The first statement (i) on the isometric division was also stated (informally) in [HN, p. 1547] as a result of Bochner's improved inequality in [BE].) See [Mai]. The lack of smooth structure (precisely, parallel vector fields and de Rham decomposition) prevents us from following the simple proof of [CZ]. To overcome these difficulties, we consider the elevation U of the eigenfunction u in the L2-Wasserstein space, defined by U(µ) := R.

In Section 3 we show that the liftU of the eigenfunction is affine along the lines of [Ke]. We then apply the theory in [AT1] to obtain the regular Lagrangian flow (Ft)t∈R of −∇u in Section 4 (this step is not straightforward since u is unbounded), and analyze the behavior of the measure m and the distance d along the flow. With these properties of the flow, we can follow the argument in [Gi1] to prove the k = 1 case of Theorem 1.1, as we will see in Section 5.

One can in fact show (2.6) makes sense in the m-almost everywhere by replacing the left-hand side with the absolutely continuous part of the Γ2 operator, see [Sa, Gi4] for details. The first step is to show that the lifting of the eigenfunction to the L2-Wasserstein space is affine. Thanks to [GMS, Proposition 6.7], the spectrum of the Laplacian is discrete and the hypothesis λ1 = 1 implies the existence of an eigenfunction u∈D(∆) that satisfies ∆u=−u.

Then the weak form of the Bochner inequality (2.5) (also called the Γ2 inequality) is written as, provided that φ≥0,.

## Regular Lagrangian flow

Regular Lagrangian flows for gradient vector fields on RCD(K,∞)-spaces are studied in [AT1, Theorem 9.7]. However, the gradient vector field ∇u does not satisfy the assumption because ∆u is only in L∞loc(X) when L∞(X) is required. Therefore – although we cannot exactly use the results in [AT1] – we carefully follow the argument of the more general result [AT1, Theorem 8.3] to prove Theorem 4.2 by means of Statement 4.1.

We can easily construct a class of functions satisfying [AT1, (4-3)], and the inequality L2-Γ always holds (as mentioned after [AT1, Definition 5.1]). Here we have used the fact that we can replace u by ˜u∈D∞(X) which is bounded and agrees with u onB3R(x0) due to the presence of ψR. Indeed, since u is Lipschitz, ˜u can be constructed by taking a composition of a suitable switch function and u.

Then vt defined by vtm = (et)∗((¯v◦e0)η) solves the continuity equation for b with the initial condition v0 = ¯v. Combining this with the uniqueness of the continuity equation for bR starting from ¯v as claimed, we can use [AT1, Theorem 8.4] to obtainηx ∈C([0, T];X) by solving the ODE ˙ηx =bR( ηx) withηx(0) =x for (¯vm)-almost every x and satisfying. We are now able to follow almost the same argument as in [AT1, Theorem 8.3].

One can further extend this to F:

## Behavior of the distance under the flow

The truncation argument shows that we can remove the boundedness of f from the assumption in the last statement. The key fact in the proof is that (Fet(x))t∈R provides an EVI-gradient flow u, as we saw in Proposition 4.10. On the one hand, since EVI-gradient currents are also gradient currents in terms of the energy dissipation identity (a proof of this fact by Savar´e can be found in [AG]), we have for every s < t.

The properties of the gradient flow ( ˜Ft)t∈R of the eigenfunction-u obtained in the previous section allow us to largely follow the strategy of the splitting theorem in [Gi1, Gi2]. The affine property of u (Proposition 4.10) implies that Y is completely geodesic in the sense that every geodesic connecting two points in Y is contained in Y. We first prove the important mapping property along the strategy in [Gi1, Corollary 5.19] (see also [Gi2, Corollary 4.6]).

The proof follows the same lines as [Gi2, Proposition 4.15] and [Gi1, Proposition 6.5], we refer to them for details of the discussion. For reasons of compactness (eg, compare with [GH]), it suffices to show the claim for shape functions. To compare this with the same decomposition of f ◦Φ, note that by the same arguments used in Gigli's proof of the Splitting Theorem [Gi1], we have that.

Recall that the Sobolev-to-Lipschitz property, which is required in the quoted proposition, holds in RCD(K,∞) spaces as we mentioned in §2.2. 2 Remark 5.6 The discussions in Sections 3–5 can be compared to the study of spaces admitting nonconstant affine functions. It is somewhat implicit in our discussion that the sharp spectral gap prevents the appearance of "bounded" spaces such as Y ×[0,∞) (whereas Y e Y ×R may have a boundary).

This sequence, however, converges to (R,| · |,e−y2/2dy) in the weaker notion of the measured Gromov topology. We expect that the equality in this result yields the same stiffness statement as in this paper, similar to the finite dimensional situation of the isoperimetric L'evy-Gromov [CMo] inequality. Mondino, sharp and rigid isoperimetric inequalities on metric measure spaces with Ricci curvature lower bounds.

Sturm, On the equivalence of the entropic curvature dimension condition and Bochner's inequality on metric measure spaces. Gigli, A survey of the proof of the splitting theorem in spaces with non-negative Ricci curvature.