• 検索結果がありません。

A semigroup approach to Finsler geometry: Bakry–Ledoux’s isoperimetric inequality

N/A
N/A
Protected

Academic year: 2024

シェア "A semigroup approach to Finsler geometry: Bakry–Ledoux’s isoperimetric inequality"

Copied!
32
0
0

読み込み中.... (全文を見る)

全文

This Bochner inequality has the same form as the Riemannian case by means of the mixture of the nonlinear Laplacian ∆and its linearization ∆∇u. The first application of (1.2) is the L1 gradient estimate (Theorem 3.7), where we also include the non-compact case, but with some additional (probably redundant) assumptions, see the theorem below where we assume the same conditions. However, in the non-compact case there are technical difficulties and it is unclear how to remove them in this semigroup approach (see §3.4 for a discussion).

The inequality (1.3) has the same form as the Riemannian case in [BL], and it is sharp and the model space is the real line R fitted with the normal (Gaussian) distribution dm =p. We also refer to [AM] for the Gaussian isoperimetric inequality on RCD(K,∞) spaces by a refinement of the Γ calculus. They considered essentially unbranched metric measure spaces (X, d,m) satisfying CD(K, N) for K ∈ R and N ∈ (1,∞), and the sharp L´evy-Gromov type isoperimetric inequality shown of the form.

The case N =∞ is not included in [CM] for technical reasons regarding the structure of CD(K,∞)-spaces, but the same argument yields (1.3) (corresponding to N = D=∞) for reversible Finsler manifolds. Inequality (1.3) improves (1.5) in the case where N = D = ∞ and K >0 and supports the assumption that the sharp isoperimetric inequality in the irreversible case is the same as the reversible case, namely Λ−F1 in (1.5) has been removed.

Finsler manifolds

Then the Hopf–Rinow theorem ensures that any pair of points is connected by a minimal geodesic (see [BCS, Theorem 6.6.1]). The following useful fact about homogeneous functions (see [BCS, Theorem 1.2.1]) plays a fundamental role in our calculus. Define the formal Christoffel symbol γjki (v for v ∈T M\0, and the geodesic spray coefficients and the non-linear relation.

The corresponding covariant derivative of a vector field X byv ∈TxM with reference vector w∈TxM \0 is defined as.

Uniform convexity and smoothness

Weighted Ricci curvature

Nonlinear Laplacian and heat flow

Note that the space Hloc1 (M) is defined exclusively in terms of the differentiable structure of M. Since taking the gradient vector (more precisely, the Legendre transform) is a non-linear operation, our Laplacian ∆ is a non-linear operator unless F is Riemannian. Global solutions can be constructed as gradient curves of the energy functional E in the Hilbert space L2(M).

For each initial datum0 ∈H01(M) andT > 0, there exists a unique global solution u for the heat equation on [0, T]×M, and the distributional Laplacian∆ut is absolutely continuous with respect to m for all t ∈( 0,T). ii). One can take the continuous version of a global solution u, and it enjoys the Hloc2 regularity in x as well as the C1,α regularity for some α in both t and x.

Bochner–Weitzenb¨ ock formula

In (2.11) it is compensated by the fact that ∇2ues does not necessarily coincide with the Hessen of u with respect to g∇u. In the Bochner–Weitzenb¨ock formula (Theorem 2.10) in the previous section, we used the linearized Laplacian ∆∇u induced from the Riemannian structure g∇u. In the same vein, we can consider the linearized heat equation associated with the global solution of the heat equation.

This technique proved useful and we obtained gradient estimates 'a la Bakry-'Emery and Li-Yau in [OS3, §4]. In this section we discuss such a linearization in detail and improve the estimation of the L2 gradient to an L1 bound (Theorem 3.7). We will establish a measurable family of non-vanishing vector fields (Vt)t≥0 with one parameter, such that Vt = ∇ut on Mut for every t ≥ 0.

3.1) The existence and other properties of the linearized semigroup Ps,t∇u are summarized in the following statement. i) Let s = 0 without loss of generality. This unique existence follows from Theorem 4.1 and Remark 4.3 in [LM, Chapter III] (see also [RR, Theorem 11.3], where it is assumed that A(t) is continuous in t but this is in fact unnecessary). ii) The H¨ouer continuity is a consequence of the local uniform ellipticity of ∆Vt (see [OS1, Proposition 4.4]). Let us denote byPbs,t∇u the additive operator of Ps,t∇u. to see that the adjoint heat half-group solves the linearized heat equation backward in time.

This development is sometimes called the conjugate thermal semigroup, especially in Ricci's flow theory; see for example [Ch+, Chapter 5].). Therefore, we see in the same way as Ps,t∇u that ∥Pbt∇−uσ,t(ϕ)∥L2 is nonincreasing in σ and that Pbt∇−uσ,t expands to a linear contraction semigroup acting on L2( M ). We will not discuss this issue, but we carefully replace Vt with ∇ut as far as possible (using Lemma 2.12).

By a well-known technique based on the Bochner inequality (2.12) with N =∞, we obtained in [OS3, Theorem 4.1] the L2 gradient estimate of the following form. Let us emphasize that we use the non-linear semigroup (us → ut) in the LHS, while in the RHS the linearized semigroup Ps,t∇u is used. See the proof of Proposition 3.7 below which is based on a similar calculation (with the sharper inequality in Proposition3.5).

Improved Bochner inequality

The following integrated form can be shown in the same way as Theorem 2.13, for details we refer to [OS3, Theorem 3.6].

L 1 -gradient estimate

On the hypothesis (3.6)

Weighted Riemannian case

This density is a consequence of the hypoellipticity (see [BGL, Theorem 3.2.1]), which is defined by the property that every solution for L∗f =λf is smooth (see also [BGL, Definition 3.3.8] , typically A=C∞(M)). This is not the case for operators with non-smooth coefficients, making it unclear whether we can apply this method in the Finsler case (on the linearized Laplace-∆∇u).

RCD-case

Characterizations of lower Ricci curvature bounds

Now in [BGL] for the linear operator L we use the density A0 =Cc∞(M) in the domain D(L) with respect to the norm. This does not apply to operators with non-smooth coefficients, so it is not clear whether this method can be used in the Finsler case (for the linearized Laplacian ∆∇u). III) Improved Bochner inequality. In general, for linear semigroups, gradient estimates are directly equivalent to the corresponding contraction properties (see [Ku]).

In our Finsler setting, however, the lack of commutativity (see [OP]) precludes such a contraction estimate, at least in the same form (see [OS2] for details). The Ricci flow provides time-dependent Riemannian metrics obeying a kind of heat equation on the space of Riemannian metrics, while we considered time-dependent (singular) Riemannian structures g∇u to solve the heat equation. More specifically, what corresponds to our lower bound on the Ricci curvature is the super Ricci flow (super solutions of the Ricci flow equation).

What is missing in our Finsler setting is the contraction property, for which the Riemannian nature of space is necessary. This section is devoted to the isoperimetric inequality, as a geometric application of the improved Bochner inequality (Theorem 3.5). Recall that, under CF < ∞ or SF < ∞, the forward completeness is equivalent to the backward completeness according to Lemma 2.4.

In (4.1) the role of the unit sphere is played by the real line R, equipped with the Gaussian measure p. It is known that the curvature bound Ric∞ ≥K (or CD(K,∞)) implies the log-Sobolev inequality, Z. We then show that the Poincar´e inequality (4.2) is the exponential decay of the variance and a kind of ergodicity along heat flow (similar to [BGL, §4.2]), which is one of the key ingredients in the proof of Theorem 4.1 (see the proof of Corollary 4.5).

Therefore e2Kt/SF Varm(ft) is non-increasing int, this completes the proof of the first assertion.

Key estimate

Proof of Theorem 4.1

Bobkov, an isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gaussian space. Mondino, Sharp and rigid isoperimetric inequalities in metric measure spaces with lower Ricci curvature bounds. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces.

Milman, Beyond traditional curvature dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension. Savar'e, Self-improvement of the Bakry-'Emery condition and Wasserstein contraction of the heat flow in RCD(K,∞) metric gauge spaces.

参照

関連したドキュメント