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Cheeger's constant and the first eigenvalue of a closed Finsler manifold

In this paper, we consider Cheeger's constant and the first eigenvalue of the nonlinear Laplacian on closed Finsler manifolds. Being based on these, we establish Cheeger's inequality and Buser's inequality for closed Finsler manifolds.


Introduction
The study of the eigenvalues of Laplacian is a classical and important problem in Riemannian geometry, which highlights the interplay of the geometry-topology of the manifold with the analytic properties of functions. In order to bound below the first eigenvalue λ 1 (M ) of a closed Riemannian manifold (M n , g), Cheeger [10]  Moreover, even if M has a boundary, Cheeger's inequality still holds if λ 1 (M ) is subject to the Neumann boundary condition or the Dirichlet boundary condition [8,17]. This inequality has found a number of applications, e.g., [6,8,15]. It is an important result due to Buser [7] that (M ) is actually equivalent to manifolds [18,20,21,24], but it is quasilinear and dependent on the measure. While the measure on a Finsler manifold can be defined in various ways and essentially different results may be obtained, e.g., [1,2]. In general, the eigenfunctions of this Laplacian are not smooth but C 1,α [14]. Hence, it seems indeed difficult to compute the first eigenvalue even for a the Eucildean sphere S n equipped with a Randers metric F = α + β, where α is the canonical Riemannian metric and β is a 1-form on S n . The purpose of this paper is to investigate the relationship between Cheeger's constant and the first eigenvalue of such Laplacian. Let (M, F, dµ) be a closed Finsler n-manifold, where dµ be any measure on M . According to [18,19], the first nontrivial eigenvalue λ 1 (M ) is defined as where F * is the dual metric of F and H 0 (M ) := {f ∈ W 1 2 (M ) : M f dµ = 0}. It follows from [14,18,19] that λ 1 (M ) is the smallest positive eigenvalue of Shen's Laplacian. Inspired by [11,18,19], we define Cheeger's constant of a closed Finsler manifold as where Γ varies over compact (n − 1)-dimensional submanifolds of M which divide M into disjoint open submanifolds D 1 , D 2 of M with common boundary ∂D 1 = ∂D 2 = Γ and A ± (Γ) denote the areas of Γ induced by the outward and inward normal vector fields n ± . In general, A + (Γ) = A − (Γ). In fact, one can construct examples in which the ratio of these two areas can be arbitrarily lager (see [11]). Recall that the reversibility λ F of F is defined by λ F := sup y =0 F (y)/F (−y) (cf. [16]). Clearly, λ F ≥ 1. If λ F = 1, F is called a reversible metric, in which case A + (·) = A − (·). First, we have the following Cheeger type inequality.
Theorem 1.1. Let (M, F, dµ) be a closed Finsler manifold with the reversibility λ F ≤ λ. Then It should be remarkable that if M has a boudary, Chen [11] proved the above inequality still holds if λ 1 subject to the Dirichlet boundary condition. Recall the uniform constant [12] Λ F is defined as In the Riemannian case, λ F = Λ F = 1. Hence, Theorem 1.1 implies (1.1) while Theorem 1.3 implies (1.2). In particular, for a Randers metric F = α + β, the uniform constant . For the Busemann-Hausdorff measure or the Holmes-Thompson measure, the S-curvature of a Berwald metric always vanishes (see Theorem 6.5 below). Then we have the following corollary.

Preliminaries
In this section, we recall some definitions and properties about Finsler manifolds. See [4,18] for more details.
Let (M, F ) be a (connected) Finsler n-manifold with Finsler metric F : where G i is the geodesic coefficients. A smooth curve γ(t) in M is called a (constant speed) geodesic if it satisfies Define the Ricci curvature by Ric(y) : Set S x M := {y ∈ T x M : F (x, y) = 1} and SM := ∪ x∈M S x M . The reversibility λ F and the uniformity constant Λ F of (M, F ) are defined by The dual Finsler metric F * on M is defined by The Legendre transformation L : T M → T * M is defined by In particular, F * (L(X)) = F (X). Now let f : M → R be a smooth function on M .
The gradient of f is defined by ∇f = L −1 (df ). Thus we have df (X) = g ∇f (∇f, X).
Let dµ be a measure on M . In a local coordinate system (x i ), express dµ = σ(x)dx 1 ∧ · · · ∧ dx n . In particular, the Busemann-Hausdorff measure dµ BH and the Holmes-Thompson measure dµ HT are defined by For y ∈ T x M \{0}, define the distorsion of (M, F, dµ) as And S-curvature S is defined by where γ(t) is the geodesic withγ(0) = y. Define the canonical energy functional E on H 0 (M )\{0} by

A Cheeger type inequality for Finsler manifolds
λ is an eigenvalue of (M, F, dµ) if there is a function u ∈ H 0 (M ) − {0} such that d u E = 0 with λ = E(u). In this case, u is called an eigenfunction corresponding to λ. The first eigenvalue of (M, F, dµ), λ 1 (M ), is defined by which is the smallest positive critical value of E.
Remark 1. It should be noticeable that u is an eigenfunction corresponding to λ if and only if ∆u + λu = 0 (in the weak sense), where ∆ := div • ∇ is the Laplacian induced by Shen [14,18,19].
where Γ varies over compact (n − 1)-dimensional submanifolds of M which divide M into disjoint open submanifolds D 1 , D 2 of M with common boundary ∂D 1 = ∂D 2 = Γ.
To prove Theorem 1.1, we need the following co-area formula.
Theorem 3.3 then yields the following lemma. where Proof. Without loss of generality, we assume that f is nonconstant. For almost every t ∈ [min f, max f ], Ω(t) is a domain in M , with compact closure and smooth boundary. Note that n := ∇f F (∇f ) is a unit normal vector field along ∂Ω(t). (1). It follows Theorem 3.3 that (2). Theorem 3.3 now yields Proof of Theorem 1.1. Given a smooth function f on M , let α be a median of f , i.e., Set f + := max{f − α, 0} and f − := min{f − α, 0}. By the definition of median, one can check that for any t > 0, Thus, the above inequalities together with Lemma 3.4 yield Hence, By a Croke type isoperimetric inequality [23], one has the following result. Also refer to [24, Theorem 6.2, Prposition 6.4] for a reversible version of the isoperimetric inequality.
where diam(M ) denotes the diameter of M .
Theorem 3.5 together with Theorem 1.1 now yields a Finslerian version of Yau's lower estimate for the first eigenvalue [22]. Theorem 3.6. Let (M, F, dµ) be a closed Finlser manifold with Ric ≥ (n − 1)k, where dµ denotes either the Busemann-Hausdorff measure or the Holmes-Thompson measure. Then That is, λ 1 (M ), of a closed Finsler manifold, can be bounded from below in terms of the diameter, volume, uniform constant and a lower bound for the Ricci curvature.

Volume comparison
In this section, we will study the properties of the polar coordinate system of a Finsler manifold, which is useful to show Theorem 1.3. Refer to [20,24] for more details.
Let (M, F, dµ) be a forward complete Finsler n-manifold. In the rest of this paper, we always assume that dµ is either the Busemann-Hausdorff measure or the Homles-Thompson measure. Given p ∈ M , denote by (r, y) = (r, θ α ), 1 ≤ α ≤ n, the polar coordinates about p. Express where dν p is the measure on S p M induced by F .
Here, V n,k (r) (resp. A n,k (r)) is the volume (resp. area) of ball (resp. sphere) with radius r in the Riemannian space form of constant curvature k, that is, Proof. It is easy to check that Λ −n ≤ e −τ (y) ≤ Λ n , for all y ∈ SM . By [20,24], for each y ∈ S p M , we have ∂ ∂r Hence, Then (1), (2)  Note that dA + | (r,y) :=σ(r, y)dν p (y) is the measure on S + p (r) induced by ∇r. Then we have the following result.

A Buser type isoperimetric inequality for starlike domains
In this section, we will extend Buser's isoperimetric inequality[7, Lemma 5.1] to Finsler setting. However, the original method of Buser's cannot be used directly, since the Finsler metrics considered here can be nonreversible. To overcome this difficulty, we introduce the "reverse" of a Finsler metric. The reverse of a Finsler metric F is defined byF (y) := F (−y).
A direct calculation yields the following two lemmas.
Lemma 5.1. For each y = 0, we havẽ whereG i (resp. G i ) is the spray ofF (resp. F ) and Ric (resp. Ric) is the Ricci curvature ofF (resp. F ). Hence, if γ is a geodesic of F , then the reverse of γ is a geodesic ofF . Proof. Let n ± (resp.ñ ± ) be the unit normal vector along Γ in (M, F ) (resp. (M,F )). It is easy to check thatñ ± = −n ∓ . Then we are done by Lemma 5.2.
From above, we obtain the following key lemma.
Step 1: Suppose µ(D 1 ∩ B(r/(2 √ Λ))) ≤ αµ(D 1 ). For each q ∈ D 1 − Cut p , set q * is the last point on the minimal geodesic segment γ pq from p to q, where this ray intersects Γ. If the whole segment γ pq is contained in D 1 , set q * := p.

Lemma 4.2 now yields that
Hence, By assumption, we have . Then Fubini's theorem yields that there exist a point q 1 ∈ W 1 and a measurable set U 2 ⊂ W 2 such that (1) For each w ∈ U 2 , d(w, q * 1 ) ≤ d(q * 1 , q 1 ) and (q 1 , w) / ∈ N . (2) µ(U 2 ) ≥ 1 2 µ(W 2 ); It should be noticeable that q * 1 is dependent on the choice of w. Let w ♯ denote the first point on γ wq1 where the segment intersects Γ. Thus, for each w ∈ U 2 , . LetF denote the reverse of F . It follows from Lemma 5.1 that the reverse of the geodesic γ wq1 is a minimal geodesicγ q1w from q 1 to w in (M,F ). Note that w ♯ is the last point onγ q1w whereγ q1w intersects Γ. Let N be defined as N in (M,F ). It is easy to see that N = N . Denote byd the metric induced byF . Thus, 2μ (W 2 ); Note that Lemma 5.1 also implies that Ric ≥ (n − 1)k. A similar argument to the one in Case I together with Lemma 5.2 and Corollary 5.3 yields that By assumption, we have Hence, A n,k (r/2) V n,k (r) − V n,k (r/2) .
Step 3: From above, we obtain To obtain the best possible bound, we set Thus, .

An easy calculation then yields
.

Since
where Γ varies over smooth hypersurfaces in D satisfying (1) Γ is embedded in D; (2) Γ divides D into disjoint open sets D 1 , D 2 in D with common boundary ∂D 1 = ∂D 2 = Γ.

A Buser type inequality for Finsler manifolds
Let (φ, ϕ) := M φϕdµ. Then we have the following minimax principle.
Proof of Theorem 1.3. By Lemma 6.2, we can suppose that Ric ≥ −(n − 1), i.e., δ = 1. Given any ǫ > 0, let Γ, D 1 and D 2 be as in Definition 3.2 such that 0 ≤ I − (M ) < ǫ, where Step 1. For n ≥ 3, Γ may satisfy max p∈M d(p, Γ) < ρ, for any ρ > 0 ("the problem of hair" [6]). Hence, we will find a new setΓ to replace Γ. Let P := {p 1 , . . . , p k } be a forward complete r-package in M , that is (1) d(p i , p j ) ≥ 2r, for i = j; (2) ∪ 1≤i≤k B + pi (2r √ Λ) = M . For each p i ∈ P, the Dirichlet region of p i is defined by Property (1) and (2) imply that Lemma 4.1 yields Now it follows from Corollary 5.5 that for 0 < r < 1 2 √ Λ , Here, C 1 (n, Λ) is a positive constant depending only on n and Λ. One can easily check that Claim: For i = j, has measure zero (with respect to dµ). Note that d(p i , ·) and d(p j , ·) are smooth on int( This implies that the unique minimal geodesic from p i to x overlaps the unique minimal geodesic from p j to x. Since d( . Therefore, int(D i ∩D j ) = ∅ and D i ∩ D j has measure zero. Thus, , for i = m + 1, · · · , k.
If not, then (6.6) together with (6.4) yields which is a contradiction.
For Randers metrics, we have the following corollary.
Corollary 6.3. Let (M, α + β) be a n-dimensional closed Randers manifold. Then Proof. Since M is closed, there exists a point x ∈ M , such that β α (x) = b.
By [20,24], we can see that the upper bound for the uniform constant in Lemma 4.1 can be replaced by the lower bound for the S-curvature. Using the similar argument, one can show the following theorem. Theorem 6.4. Let (M, F, dµ) be a closed Finsler n-manifold with the Ricci curvature Ric ≥ −(n − 1)δ 2 , the S-curavture S ≥ (n − 1)η and the reversibility λ F ≤ λ. Then λ 1 (M ) ≤ C(n, λ, η) δ (M ) + 2 (M ) .
It follows from [18] that for the Busemann-Hausdorff measure, the S-curvature of a Berwald manifold always vanishes. Furthermore, we have the following Theorem 6.5. For the Holmes-Thompson measure, the S-curvature of a Berwald manifold also vanishes.
Proof. Let (M, F ) be a n-dimensional Berwald manifold and let γ y (t) be a unit speed geodesic withγ y (0) = y. Denote by P t the parallel transportation along γ y (t). Choose a basis {e i } of T γy(0) M . Then E i (t) := P t e i , 1 ≤ i ≤ n, is a basis of T γy(t) M . Let (y i ) (resp. (z i )) denote the corresponding coordinate system in T γy(0) M (resp. T γy(t) M ). Thus, z i • P t = y i .