EIGENVALUE COMPARISON ON BAKRY-EMERY MANIFOLDS

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1 EIGENVALUE COMPARISON ON BAKRY-EMERY MANIFOLDS BEN ANDREWS AND LEI NI 1. A lower bound for the first eigenvalue of the drift Laplacian Recall that (M, g, f), a triple consisting of a manifold M, a Riemannian metric g and a smooth function f, is called a gradient Ricci soliton if the Ricci curvature and the Hessian of f satisfy: (1.1) Rc ij + f ij = ag ij. It is called shrinking, steady, or expanding soliton if a > 0, a = 0 or a < 0 respectively. More generally (M, g, f) is called a Bakry-Emery manifold if the so-called Bakry-Emery Ricci tensor Rc ij + f ij ag ij for some a R. In this paper we apply the modulus of continuity estimates developed in [AC, AC, AC3] to give a different proof of an eigenvalue comparison estimate on Bakry-Emery manifol for the operator f ( ), f on strictly convex Ω M with diameter D and smooth boundary. This result was first proved in [BQ] (Theorem 14), which serves as a generalization to the earlier works of Payne- Weinberger[PW], Li-Yau[LY] and Zhong-Yang [ZY]. An more recent result of this kind was obtained in [FS]. Here in fact in Theorem 1. we extend a comparison theorem of [AC3] on the modulus of continuity to manifol with lower bound on the Bakry-Emery Ricci tensor. This new result gives sharp modulus of continuity comparison between the solution to t f and the corresponding sup-soultion to a heat equation on a certain interval. This implies the eigenvalue comparison result of Bakry-Qian, Theorem 1.1, since first eigenvalue determines the rate of convergence to equilibrium. As in [FS] applying to the soliton setting, this implies a lower diameter estimate for nontrivial gradient shrinking solitons (which improves [FS]). We remark here that the eigenvalue estimate we obtain is sharp for (M, g, f) satisfying the Bakry-Emery-Ricci lower bound Rc ij +f ij ag ij, but presumably is not so for Ricci solitons where the Bakry-Emery-Ricci tensor is constant, and so we expect that our diameter bound is also not sharp. We discuss the sharpness of the eigenvalue inequality in Section. Before we state the result, we define a corresponding 1-dimensional eigenvalue problem. On [ D, D ] we consider the functionals F(ψ) = D D e a s (ψ ), and R(ψ) = F(ψ) e a s ψ, namely the Dirichlet energy with weight e a s and its Rayleigh quotient. The associated elliptic operator is L a = d a s d. Let λ a,d be the first non-zero Neumann eigenvalue of L a, which is the minimum of R among W 1, -functions with zero average. Theorem 1.1 (Bakry-Qian). Let Ω be a compact manifold M, or a bounded strictly convex domain inside a complete manifold M, satisfying that Rc ij + f ij a g ij. Assume that D is Partially supported by Discovery Projects grant DP of the Australian Research Council. Partially supported by NSF grant DMS

2 BEN ANDREWS AND LEI NI the diameter of Ω. Then the first non-zero Neumann eigenvalue λ 1 of the operator f is at least λ a,d. Proof. The key is to adapt Theorem.1 of [AC3] to this setting. Recall that ω is a modulus ). of continuity for a function f on M if for all x and y in M, f(y) f(x) ω Theorem 1.. Let v(x, t) be a solution to (1.) v t = v X, v ( dm (x,y) with X = f. Assume also v(x, t) satisfies the Neumann boundary condition. Suppose that v(x, 0) has a modulus of continuity φ 0 (s) : [0, D ] R with φ 0(0) = 0 and φ 0 > 0 on [0, D ]. Assume further that there exists a function φ(s, t) : [0, D ] R + R such that (i) φ(s, 0) = φ 0 (s) on [0, D/]; (ii) φ t φ a s φ on [0, D/] R + ; (iii) φ (s, t) > 0 on [0, D ]; (iv) φ(0, t) 0 for each t 0. Here φ = s φ(s, t), φ = s φ(s, t). Then φ(s, t) is a modulus of the continuity of v(x, t) for all t > 0. Proof. The proof of Theorem 1. is a modification of the argument of Theorem.1 in [AC3]. Precisely, consider ( ) r(x, y) O ϵ (x, y, t) v(y, t) v(x, t) φ, t ϵe t. It suffices to show that O ϵ 0 for any ϵ > 0. The proof of this claim is via reductio ad absurdum. Assume that there exists (x 0, y 0, t 0 ) such that O ϵ (,, t) = 0 for the first time. Namely O ϵ (x, y, t) achieves its maximum over Ω Ω [0, t 0 ] at (x 0, y 0, t 0 ). The strictly convexity, the Neumann boundary condition satisfied by v(x, t), and the positivity of φ rule out the possibility that the maximum can be attained at (x 0, y 0 ) (Ω Ω). For the interior pair (x 0, y 0 ) where the maximum of O is attained, pick a frame {e i } as before at x 0 and parallel translate it along a minimizing geodesic γ(s) : [0, d] M joining x 0 with y 0. Still denote it by {e i }. Let {E i } be the frame at (x 0, y 0 ) (in T (x0,y 0)Ω Ω) defined as E i = e i e i for 1 i n 1 and E n = e n ( e n ). Direct calculations show that at (x 0, y 0, t 0 ), n t O ϵ (x, y, t) = ( f(y), γ f(x), γ ) φ j=1 E j E j +φ n 1 E i E i r(x, y) φ t + φ ϵe t. Here we have used the first variation O(,, t 0 ) = 0 at (x 0, y 0 ) which implies the identities i=1 ( v)(y, t 0 ) = φ γ (d), ( v)(x, t 0 ) = φ γ (0). Now choose the variational vector field V i (s) = e i (s), the parallel transport of e i along γ(s), the second variation computation gives that (1.3) n 1 E ie i r(x, y) i=1 d 0 Rc(γ, γ ).

3 MODULUS OF CONTINUITY 3 Hence at (x 0, y 0, t 0 ) we have that n t d O(x, y, t) φ ( f + Rc)(γ, γ ) φ t + φ ϵe t j=1 E je j < 0. 0 φ ar(x, y) φ t + φ ϵe t Here we have used d = r(x, y) and s = r(x,y). This contradicts with that at (x 0, y 0, t 0 ) since it is the first time O(x, y, t) = 0, O t 0, (x0,y 0,t 0 ) E i E i O (x0,y 0,t 0 ) 0. The above argument works well as long as (x 0, y 0 ) is not conjugate to each other along the minimizing geodesic γ(s) since we need this condition in establishing (1.3). For the case (x 0, y 0 ) is a pair of points conjugate to each other, we can evoke a trick similar to the one of Calabi to work through the argument. Let e i (s), with 1 i n 1 and 0 s d, be the parallel transport of e i along a minimizing geodesic γ(s) joining x 0 = γ(0), y 0 = γ(d). Let γ(η 1, η,, η n 1, s) = exp γ(s) (η 1 e 1 (s) + η n 1 e n 1 (s)). Let L(η 1,, η n 1 ) be the arc length of γ(η 1,, η i, ), which is greater or equal to the distance between γ(η 1,, η n 1, 0) and γ(η 1,, η n 1, d). Let Õ(η 1,, η n 1 ) v(γ(η 1,, η n 1, d), t 0 ) v(γ(η 1,, η n 1, 0), t 0 ) ( ) L(η1,, η n 1 ) φ, t ϵe t0. Then in view of the monotonicity of φ(s, t) in s, Õ(0,, 0) = O(x 0, y 0, t 0 ) O(γ(η 1,, η n 1, 0), γ(η 1,, η n 1, d), t 0 ) Õ(η 1,, η n 1 ). In the previous computation if we replace O by Õ, we can still apply the maximum principle to get a contradiction verbatim. To prove Theorem 1.1, let ω(s) be the first non-constant eigenfunction for L a, which can be chosen to be positive on (0, D ). To apply Theorem 1. as in [N] we consider (s), ωd the eigenfunction on [ D, D ] with the corresponding eigenvalue λ a,d. Let φ(x, t) = Ce λ a,d t ω D (s). Let w(x) be the first non-constant eigenfunction of f and let v(x, t) = e λ 1t w(x). Since ω D (s) is an odd function (by adding an eigenfunction ψ(s) with ψ( s) one can always obtain one), we do have φ(0, t) = 0. The possibility of choosing ( ω D ) (s) > 0 on [0, D ] can be proved as follows. By the uniqueness, we have that ) (0) > 0. It ( ωd suffices to show that ( ω D ) (s) > 0 for s (0, D ). For this, observe that e a s ( ω D ) (s) is non-increasing in s and vanishes at s = D. Finally Theorem 1. implies that for sufficient large C, φ(s, t) is a modulus of continuity of v(x, t) for all t > 0. Hence λ 1 λ a,d. The claimed result of Theorem 1.1 follows by letting D D.

4 4 BEN ANDREWS AND LEI NI. Sharpness of the lower bound In this section we show that (for n 3 for any a or for n for a 0) the lower bound λ 1 λ a given in Theorem 1.1 is sharp: Precisely, for each ε > 0 we construct a Bakry-Emery manifold (M, g, f) with diameter D and λ 1 < λ a,d + ε. We will construct a smooth manifold M which is approximately a thin cylinder with hemispherical caps at each end. Let γ be the curve in R with curvature k given as function of arc length as follows for suitably small positive r and δ > 0 small compared to r: 1 r, (.1) k(s) = φ ( s πr δ ) 1 r, s [ 0, πr δ] ; s [ πr πr δ, + δ] ; ], 0, s [ πr + δ, D extended to be even under reflection in both s = 0 and s = D/. This correspon to a pair of line segments parallel to the x axis, capped by semicircles of radius r and smoothed at the joins. We write the corresponding embedding (x(s), y(s)). Here φ is a smooth nonincreasing function with φ(s) = 1 for s 1, φ(s) = 0 for s 1, and satisfying φ(s) + φ( s) = 1. We choose the point corresponding to s = 0 to have y(0) = 0 and y (0) = 1. The manifold M will then be the hypersurface of rotation in R n+1 given by {(x(s), y(s)z) : s R, z S n 1 }. On M we choose the function f to be a function of s only, such that a ( ) [ 1 D πr, s 0, πr ( ) δ] ; (.) f s πr = φ δ a ( ) ( ( )) 1 D s πr πr + 1 φ δ a, s [ πr πr δ, + δ] ; a, s [ πr + δ, ] D, with f (0) = 0 (the value of f(0) is immaterial). Note that this choice gives f (D/) = 0. We extend f to be even under reflection in s = 0 and s = D/. With these choices we compute the Bakry-Emery-Ricci tensor and verify that the eigenvalues are no less than a for suitable choice of r. The eigenvalues of the second fundamental form are k(s) (in the s direction) and 1 (y ) y 1 (y ) in the orthogonal directions. Therefore the 1 (y ) in the s direction, and k y +(n ) 1 (y ) y Ricci tensor has eigenvalues (n 1)k y in the orthogonal directions. We can also compute the eigenvalues of the Hessian of f: The curves of fixed z in M are geodesics parametrized by s, the the Hessian in this direction is just f as given above. Since f depen only on s we also have that f( s, e i ) = 0 for e i tangent to S n 1, and f(e i, e j ) = y y f δ ij. The identities y(s) = s 0 cos (θ(τ)) dτ and y (s) = cos (θ(s)) where θ(s) = s k(τ) dτ 0 applied to (.1) imply that y = { r sin(s/r), s [ 0, πr δ] ; cos(s/r), s [ 0, πr r(1 + o(δ)), s [ πr δ, ] D y δ] ; = o(δ), s [ πr πr, δ, + δ] ; 0, s [ πr + δ, ] D, as δ approaches zero. This gives the following expressions for the Bakry-Emery Ricci tensor Rc f = Rc + f: a + n 1 r ad πr, s [ ( ) 0, πr δ] ; πr ( Rc f ( s, s ) = a + φ n 1 ) δ r (1 + o(δ)) ad πr s [ πr πr δ, + δ] ; a, s [ πr + δ, ] D,

5 MODULUS OF CONTINUITY 5 and n 1 r + as(1 D πr ) r tan(s/r), s [ 0, πr δ] ; Rc f (e, e) = n r + o(δ), s [ πr πr δ, + δ] ; n r (1 + o(δ)) s [ πr + δ, ] D, while Rc f ( s, e) = 0, for any unit vector e tangent to S n 1. In particular we have Rc f ag for sufficiently small r and δ for any a R if n 3, and for a < 0 if n =. Note also that the diameter of the manifold M is D(1 + o(δ)). Having constructed the manifold M, we now prove that for this example the first nontrivial eigenvalue λ 1 of f can be made as close as desired to λ a,d by choosing r and δ small. Theorem 1.1 gives the upper bound λ 1 λ a,d(1+o(δ)) = λ a,d + o(δ). To prove an upper bound we can simply find a suitable test function to substitute into the Rayleigh quotient which defines λ 1 : We set { πr w(s D/), + δ s D ( πr + δ); ψ(s, z) = w ( D πr δ), 0 s πr + δ, and D πr δ s D. where w is the solution of w asw + λ a,d πr δ w = 0 with w(0) = 0 and w ( D 0 and w (0) = 1. This choice gives R(ψ) = λ a,d πr δ { s D/ D π r δ } ψ e f dv ol(g) { s D/ D/} ψ e f dv ol(g) πr λ a,d πr δ. δ) = It follows that λ 1 λ a,d as r and δ approach zero, proving the sharpness of the lower bound in Theorem 1.1. Remark.1. If we allow manifol with boundary the construction is rather simpler: Simply take a cylinder rs n [ D/, D/] for small r, with quadratic potential f = a s, and substitute the test function ψ(z, s) = w(s) defined above. 3. A linear lower bound Here we use the notation from Section 1. Concerning the lower estimate of λ a,d, at least for a 0, if we apply Theorem 1. to the trivial case M = R with φ(x, t) = e ( π D ) t sin( π D s), and letting D D we get λ a,d π D. For a more precise estimate, observe that y = ( ω D ) satisfies ( (3.1) e a s y ) = (a λa,d )e a s y. This immediately implies that λ a,d a+µ a,d, where µ a,d is the Dirichlet eigenvalue of the operator d a s d. On the other hand, if y is the first Dirichlet function on [ D/, D/], which is an even function, let ω = s y. Direct calculation also shows that ω is a Neumann 0 eigenfunction, hence we have the upper bound λ a,d a+µ a,d. This establishes the identity: (3.) λa,d = a + µ a,d. The following result and its consequence improve the main results of [FS]. It applies to a compact Riemannian manifold (M, g) satisfying Rc (n 1)K for some K > 0, and concludes that λ 1 (M) n 1 K + π D hol with D being the diameter of the manifold M. This improves earlier works of [L], [Y], etc.

6 6 BEN ANDREWS AND LEI NI Proposition 3.1. When a > 0, the first nonzero Neumann eigenvalue λ a,d is bounded from below by a + π D. Moveover, the a in the lower bound is the largest possible. In particular λ 1 (Ω), with Ω being a convex domain in any Riemannian manifold with Rc ij + f ij ag ij, is bounded from below by a + π D. Proof. We present two proofs for the lower bound via two types of normalization. The first is to reduce the problem to finding the first nontrivial Neumann eigenvalue λ, a D for d the Hermite equation: s d on the interval [ a D Namely the first one normalizes the a., a D ] since λ a,d = a λ, a D. For the operator d s d D on interval [, D ] the equation (3.) implies that λ, D = + µ, D, with µ being the first Dirichlet eigenvalue. Now we may introduce the the transformation w = e s φ. Direct calculation shows that φ is the first Dirichlet eigenfunction of d s d if and only if w is the eigenfunction of the harmonic oscillator: d w s w = (µ + 1)w with w vanishing on the boundary. By Corollary 6.4 of [N] we have that µ + 1 π D. Combining them together we have that λ, D 1 + π. Letting D = a D D we get the claimed lower bound for λ a,d. The second one is via the normalization on D. Precisely by the change of variable, D 4 µ D d a,d = µ D 4 a,. Now let ā = 4 a and consider the operator ās d. The transformation φ e ā 4 s φ relates µā, with the first eigenvalue µ(b) of the harmonic oscillator d bs with b = ā 4 via the equation: (3.3) µ(b) = µā, + ā. For the harmonic oscillator d bs on ( 1, 1), µ(b) = inf ψ =1,ψ(±1)=0 1 1 (ψ ) + bs ψ. Note that the functional is increasing and concave in b. Hence we have that π ( 4 = µ(0) µ(b) µ(0) + µ (0)b = π ) π b. Together with (3.3), µ a,d = 4 D µā, it implies that λ a,d = µ a,d + a satisfies a + π D λ a,d a ( + π π D ) a D π. Corollary 3.1. If (M, g, f) is a nontrivial (namely with nonconstant f) gradient shrinking soliton satisfying (1.1) with a > 0. Then Diameter(M, g) 3a π.

7 MODULUS OF CONTINUITY 7 Proof. The result follows from the above lower estimate on the first Neumann eigenvalue, applying to the case that Ω = M, and the observation, Lemma.1 of [FS], that a is an eigenvalue of the operator f, ( ). This result clearly is not sharp. See [BL] for a related upper bound on the diameter. A better eigenvalue lower bound (and hence a better diameter lower bound) will follow from a better understanding of the first Dirichlet eigenvalue of the harmonic oscillator. We investigate this in the next section. 4. The harmonic oscillator on bounded symmetric intervals In this section we will investigate the sharp lower bound given by the eigenvalue of the one-dimensional harmonic oscillator on a bounded symmetric interval: Recall from section 3 that the first Neumann eigenvalue λ,d is equal to ˆλ 1,D + 1, where ˆλ b,d is defined by the existence of a solution of the eigenvalue problem w + (ˆλb,D bs ) w = 0, s [ D/, D/]; w(d/) = w( D/) = 0; w(x) > 0, s ( D/, D/). The solution of the ordinary differential equation w s w + λw = 0 (with w (0) = 0), which is also called Weber s equation, can be written in terms of confluent hypergeometric functions: We have ( 1 w(s) = e s U 4 λ 8, 1 ), s where U is the confluent ( hypergeometric ) function of the first kind. Thus ˆλ 1,D is the first root 1 of the equation U 4 λ 8, 1, D = 0. Since U is strictly monotone in the first argument, the solution is an analytic function of D. Noting that ˆλ 1,D = π ˆλ D D 4 π4,π, we use a perturbation argument to compute the Taylor expansion for ˆλ b,π as a function of b = a 4 about b = 0 (this provides an expansion for ˆλ 1,D about D = 0). That is, we consider the solution u of the eigenvalue problem u + ( λ bs ) u = 0, s [ π/, π/]; u(π/) = u( π/) = 0; u(s) > 0, s ( π/, π/). The solution for b = 0 is of course given by u(s) = cos(s). The perturbation expansion produces a solution of the form 3k u(s, b) = b k ( αk,j s j cos s + β k,j s k sin s ), k=0 j=0 with λ = k=0 λ kb k. This expansion is unique provided we specify that u is even, α 0,0 = 1, β 0,1 = 0, and α k,0 = β k,0 = 0 for k > 0. The first few terms in the expansion for λ are given by ( π ˆλ b,π = ) ( π 4 b π ) ( π b ( π π π π π π ) b 4 + O(b 5 ). ) b 3

8 8 BEN ANDREWS AND LEI NI Figure 1. The eigenvalue ˆλ b,π for Weber s equation y + (λ bs )y = 0, y(±π/) = 0 [solid curve]; shown also are the lower boun ˆλ 1 and ˆλ b [dashed curves]. We note that there is also a useful lower bound for ˆλ b,π, which we can arrive at as follows: The inclusion of [ D/, D/] in R implies ˆλ 1,D lim d ˆλ1,d = 1, with eigenfunction u(s) = e s /. Therefore we also have ˆλ b,π = bˆλ 1,πb 1/4 b. This translates to an estimate for the drift eigenvalue λ a,π appearing in Theorem 1.1: We have λ a,π = a + ˆλ a /4,π, giving the following Taylor expansion: λ a,π = 1 + a ( π ) ( a π π ) ( a 4 π π π 3 11 ) a ( π π π π ) a O(a10 ).

9 MODULUS OF CONTINUITY 9 In particular the lower bound ˆλ 1 translates to λ 1 + a/, and the lower bound ˆλ b translates to λ a/ + a /4 = a. Finally, by scaling we obtain the following: λ a,d = π D + a ( π ) D a ( π 4 4π π ) D 6 a π 6 ( π π π 3 11 ) D 10 a π 1 0 ( π π π π ) D 14 a π 14 + O(D 18 a 10 ) as D a 0. An interesting consequence of the Taylor expansion (combined with the fact that the estimate λ 1 λ a,d is sharp as proved in section ) is the following: Proposition 4.1. The constant a/ in the lower bound λ 1 π D + a This follows from the Taylor expansion for small values of ad. is the largest possible. We note that the sharp diameter bound (given by the value of a where the dotted line λ = a intersects with the solid curve in Figure ) is not dramatically different from the one given in Corollary 3.1 (where the dotted line intersects the dashed line λ = 1 + a/). Since the eigenvalue estimate λ 1 λ a,d appears from the examples in section to be sharp only in situations which are far from gradient solitons, we expect that neither of these diameter boun is close to the sharp lower diameter bound for a nontrivial gradient Ricci soliton. Acknowledgments. We thank Randy Bank, Chongshen Cao, Bruce Driver, Elliot Lieb for helpful discussions, the referee for helpful comments, particularly for the second proof of Proposition 3.1. References [AC1] Ben Andrews and Julie Clutterbuck, Lipschitz boun for solutions of quasilinear parabolic equations in one space variable, J. Differential Equations 46 (009), no. 11, [AC], Time-interior gradient estimates for quasilinear parabolic equations, Indiana Univ. Math. J. 58 (009), no. 1, [AC3], Proof of the fundamental gap conjecture, J. Amer. Math. Soc. 4 (011), no. 3, [BL] Dominique Bakry and Michel Ledoux, Sobolev inequalities and Myers diameter theorem for an abstract Markov generator, Duke Math. J. 85 (1996), no. 1, [BQ] Dominique Bakry and Zhongmin Qian, Some new results on eigenvectors via dimension, diameter, and Ricci flow, Adv. Math. 155 (000), [FS] A Futaki and Y. Sano, Lower diameter boun for compact shrinking Ricci solitons, available at arxiv: [LY] Peter Li and Shing Tung Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp [L] Jun Ling, The first eigenvalue of a closed manifold with positive Ricci curvature, Proc. Amer. Math. Soc. 134 (006), no. 10, [N] Lei Ni, Estimates on the modulus of expansion for vector fiel solving nonlinear equations, available at arxiv: [PW] L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal. 5 (1960), [Y] DaGang Yang, Lower bound estimates of the first eigenvalue for compact manifol with positive Ricci curvature, Pacific J. Math. 190 (1999), no., [ZY] Jia Qing Zhong and Hong Cang Yang, On the estimate of the first eigenvalue of a compact Riemannian manifold, Sci. Sinica Ser. A 7 (1984), no. 1,

10 10 BEN ANDREWS AND LEI NI Figure. The eigenvalue λ a,π for the drift Laplacian equation y asy + λy = 0, y (±π/) = 0 [solid curve]; shown also are the lower boun λ 1 + a and λ a [dashed lines], and the line λ = a corresponding to non-einstein gradient Ricci solitons [dotted line]. Mathematical Sciences Institute, Australia National University; Mathematical Sciences Center, Tsinghua University; and Morningside Center for Mathematics, Chinese Academy of Sciences. address: Ben.Andrews@anu.edu.au Department of Mathematics, University of California at San Diego, La Jolla, CA 9093, USA address: lni@math.ucsd.edu