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Heat Flow on Finsler Manifolds

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Heat Flow on Finsler Manifolds Heat Flow on Finsler Manifolds Shin-ichi Ohta,∗ Karl-Theodor Sturm August 4, 2008 This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F (x, ) : T M R on each tangent space. Mostly, we · x → + will require that this norm is strongly convex and smooth and that it depends smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: either as gradient flow on L2(M,m) for the energy • 1 (u)= F 2( u) dm; E 2 ∇ ZM or as gradient flow on the reverse L2-Wasserstein space (M) of probability measures on • P2 M for the relative entropy Ent(u)= u log u dm. ZM Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove 1,α-regularity for solutions to the (nonlinear) heat equation on C the Finsler space (M,F,m). Typically, solutions to the heat equation will not be 2. More- C over, we derive pointwise comparison results ´ala Cheeger-Yau and integrated upper Gaussian estimates ´ala Davies. 1 Finsler manifolds 1.1 Finsler structures Throughout this paper, a Finsler manifold will be a pair (M, F ) where M is a smooth, connected n-dimensional manifold and F : T M R is a measurable function (called Finsler structure) → + with the following properties: (i) F (x,cξ)= cF (x,ξ) for all (x,ξ) T M and all c> 0. ∈ (ii) For each point x M, there are a local coordinate system (xi)n on a neighborhood U of ∈ i=1 x and positive numbers λ and λ∗ such that, for almost every x U, the function F 2(x, ) ∈ · on T M 0 is twice differentiable and the (n n)-matrix x \ { } × ∂2 1 g (x,ξ) := F 2(x,ξ) (1.1) ij ∂ξi∂ξj 2 ∗Partly supported by the JSPS fellowship for research abroad and the Grant-in-Aid for Young Scientists (B) 20740036. 1 is uniformly elliptic on U in the sense that n n n 1 λ∗ (ηi)2 g (x,ξ)ηiηj (ηi)2 (1.2) ≤ ij ≤ λ Xi=1 i,jX=1 Xi=1 holds for all ξ T M 0 and all η T M. Here (xi,ξi)n denotes the local coordinate ∈ x \ { } ∈ x i=1 system on π−1(U) T M given by ξ = n ξi(∂/∂xi). We will say that such a point x is ⊂ i=1 regular. P The uniform ellipticity (1.2) in particular implies n n 1 λ∗ (ηi)2 F 2(x, η) (ηi)2 (1.3) ≤ ≤ λ Xi=1 Xi=1 and thus the existence of positive constants κ and κ∗ with n 1 κ∗F 2(x, η) g (x,ξ)ηiηj F 2(x, η)2 (1.4) ≤ ij ≤ κ i,jX=1 for almost all x U and all ξ, η T M 0 . This (coordinate-free) inequality in turn implies ∈ ∈ x \ { } ξ + η 1 1 1 F 2 x, F 2(x,ξ)+ F 2(x, η) F 2(x, η ξ), 2 ≥ 2 2 − 4κ − ξ + η 1 1 κ∗ F 2 x, F 2(x,ξ)+ F 2(x, η) F 2(x, η ξ) (1.5) 2 ≤ 2 2 − 4 − for all x U and ξ, η TxM (see [BCL], [Oh3]). For any subset Ω M, the largest constants ∗ ∈ ∈ ⊂ ∗ κ, κ (0, 1] such that (1.4) holds for all x Ω will be denoted by κΩ and κΩ. The constants ∈∗ ∈ 1/ κΩ and 1/√κΩ are also known as 2-uniform convexity and smoothness constants. Let us remark that κ = 1 (or κ∗ = 1) if and only if F (x, ) is a Hilbert norm for each x Ω. p Ω Ω · ∈ A nonnegative function on Rn is called Minkowski norm — and the pair (Rn, ) is k · k k · k then called Minkowski space — if x > 0, cx = c x and x + y x + y hold for all k k k k k k k k ≤ k k k k x,y Rn 0 and c> 0. Thus a Finsler structure F on M induces for a.e. x M a Minkowski ∈ \{ } ∈ norm F (x, ) on the tangent space T M. · x Observe that there is a one-to-one correspondence between Minkowski norms : Rn R k · k → + and convex, bounded open sets B Rn containing the origin: given , B will be the ⊂ k · k open unit ball x Rn : x < 1 ; given B, the associated Minkowski norm is defined by { ∈ k k } x = inf c> 0 : c−1x B . Obviously, will even be a norm if and only if B is symmetric k k { ∈ } k · k (i.e., x B if and only if x B). ∈ − ∈ The reverse Finsler structure ←−F of F is defined by ←−F (x,ξ) := F (x, ξ). We say that F is − reversible (or absolutely homogeneous) if ←−F = F . Usually in differential geometry, Finsler manifolds are assumed to be smooth in the sense that F is smooth on T M 0 . Note that we never require smoothness at the zero section. Requiring \ { } that F 2(x, ) is 2 on all of T M implies that F (x, ) is a Hilbert norm (see [Sh1, Proposition · C x · 2.2]). 1.2 The Legendre transform Given a Finsler structure F on M, we define the dual structure F ∗ : T ∗M R by → + F ∗(x, α) = sup αξ : ξ T M, F (x,ξ) 1 { ∈ x ≤ } for (x, α) T ∗M. For each regular x M, F ∗(x, ) is a Minkowski norm on T ∗M. If F ∗(x, ) is ∈ ∈ · x · twice differentiable on T ∗M 0 , then we set x \ { } ∂2 1 g∗ (x, α) := F ∗2(x, α) (1.6) ij ∂αi∂αj 2 2 for α = n αidxi T ∗M in a local coordinate system (xi)n . i=1 ∈ x i=1 The Legendre transform or transfer map J ∗ : T ∗M T M assigns to each α T ∗M the unique P → ∈ x maximizer of the function 1 1 ξ αξ F 2(x,ξ) F ∗2(x, α) (1.7) 7→ − 2 − 2 on TxM. (The last term is unnecessary but inserted for the sake of symmetry.) The uniqueness is guaranteed by the strict convexity of F (x, ). The vector J ∗(x, α) can be characterized as · the unique vector ξ T M with F (x,ξ) = F ∗(x, α) and αξ = F ∗(x, α)F (x,ξ). We can define ∈ x J : T M T ∗M in an analogous way and then J(x,ξ) is the unique maximizer of (1.7) as a → function of α. We recall several standard properties of the Legendre transform which can be found in [BCS, 14.8] for instance. § Lemma 1.1 (i) It holds that J ∗ = J −1. (ii) For any α = n αidxi T ∗M and ξ = n ξi(∂/∂xi) T M, we have i=1 ∈ x i=1 ∈ x n n P ∂ 1 ∂P ∂ 1 J ∗(x, α)= F ∗2(x, α) , J(x,ξ)= F 2(x,ξ) dxi. ∂αi 2 ∂xi ∂ξi 2 Xi=1 Xi=1 ∗ ∗ (iii) If x M is regular, then gij(x, α) in (1.6) is well-defined for all α Tx M 0 and we have∈ ∈ \ { } n J(x,ξ)= g(x,ξ)ξ = g (x,ξ)ξidxj T ∗M, ij ∈ x i,jX=1 n ∂ J ∗(x, α)= g∗(x, α)α = g∗ (x, α)αi T M. ij ∂xj ∈ x i,jX=1 ∗ ∗ In particular, gij(x, α) is the inverse matrix of gij(x, J (x, α)). (iv) If x M is regular, then ∈ J(x, η) J(x,ξ) (η ξ) κ∗F 2(x, η ξ) (1.8) − − ≥ − for all ξ, η T M. ∈ x (v) The dual structure F ∗ satisfies estimates analogous to (1.2), (1.3), (1.4), (1.5) and (1.8) with λ∗ and κ∗ in the place of λ and κ, respectively, and vice versa. ∗ Proof: Here the existence of gij(x, α) in (iii) is merely a consequence of the inverse func- ∗ tion theorem (for J and J ). Since g = ∂ξJ by (ii), the mean value theorem implies that (J(x, η) J(x,ξ))(η ξ) = (η ξ)g(x, γ)(η ξ) for some γ on the segment between ξ and η. − − − − Using (1.4) the RHS can thus estimated from below by κ∗F 2(x, η ξ). − Note that at the origin J ∗(x, ) is continuous but not differentiable (even if F is smooth on · T M 0 ). \ { } ∗ Remark 1.2 Fixing a coordinate system, we may identify both TxM and Tx M with the Eu- clidean space Rn. Given a vector ξ T M of length F (x,ξ) = 1, the vector J ∗(x,ξ) corresponds ∈ x to the unit normal vector at the point ξ at the unit sphere in TxM (Figure 1). For each regular x M and ξ T M 0 , the map ∈ ∈ x \ { } 1/2 n F ξ(x, ) : η g (x,ξ)ηiηj (1.9) · 7→ ij i,jX=1 3 ξ J ∗(ξ) . unit sphere for F Figure 1: Tangent defines a Hilbert norm on TxM. It can be regarded as the best Hilbert norm approximation of the norm F (x, ) in directions close to ξ.
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