, but I could not see so far if it can help.

So I would like to ask the community for some result/reference on the subject. Maybe this is a really naive questions, but I’ve got no intuition so far.

Thanks for all members of the community.

Answer

Attribution
Source : Link , Question Author : matgaio , Answer Author : Community

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