On dimensions of tangent cones in limit spaces with lower Ricci curvature bounds
We show that if is a limit of -dimensional Riemannian manifolds with Ricci curvature bounded below and is a limit geodesic in then along the interior of same scale measure metric tangent cones are Hölder continuous with respect to measured Gromov-Hausdorff topology and have the same dimension in the sense of Colding-Naber.
1991 Mathematics Subject Classification:53C20
In this paper we obtain new continuity results for tangent cones along interiors of limit geodesics in Gromov-Hausdorff limits of manifolds with lower Ricci curvature bounds.
Our main technical result is the following
For any and ,
there exist , and such that the following holds:
Suppose that is a complete -dimensional Riemannian manifold with and let be a unit speed minimizing geodesic. Then for any with and any there exist subsets () with
and a -Bilipschitz onto map , that is, is bijective and
for any with .
Let () be the renormalized volume measures at . It’s then obvious that under the assumptions of the theorem we have
for some universal .
Let where . By passing to a subsequence we can assume that the renormalized volume measures on converge to a measure on [CC97]. For a point let be a tangent cone at corresponding to some .
Again, up to passing to a subsequence we can assume that the renormalized measures converge to a renormalized measure on (Note that .
Given we will call tangent cones together with the limit measures same scale if they come from the same rescaling sequence .
Using precompactenss and a standard Arzela-Ascoli type argument Theorem 1.1 easily yields
For any and , there exist and such that the following holds:
Let where . Let be a unit speed geodesic which is a limit of geodesics in . Let be a renormalized limit volume measure on .
Then for any with there exist subsets () in the unit ball around the origin in the same scale tangent cones () such that
and there exists a map satisfying
is -Bilipschitz onto;
In particular, () is absolutely continuous with respect to ().
In [CN12] Colding and Naber show that under the assumptions of Corollary 1.2 same scale tangent cones along vary Hölder continuously in . Corollary 1.2 implies that Hölder continuity of tangent cones also holds in measure-metric sense with respect to the renormalized limit volume measures on the tangent cones. This does not follow from the results of [CN12] which do not address measured continuity. Since same scale tangent cones do not need to exists for all for any given scaling sequence, we state the Hölder continuity quanitatively using Sturm distance which metrizes the measured Gromov-Hausdorff topology on the class of spaces in question [Stu06, Lemma 3.7].
There exist such that the following holds.
Let where . Let be a unit speed geodesic which is a limit of geodesics in . Then for any with we have that
where are same scale tangent cones and is the unit ball around the vertex in .
Let be a limit of -manifolds with Ricci curvature bounded below. Recall that a point is called -regular if every tangent cone is isometric to . The collection of all -regular points is denoted by . (When the space in question is clear we will sometimes simply write ).
The set of regular points of is the union
The set of singular points is the complement of the set of regular points. It was proved in [CC97] that with respect to any renormalized limit volume measure on . Moreover, by [CC00b, Theorem 4.15], and is absolutely continuous on with respect to the -dimensional Hausdorff measure. In particular,
It was further shown in [CN12, Theorem 1.18] that there exists unique integer such that
Altogether this implies that there exists unique integer such that
Moreover, it can be shown (Theorem 1.9 below) that this is equal to the largest integer for which is non-empty. Following Colding and Naber we will call this the dimension of and denote it by . (Note that it is not known to be equal to the Hausdorff dimension of in the collapsed case).
Corollary 1.2 immediately implies
Under the assumptions of Corollary 1.2 the dimension of same scale tangent cones is constant for .
Since is Lipschitz we have . Since is absolutely continuous with respect to the -dim Hausdorff measure on and this implies that . This is a contradiction since . ∎
Note that a “cusp” can exist in the limit space of manifolds with lower Ricci curvature bound, for example, a horn [CC97, Example 8.77]. Theorem 1.5 indicates that a ”cusp” cannot occur in the interior of limit geodesics. In particular, it provides a new way to rule out the trumpet [CC00a, Example 5.5] and its generalizations [CN12, Example 1.15]. Moreover, it shows that the following example cannot arise as a Gromov-Hausdorff limit of manifolds with lower Ricci bound, even through the tangent cones are Hölder (in fact, Lipschitz) continuous along the interior of geodesics. This example cannot be ruled out by previously known results.
Then for and . Let be the double of along its boundary. Then all points not on the -axis are in and along the -axis we have that for , (double of (i.e. it’s a cone ) degenerating to .
So but for . Lastly, any segment of the geodesic is unique shortest between its end points and hence it’s a limit geodesic if is a limit of manifolds with . Hence Theorem 1.5 is applicable to and therefore is not a limit of -manifolds with .
Note that one can further smooth out the metric on along axis to obtain a space with similar properties but which in addition is a smooth Riemannian manifold away from the -axis. In particular is non-branching.
Next we want to mention several semicontinuity results about the Colding-Naber dimension which further suggest that this notion is a natural one.
Let be the space of pointed Gromov-Hausdorff limits of manifolds with . Recall the following notions from [CC97]
such that some tangent cone splits off isometrically as .
such that every tangent cone splits off isometrically as .
such that there exist , and such that .
By [CC97, Lemma 2.5] there exists such that if for some then for all sufficiently small .
Suppose , and . Then which obviously implies that for all large as well. By above this implies that for all large .
[Hon13b, Prop 3.78] Let and . Let . Then . In other words, the dimension function is lower semicontinuous on with respect to the Gromov-Hausdorff topology.
Let . Then is equal to the largest for which .
Another immediate consequence of Theorem 1.8 is the following
[Hon13b, Prop 3.78] Let . Then for any and any tangent cone it holds that
It is obvious from (1.3) that for any we have . However, as was mentioned earlier, the following natural question remains open.
Let . Is it true that ?
1.1. Idea of the proof of Theorem 1.1
Let be a unit speed shortest geodesic in an -manifold with . In [CN12] Colding and Naber constructed a parabolic approximation to given as the solution of the heat equation with initial conditions given by , appropriately cut off near the end points of and outside a large ball containing . They showed that provides a good approximation to on an -neighborhood of . In particular, they showed that
for all . They used this to show that for any most points in remain -close to under the reverse gradient flow of for a definite time . In section 3 we show that the same holds true for the reverse gradient flow of . Next, the standard weak type 1-1 inequality for maximum function applied to the inequality (1.6) implies that
as well. This implies that for every in a subset in of almost full measure the integral is small (see estimate (4.4)). Using a small modification of a lemma from [KW11] this implies that for any such point and any most points in remain -close to for all under the flow . This then easily implies that is Bilipschitz on using Bishop-Gromov volume comparison and triangle inequality.
In this section we will list most of the technical tools needed for the proof of Theorem 1.1. Throughout the rest of the paper, unless indicated otherwise, we will assume that all manifolds involved are -dimensional complete Riemannian satisfying
2.1. Segment inequality
We will need the following result of Cheeger and Colding:
Theorem 2.1 (Segment inequality).
[CC96, Theorem 2.11] Given and there exists such that the following holds.
Let be a nonnegative measurable function. Then for any and it holds
where denotes a minimal geodesic from to .
2.2. Generalized Abresch-Gromoll Inequality
Let be a minimizing unit speed geodesic with where . To simplify notations and exposition from now on we will assume that . Let , and let be the excess function.
Theorem 2.2 (Generalized Abresh-Gromoll Inequality).
[CN12, Theorem 2.8] There exist , such that for any , it holds
2.3. Parabolic approximation for distance functions
Fix and let be parabolic approximations to constructed in [CN12]. They are given by the solutions to the heat equations
for appropriately constructed cutoff function . We will need the following properties of established in [CN12].
[CN12, Lemma 2.10] There exists such that
[CN12, Theorem 2.19] There exist such that for all there exists such that the following properties are satisfied
for any with
2.4. First Variation formula
We will need the following lemma (cf. [CN12, Lemma 3.4] ).
Let be a smooth vector field on and let be smooth curves. Let . Then
where is a shortest geodesic from to . Here means the norm of the full covariant derivative of i.e. norm of the map . In particular, if is smooth and , then
The lemma easily follows from the first variation formula for distance functions and the triangle inequality. ∎
2.5. Maximum function
Let be a nonnegative function. Consider the maximum function for . We’ll set .
The following lemma is well-known [Ste93, p. 12].
Lemma 2.6 (Weak type 1-1 inequality).
Suppose has and let be a nonnegative function. Then the following holds.
If with then is finite almost everywhere.
If then for any .
If with then and .
This lemma easily generalizes to functions defined on subsets as follows:
Let and be measurable. Let be measurable such that where . Here denotes the -neighborhood of . Then
Let . Obviously, for any . The result follows by applying Lemma 2.6 (iii) to . ∎
3. Gradient flow of the parabolic approximation
Let be the reverse gradient flow of (i.e. the gradient flow of ) and let be the reverse gradient flow of . We first want to show that for most points we have that for all and for some uniform .
Note that this (and more) is already known for by [CN12]. Following Colding-Naber we use the following
For define the set . Similarly, we define .
An important technical tool used to prove the main results of [CN12] is the following
Unlike Colding-Naber we prefer to work with the gradient flow of the parabolic approximation rather than the gradient flows of , because the gradient flow of provides better distance distortion estimates since in that case the two terms outside the integral in Lemma 2.5 vanish and the resulting inequality scales better in the estimates involving maximum function (see Lemma 4.2 below). Therefore, our first order of business is to establish the following lemma which says that Proposition 3.2 holds for the gradient flow of as well:
There exists and such that if and then we have
The proof of Proposition 3.2 uses bootstrapping in starting with infinitesimally small (depending on !) (cf. Lemma 4.2 below) for which the claim easily follows from Bochner’s formula applied to along . We don’t utilize bootstarpping in and instead use that the result has already been established for the gradient flow of .
Let be small (how small it will be chosen later). Let
We wish to show that contains for some uniform . Obviously is open in so it’s enough to show that it’s also closed. To establish this it’s enough to show that if and then .
For any we define to be the characteristic function of the set . The same argument as in [CN12] shows that
Indeed, we have
where the last inequality follows from the fact that by Lemma 2.3 and hence the Jacobian of satisfies
Similar inequality holds for by Bishop-Gromov volume comparison. Since , by definition, by the segment inequality (Theorem 2.1 ) we have
where the last inequality follows by Bishop-Gromov. Thus,
for some universal and therefore
and for let us define