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Nonlinear geometric analysis on Finsler manifolds

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One of the main challenges, compared to the Riemann case, is in the non-linearity of the Laplacian and the associated heat equation. The Γ-calculus, developed by Bakry and his associates (see [BE, Bak] and the recent book [BGL]), is a successful theory of the analysis of linear diffusion operators and the associated semigroups by means of the integration by parts and a type of Bochner inequality. We will discuss three applications of the Γ calculus: gradient estimates, functional inequalities and isoperimetric inequalities.

We will outline some proofs to show the basic idea of ​​the Γ-calculus. The only exception is the admissible range of the exponent p in the Sobolev inequality (see Note 5.12). One of the most important differences is the lack of heat flow contraction properties ([OS2]).

Finsler structures

We refer to [BCS, Sh2, SS] for the basics of Finsler geometry and related comparative geometric studies. We will also fix an arbitrary positive C∞-measure m on M from subsection 2.5. c) Although we will only consider under Definition 2.1, C∞-regularity (1) and strong convexity (3) can occasionally be weakened in different ways (see for example [OS1]). Let us continue the study of strong convexity in the remainder of this subsection.

In fact, the unit sphere of F|TxM (which is positively curved due to Remark 2.2(b)) is tangent to that ofgv atv/F(v) up to second order. We note and emphasize that α(v)≥ −F∗(α)F(v) however does not hold in general due to the non-reversibility of F. The strong convexity is related to the following quantities, these are fundamental in the geometry of Banach spaces (see [BCL, Oh2]).

Asymmetric distance and geodesics

It is easily seen that the constants SF and CF control the reversibility constant, defined by Note that γjki has the same form as the Riemannian Christoffel symbol, while it is a (0-homogeneous) function on T M\0 and cannot be reduced to a function on M. We say that (M, F) is forward complete if exponential map is defined on the entire TM. In other words, every geodesic η M expands infinitely in the forward direction to the geodesic ¯η : [0,∞)−→ M.

Backward completeness is not necessarily equivalent to forward completeness (in the non-compact case). If (M, F) is forward or backward complete, then the Hopf–Rinow theorem ensures that every pair of points is associated with a minimal geodesic (see [BCS, Theorem 6.6.1]).

Covariant derivative

The geodesic equation Dηη˙˙η˙ ≡0 concerns the special case of W =V, where we have another contraction.

Curvatures

This is one of the main obstacles to developing the theory of Finsler-Ricci flow. Then, for any w∈TxM linearly independent ofv, the flag curvature K(v, w) coincides with the section curvature of the 2-planev∧w with respect to the Riemannian metric gV onU. In particular, we find that the sectional curvature of v∧w with respect to togV does not depend on the choice of V, where the assumption of V (all integral curves are geodesic) plays the essential role.

The proof of Theorem 2.10 can be found in [Sh2, §6], which inspired the structure of this section. The most fundamental is the following (obtained in [Au]), we will see even more (regarding the weighted Ricci curvature) in the next subsection. Then V is a C∞ vector field on an open set U ⊂M\{x}and all integral curves are geodesic.

Weighted Ricci curvature

There are several known ways of generalizing the Riemannian volume measure, canonical in their own rights, leading to the various measures. If we choose m= volg, then RicN = Ric regardless of the choice of N. c) Multiplying the measure m by a positive constant does not change the weighted Ricci curvature. In the Riemann case, the study of Ric∞ goes back to Lichnerowicz [Li], he showed a partition theorem of the Cheeger-Gromoll type (see [FLZ, WW, Oh8] for some generalizations).

Some historical accounts of related works on N < 0 in convex geometry and partial differential equations can be found in [Mi2, Mi3]. The weighted version of the Bonnet–Myers theorem (Theorem 2.12) can be shown in the same way as the unweighted case. From this estimate N can be considered as (the upper limit of) the dimension (although this interpretation prevents us from considering N < 0).

As with Theorems, the proof of Theorem 2.17 can be reduced to the (weighted) Riemann situation using the Riemann metric induced from the unit velocity geodesic arising from x. The main focus of this section is the Bochner-Weitzenb¨ock formula (Theorem 3.3), which is the indispensable ingredient of our nonlinear analogue of the Γ-calculus. For a differentiable function u : M −→ R, the gradient vector at x is defined as the Legendre transform of the derivative of u: ∇u(x) := L∗(Du(x)) ∈ TxM.

Then we define the distributional Laplacian u∈Hloc1 (M) with ∆u:= divm(∇u) in the weak sense, that. Note that the space Hloc1 (M) is defined exclusively with respect to the differentiable structure M. In the Riemannian case, the Laplacian ∆m associated with the measure m= eΦvolg can be written as where ∆g is the usual Laplacian with respect to the rigid.

It is also possible to define the Laplacian associated only with F and consider our Laplacian ∆ as the one weighted with respect to m.

Bochner–Weitzenb¨ ock formula

Such a definition can be found, for example, in [Lee] (see also [Oh7]), but it is more complicated than our simple definition. In (3.2) it is compensated by the fact that ∇2u does not necessarily coincide with the Hessian ofu with respect to g∇u (unless all integral curves of ∇u are geodesic).

Heat equation

Weitzenb¨ock's formula for a more general class of Hamiltonian systems (by neglecting the positive 1-homogeneity of F, see [Lee, Oh7]). The global solutions are constructed as gradient curves of the energy functional E in the Hilbert space L2(M). As for regularity, the classical theory of partial differential equations holds because our Laplacian is locally uniformly elliptic due to the strong convexity of F .

We summarize the existence and regularity properties in the next theorem (see [OS1, §§3, 4] for details, we note that our C∞-smooth F and m clearly enjoy the mild smoothness assumption in [OS1, (4.4)] ). It was also proved in [OS1] that the heat flow is considered as the gradient flow of the relative entropy (see the proof of Proposition 5.3 below) in the L2-Wasserstein space with respect to ←− F (this result is well beyond the scope of this examination). In the Bochner–Weitzenb¨ock formula (Theorem 3.3) 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 semigroup associated with a global solution to the heat equation. This technique proves useful, and we obtain some gradient estimates as our first applications of the nonlinear Γ calculus. We will find that our arguments hereafter rely only on the Bochner inequality, the integration of parts, and the (nonlinear and linearized) heat semigroups.

We will fix a measurable one-parameter family of non-expanding vector fields (Vt)t≥0 such that Vt = ∇ut on Mut for each t≥0. 4.1) The existence and other properties of the linearized semigroup Ps,t∇u are summarized as follows ([OS1, Oh11]). We denote by Pbs,t∇u the associated operator Ps,t∇u. to see that the associated heat semigroup solves the linearized heat equation backward in time. This development is sometimes called the conjugate thermal semigroup, especially in Ricci flow theory, see for example [Ch+, Chapter 5].).

Therefore, in the same way as Ps,t∇u, we find that Pbt∇−uσ,t extends to the linear contraction semigroup acting on L2(M).

Gradient estimates

2 In the linear setting, the Bochner inequality is known to have the self-improving property ([BQ, Theorem 6], [BGL]). Using this improved inequality instead of (3.4), we obtain an estimate of the gradient L1 (which implies an estimate of the gradient L2 via Jensen's inequality, see [Oh11]).

Characterizations of lower Ricci curvature bounds

We start with the Poincar´e–Lichnerowicz (spectral gap) inequality under the curvature bound RicN ≥ K > 0. For simplicity, we will assume that M is compact (this is automatically true when N ∈[n,∞) and M is complete ), and normalize mas m(M) = 1. 2 Now the Poincar´e-Lichnerowicz inequality is obtained by a technique somewhat related to the proof of Theorem 4.3.

Logarithmic Sobolev inequality

See [Oh12] and [BGL] for the omitted calculations in the proofs of the theorem above and the theorem below. Integrating this inequality multiplied by the test function eh, using (5.5) with a= 1/2 and rearranging, we get. Going back to [OV] by the standard implication, we have the Talagrand inequality as a corollary.

This is because Talagrand's inequality implies the normal concentration of m ([Led]), while the model space in [Mi2, Oh9] only has the exponential concentration.

Sobolev inequalities

This last section is devoted to a geometric implementation of the Γ calculus, the isoperimetric Gaussian inequality obtained in [Oh11]. This had been an obstacle to generalizations to Finsler manifolds as well as less smooth spaces such as metric spaces. In 2014, Klartag [Kl] gave a nice alternative proof of the isoperimetric L´evy–Gromov inequality, still on weighted Riemannian manifolds, but without the regularity.

Inspired by [Kl], Cavalletti-Mondino generalized the localization method to essentially unbranched metric measure spaces satisfying the curvature-dimension condition, and showed the isoperimetric inequality (6.2) in [CM1] and various functional inequalities in [CM2]. It seems plausible to expect that Λ−F1in (6.3) will be removed, that is, non-reversible Finsler manifolds enjoy the same isoperimetric inequality as reversible Finsler manifolds. This is the only case where we still have an alternative proof of (6.1), based on the Γ-calculus ([BL]).

The inequality (6.6) has the same form as the Riemannian case in [BL], i.e. it is sharp and a model space is the real line R equipped with the normal (Gaussian) distribution dm=√. Bobkov, An isoperimetric inequality in the discrete cube and an elementary proof of the isoperimetric inequality in Gaussian space. Mondino, sharp and rigid isoperimetric inequalities on metric measure spaces with lower Ricci curvature limits.

Mondino, Sharp geometric and functional inequalities in metric space spaces with lower Ricci curvature limits. Sturm, On the equivalence of the entropic curvature dimension condition and Bochner's inequality on metric measure spaces. Gigli, Non-smooth differential geometry – A tailor-made approach for spaces with a Ricci curvature bounded from below.

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