Spectral bounds for the torsion function M. van den Berg School of Mathematics, University of Bristol University Walk, Bristol BS8 1TW United Kingdom 7 mamvdb@bristol.ac.uk 1 0 30 March 2017 2 r a M 0 3 ] P S . h t a m Abstract [ Let Ω be an open set in Euclidean space Rm, m = 2,3,..., and let v Ω 2 denote the torsion function for Ω. It is known that vΩ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting v in L2(Ω), denoted by λ(Ω), is bounded away from 0. It is shown that 2 7 the previously obtained bound (cid:107)vΩ(cid:107)L∞(Ω)λ(Ω) ≥ 1 is sharp: for m ∈ 1 {2,3,...}, and any (cid:15) > 0 we construct an open, bounded and connected 2 set Ω(cid:15) ⊂ Rm such that (cid:107)vΩ(cid:15)(cid:107)L∞(Ω(cid:15))λ(Ω(cid:15)) < 1+(cid:15). An upper bound for 0 v is obtained for planar, convex sets in Euclidean space M =R2, which Ω . is sharp in the limit of elongation. For a complete, non-compact, m- 1 dimensional Riemannian manifold M with non-negative Ricci curvature, 0 7 and without boundary it is shown that vΩ is bounded if and only if the 1 bottomofthespectrumoftheDirichlet-Laplace-Beltramioperatoracting : in L2(Ω) is bounded away from 0. v i X r a Keywords: Torsion function; Dirichlet Laplacian; Riemannian manifold; non- negative Ricci curvature. AMS 2000 subject classifications. 58J32; 58J35; 35K20. Acknowledgement. MvdBacknowledgessupportbyTheLeverhulmeTrustthrough International Network Grant Laplacians, Random Walks, Bose Gas, Quantum Spin Systems. 1 1 Introduction LetΩbeanopensetinRm,andlet∆betheLaplaceoperatoractinginL2(Rm). Let(B(s),s≥0,P ,x∈Rm)beBrownianmotiononRm withgenerator∆. For x x ∈ Ω we denote the first exit time, and expected lifetime of Brownian motion by T =inf{s≥0:B(s)∈/ Ω}, Ω and v (x)=E [T ], x∈Ω, (1) Ω x Ω respectively, where E denotes the expectation associated with P . Then v is x x Ω the torsion function for Ω, i.e. the unique solution of −∆v =1, v ∈H1(Ω). (2) 0 The bottom of the spectrum of the Dirichlet Laplacian acting in L2(Ω) is denoted by (cid:90) |Dϕ|2 λ(Ω)= inf Ω(cid:90) . (3) ϕ∈H01(Ω)\{0} ϕ2 Ω Itwasshownin[1],[2]that(cid:107)v (cid:107) isfiniteifandonlyifλ(Ω)>0. Moreover, Ω L∞(Ω) if λ(Ω)>0, then λ(Ω)−1 ≤(cid:107)v (cid:107) ≤(4+3mlog2)λ(Ω)−1. (4) Ω L∞(Ω) The upper bound in (4) was subsequently improved (see [11]) to 1 (cid:107)v (cid:107) ≤ (m+cm1/2+8)λ(Ω)−1, Ω L∞(Ω) 8 where c=(5(4+log2))1/2. InTheorem1belowweshowthatthecoefficient1ofλ(Ω)−1 intheleft-hand side of (4) is sharp. Theorem 1. For m∈{2,3,...}, and any (cid:15)>0 there exists an open, bounded, and connected set Ω ⊂Rm such that (cid:15) (cid:107)v (cid:107) λ(Ω )<1+(cid:15). (5) Ω(cid:15) L∞(Ω(cid:15)) (cid:15) The set Ω is constructed explicitly in the proof of Theorem 1. (cid:15) It has been shown by L. E. Payne (see (3.12) in [9]) that for any convex, open Ω⊂Rm for which λ(Ω)>0, π2 (cid:107)v (cid:107) λ(Ω)≥ , (6) Ω L∞(Ω) 8 with equality if Ω is a slab, i.e. the connected, open set, bounded by two parallel (m − 1)-dimensional hyperplanes. Theorem 2 below shows that for any sufficiently elongated, convex, planar set (not just an elongated rectangle) (cid:107)v (cid:107) λ(Ω) is approximately equal to π2. We denote the width and the Ω L∞(Ω) 8 diameter of a bounded open set Ω by w(Ω) (i.e. the minimal distance of two parallel lines supporting Ω), and diam(Ω) = sup{|x − y| : x ∈ Ω, y ∈ Ω} respectively. 2 Theorem 2. If Ω is a bounded, planar, open, convex set with width w(Ω), and diameter diam(Ω), then π2 (cid:32) (cid:18) w(Ω) (cid:19)2/3(cid:33) (cid:107)v (cid:107) λ(Ω)≤ 1+7·32/3 . Ω L∞(Ω) 8 diam(Ω) In the Riemannian manifold setting we denote the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator by (3). We have the following. Theorem 3. Let M be a complete, non-compact, m-dimensional Riemannian manifold,withoutboundary,andwithnon-negativeRiccicurvature. Thereexists K <∞, depending on M only, such that if Ω⊂M is open, and λ(Ω)>0, then λ(Ω)−1 ≤(cid:107)v (cid:107) ≤2(3m+8)/4·3m/2K2λ(Ω)−1, (7) Ω L∞(Ω) where K is the constant in the Li-Yau inequality in (35) below. The proofs of Theorems 1, 2, and 3 will be given in Sections 2, 3 and 4 respectively. Belowwerecallsomebasicfactsontheconnectionbetweentorsionfunction and heat kernel. It is well known (see [5], [6], [7]) that the heat equation ∂u(x;t) ∆u(x;t)= , x∈M, t>0, ∂t has a unique, minimal, positive fundamental solution p (x,y;t), where x∈M, M y ∈M,t>0. Thissolution,theheatkernelforM,issymmetricinx,y,strictly positive, jointly smooth in x,y ∈ M and t > 0, and it satisfies the semigroup property (cid:90) p (x,y;s+t)= dz p (x,z;s)p (z,y;t), M M M M for all x,y ∈ M and t,s > 0, where dz is the Riemannian measure on M. See, for example, [10] for details. If Ω is an open subset of M, then we denote the unique, minimal, positive fundamental solution of the heat equation on Ω by p (x,y;t), where x∈Ω,y ∈Ω,t>0. This Dirichlet heat kernel satisfies, Ω p (x,y;t)≤p (x,y;t), x∈Ω,y ∈Ω,t>0. Ω M Define u :Ω×(0,∞)(cid:55)→R by Ω (cid:90) u (x;t)= dyp (x,y;t). Ω Ω Ω Then, (cid:90) u (x;t)= dyp (x,y;t)=P [T >t], Ω Ω x Ω Ω and by (1) (cid:90) ∞ (cid:90) ∞ (cid:90) v (x)= dtP [T >t]= dt dyp (x,y;t). (8) Ω x Ω Ω 0 0 Ω It is straightforward to verify that v as in (8) satisfies (2). Ω 3 2 Proof of Theorem 1 We introduce the following notation. Let C = (−L,L)m/2 be the open cube L 2 2 with measure Lm, and delete from C , Nm closed balls with radii δ, where L each ball B(c ;δ) is positioned at the centre of an open cube Q with measure i i (L/N)m. These open cubes are pairwise disjoint, and contained in C . Let L 0<δ < L , and put 2N Ω =C −∪NmB(c ;δ). δ,N,L L i=1 i Below we will show that for any (cid:15)>0 we can choose δ,N such that (cid:107)v (cid:107) λ(Ω )<1+(cid:15). Ωδ,N,L L∞(Ωδ,N,L) δ,N,L Figure 1: Ω with m=2,N =10,δ = L . δ,N,L 8N In Lemma 4 below we show that λ(Ω ) is approximately equal to the δ,N,L first eigenvalue, µ , of the Laplacian with Neumann boundary con- 1,B(0;δ),L/N ditions on ∂C , and with Dirichlet boundary conditions on ∂B(0;δ). The L/N requirement µ not being too small stems from the fact that the ap- 1,B(0;δ),L/N proximation of replacing the Neumann boundary conditions on C is a surface L effect which should not dominate the leading term µ . 1,B(0;δ),L/N Lemma 4. If δ ≤ L , N ≥10, and N ≤µ , then 4N L2 1,B(0;δ),L/N (cid:18)5(cid:19)m(cid:18)N 1 (cid:19) λ(Ω )≤µ +32m + µ δ,N,L 1,B(0;δ),L/N 4 L2 N1/2 1,B(0;δ),L/N Proof. Let ϕ be the first eigenfunction (positive) corresponding to 1,B(0;δ),L/N µ , and normalised in L2(C − B(0;δ)). In order to prove the 1,B(0;δ),L/N L/N lemma we construct a test function by periodically extending ϕ to 1,B(0;δ),L/N all cubes Q ,...Q of Ω . We denote this periodic extension by f. We 1 Nm δ,N,L define C =C . L,N L(1−2) N 4 SoC isthesub-cubeofC withtheouterlayerofcubesofsizeL/N removed. L,N L Let (cid:18) (cid:19) dist(x,C ) f˜= 1− L,N f. L/(4N) + Then f˜∈H1(Ω ), and 0 δ,N,L (cid:90) (cid:107)f˜(cid:107) ≥ f2 =(N −2)m, (9) L2(Ωδ,N,L) CL,N since f restricted to any of the cubes Q in Ω is normalised. Furthermore i δ,N,L (cid:18) dist(x,C )(cid:19)2 (cid:18)(cid:18)4N(cid:19)2 8N (cid:19) |Df˜|2 ≤ 1− L,N |Df|2+1 f2+ f|Df| L/(4N) CL−CL,N L L (cid:18)4N(cid:19)2 8N ≤|Df|2+ 1 f2+ 1 f|Df|. L CL−CL,N L CL−CL,N Hence (cid:90) (cid:90) (cid:18)4N(cid:19)2(cid:90) |Df˜|2 ≤ |Df|2+ f2 L Ωδ,N,L Ωδ,N,L CL−CL,N 8N(cid:18)(cid:90) (cid:19)1/2(cid:18)(cid:90) (cid:19)1/2 + |Df|2 f2 L CL−CL,N CL−CL,N =Nmµ +(cid:0)Nm−(N −2)m(cid:1)(cid:18)(cid:18)4N(cid:19)2+ 8N(cid:0)µ (cid:1)1/2(cid:19) 1,B(0;δ),L/N L L 1,B(0;δ),L/N ≤Nmµ +(cid:0)Nm−(N −2)m(cid:1)(cid:18)(cid:18)4N(cid:19)2+8N1/2µ (cid:19), 1,B(0;δ),L/N L 1,B(0;δ),L/N (10) wherewehaveusedthelasthypothesisinthelemma. By(9),(10),theRayleigh- Ritz variational formula, and the hypothesis N ≥10, λ(Ω )≤µ δ,N,L 1,B(0;δ),L/N + Nm−(N −2)m(cid:18)(cid:18)4N(cid:19)2+(cid:0)8N1/2+1(cid:1)µ (cid:19) (N −2)m L 1,B(0;δ),L/N (cid:18)5(cid:19)m(cid:18)N 1 (cid:19) ≤µ +32m + µ . (11) 1,B(0;δ),L/N 4 L2 N1/2 1,B(0;δ),L/N To obtain an upper bound for (cid:107)v (cid:107) , we change the Dirichlet Ωδ,N,L L∞(Ωδ,N,L) boundary conditions on ∂C to Neumann boundary conditions. This increases L the corresponding heat kernel, torsion function, and L∞ norm respectively. By periodicity, we have that (cid:107)v (cid:107) ≤(cid:107)v˜ (cid:107) , (12) Ωδ,N,L L∞(Ωδ,N,L) CL/N−B(0;δ) L∞(CL/N−B(0;δ)) where v˜ is the torsion function with Neumann boundary conditions CL/N−B(0;δ) on∂C ,andDirichletboundaryconditionson∂B(0;δ). Denotethespectrum L/N 5 of the corresponding Laplacian by {µ := µ ,j = 1,2,...}, and let j j,B(0;δ),L/N {ϕ :=ϕ ,j =1,2,...} denote a corresponding orthonormal basis of j 1,B(0;δ),L/N eigenfunctions. We denote by π (x,y;t),x ∈ C −B(0;δ),y ∈ C − δ,N/L L/N L/N B(0;δ),t>0 the corresponding heat kernel. Then ∞ (cid:88) π (x,y;t)= e−tµjϕ (x)ϕ (y), (13) δ,N/L j j j=1 and v˜ (x) CL/N−B(0;δ) (cid:90) ∞ (cid:90) (cid:18)ϕ (y) ϕ (y)(cid:19) = dt dy π (x,y;t) 1 +1− 1 δ,N/L (cid:107)ϕ (cid:107) (cid:107)ϕ (cid:107) 0 CL/N−B(0;δ) 1 1 1 ϕ (x) (cid:90) ∞ (cid:90) (cid:18) ϕ (y)(cid:19) = 1 + dt dy π (x,y;t) 1− 1 µ (cid:107)ϕ (cid:107) δ,N/L (cid:107)ϕ (cid:107) 1 1 0 CL/N−B(0;δ) 1 1 (cid:90) T (cid:90) ≤ + dt dy π (x,y;t) µ δ,N/L 1 0 CL/N−B(0;δ) (cid:90) ∞ (cid:90) (cid:18) ϕ (y)(cid:19) + dt dy π (x,y;t) 1− 1 δ,N/L (cid:107)ϕ (cid:107) T CL/N−B(0;δ) 1 1 (cid:90) ∞ (cid:90) (cid:18) ϕ (y)(cid:19) ≤ +T + dt dy π (x,y;t) 1− 1 , (14) µ δ,N/L (cid:107)ϕ (cid:107) 1 T CL/N−B(0;δ) 1 where (cid:107)ϕ (cid:107) = (cid:107)ϕ (cid:107) . By (13), we have that the third term in 1 1 L∞(CL/N−B(0;δ)) the right-hand side of (14) equals (cid:88)∞ µ−1e−Tµjϕ (x)(cid:90) dy ϕ (y)(cid:18)1− ϕ1(y)(cid:19). (15) j j j (cid:107)ϕ (cid:107) j=1 CL/N−B(0;δ) 1 The term with j =1 in (15) is bounded from above by (cid:90) (cid:18) ϕ (cid:19) µ−1(cid:107)ϕ (cid:107) (cid:107)ϕ (cid:107) 1− 1 1 1 1 (cid:107)ϕ (cid:107) CL/N−B(0;δ) 1 (cid:90) =µ−1(cid:107)ϕ (cid:107) (cid:0)(cid:107)ϕ (cid:107)−ϕ (cid:1) 1 1 1 1 CL/N−B(0;δ) (cid:18) (cid:18)L(cid:19)m (cid:19) ≤µ−1 (cid:107)ϕ (cid:107)2 −1 , 1 1 N whereweusedthefactthat1=(cid:82) ϕ2 ≤(cid:107)ϕ (cid:107)(cid:82) ϕ . Itwas CL/N−B(0;δ) 1 1 CL/N−B(0;δ) 1 shown on p.586, lines -3,-4, in [3] (with appropriate adjustment in notation) that (cid:18)N(cid:19)m(cid:18) mL2 (cid:19)−1 (cid:107)ϕ (cid:107)2 ≤ 1−sµ − ,s≥0, 1 L 1 3esN2 providedthelasttermintheroundbracketsisnon-negative. Theoptimalchoice for s gives that (cid:18)N(cid:19)m(cid:18) (4mµ )1/2L(cid:19)−1 3eN2 (cid:107)ϕ (cid:107)2 ≤ 1− 1 , µ < . 1 L (3e)1/2N 1 4mL2 6 By further restricting the range for µ , we have that the first term with j = 1 1 in (15) is then bounded from above by 2L(mµ /(3eN2))1/2 (2m)1/2L 3eN2 µ−1 1 ≤ , µ ≤ . (16) 1 1−2L(mµ1/(3eN2))1/2 µ1/2N 1 16mL2 1 The terms with j ≥2 in (15) give, by Cauchy-Schwarz for both the series in j, and the integral over C −B(0;δ), a contribution L/N (cid:12)(cid:12)(cid:12)(cid:12)(cid:88)∞ µ−j1e−Tµjϕj(x)(cid:90) ϕj(cid:18)1− (cid:107)ϕϕ1(cid:107)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) j=2 CL/N−B(0;δ) 1 ∞ (cid:90) (cid:88) ≤µ−1 e−Tµj|ϕ (x)| |ϕ | 2 j j j=2 CL/N−B(0;δ) (cid:18)L(cid:19)m/2(cid:18)(cid:88)∞ (cid:19)1/2(cid:18)(cid:88)∞ (cid:19)1/2 ≤µ−1 e−Tµj e−Tµj|ϕ (x)|2 2 N j j=2 j=2 ≤µ−1(cid:18)L(cid:19)m/2(cid:18)(cid:88)∞ e−Tµj(cid:19)1/2(cid:0)π (x,x;T)(cid:1)1/2. (17) 2 N δ,N/L j=2 To bound the first series in (17), we note that the µ ’s are bounded from below j by the Neumann eigenvalues of the cube C . So choosing T = (L/N)2 we L/N get that (cid:18)(cid:88)∞ e−L2µj/N2(cid:19)1/2 ≤(cid:18)1+(cid:88)∞ e−π2j2(cid:19)m/2 ≤(cid:18)4(cid:19)m/2. 3 j=2 j=1 Similarly to the proof of Lemma 3.1 in [3], we have that (cid:0)π (x,x;L2/N2)(cid:1)1/2 ≤(cid:0)π (x,x;L2/N2)(cid:1)1/2 δ,N/L 0,N/L ≤(cid:18)N(cid:19)m/2(cid:18)1+2(cid:88)∞ e−π2j2(cid:19)m/2 L j=1 (cid:18)4(cid:19)m/2(cid:18)N(cid:19)m/2 ≤ . (18) 3 L Finally, µ ≥ π2N2, together with (12), (14), (16), (17), (18), and the choice 2 L2 T =(L/N)2 gives that (2m)1/2L (cid:18)4(cid:19)mL2 3eN2 (cid:107)v (cid:107) ≤µ−1+ + , µ ≤ . (19) Ωδ,N,L L∞(Ωδ,N,L) 1 µ1/2N 3 N2 1 16mL2 1 Proof of Theorem 1. Let 1<α<2. By (11) and (19), we have that (cid:18) (cid:18)5(cid:19)m(cid:18)N 1 (cid:19)(cid:19) λ(Ω )(cid:107)v (cid:107) ≤ µ +32m + µ δ,N,L Ωδ,N,L L∞(Ωδ,N,L) 1 4 L2 N1/2 1 (cid:18) (2m)1/2L (cid:18)4(cid:19)mL2(cid:19) × µ−1+ + , (20) 1 µ1/2N 3 N2 1 7 provided N 3eN2 ≤µ ≤ . L2 1 16mL2 First consider the planar case m = 2. Recall Lemma 3.1 in [3]: for δ < L/(6N), N2 (cid:18) L (cid:19)−1 8πN2 (cid:18) L (cid:19)−1 log ≤µ ≤ log . (21) 100L2 2δN 1,B(0;δ),L/N (4−π)L2 2δN Let L δ∗ :=δ∗(α,N,L)= e−N2−α, (22) 2N where 1 < α < 2. Let N ∈ N be such that for all N ≥ N , δ∗ < L/(6N). We 1 1 now use (21) to see that there exists C >1 such that Nα Nα C−1 ≤µ ≤C . (23) L2 1,B(0;δ∗),L/N L2 (In fact C =max{100,8π/(4−π)}). We subsequently let N ∈N be such that 2 for all N ≥N , 2 N Nα Nα 3eN2 ≤C−1 ≤C ≤ . L2 L2 L2 16mL2 By (20), (23), and all N ≥max{N ,N } we have that 1 2 λ(Ω )(cid:107)v (cid:107) ≤1+C(cid:0)N1−α+N(α−2)/2(cid:1), (24) δ∗,N,L Ωδ∗,N,L L∞(Ωδ∗,N,L) where C depends on C and on m only. Finally, we let N ∈N be such that for 3 all N ≥N , 3 C(cid:0)N1−α+N(α−2)/2(cid:1)<(cid:15). We conclude that (5) holds with Ω = Ω with δ∗ given by (22), and (cid:15) δ∗,N,L N ≥max{N ,N ,N }. 1 2 3 Next consider the case m = 3,4,.... We apply Lemma 3.2 in [3] to the case K = B(0;δ), and denote the Newtonian capacity of K by cap(K). Then cap(B(0;δ)) = κ δm−2, where κ is the Newtonian capacity of the ball with m m radius 1 in Rm. Then Lemma 3.2 gives that there exists C ≥1 such that (cid:18)N(cid:19)m (cid:18)N(cid:19)m C−1 δm−2 ≤µ ≤C δm−2, (25) L 1,B(0;δ,L/N) L provided 1 κ δm−2 ≤ (L/N)m−2. (26) m 16 We choose δ∗ :=δ∗(α,N,L)=LN(α−m)/(m−2). (27) This gives inequality (23) by (25). The requirement (26) holds for all N ≥N , 1 whereN isthesmallestnaturalnumbersuchthatN2−α ≥16κ . Theremain- 1 1 m der of the proof follows the lines below (23) with the appropriate adjustment of constants, and the choice of δ∗ as in (27). (cid:3) We note that the choice α = 4 in either (22) or in (27) gives, by (24), the 3 decay rate λ(Ω )(cid:107)v (cid:107) ≤1+2CN−1/3. (28) δ∗,N,L Ωδ∗,N,L L∞(Ωδ∗,N,L) 8 3 Proof of Theorem 2 In view of Payne’s inequality (6) it suffices to obtain an upper bound for (cid:107)v (cid:107) λ(Ω). We first observe, that by domain monotonicity of the tor- Ω L∞(Ω) sion function, v is bounded by the torsion function for the (connected) set Ω bounded by the two parallel lines tangent to Ω at distance w(Ω). Hence w(Ω)2 (cid:107)v (cid:107) ≤ . (29) Ω L∞(Ω) 8 Inordertoobtainanupperboundforλ(Ω),weintroducethefollowingnotation. Foraplanar,open,convexset,withfinitemeasure,weletz ,z betwopointson 1 2 the boundary of Ω which realise the width. That is there are two parallel lines tangentto∂Ω,atz andz respectively,andatdistancew(Ω). Letthex-axisbe 1 2 perpendicular to the vector z z , containing the point 1(z +z ). We consider 1 2 2 1 2 the family of line segments parallel to the x-axis, obtained by intersection with Ω, andletl ,l betwopointsontheboundaryofΩwhichrealisethemaximum 1 2 lengthLofthisfamily. Thequadrilateralwithvertices, z ,z ,l ,l iscontained 1 2 1 2 in Ω. This quadrilateral in turn contains a rectangle with side-lengths h, and (cid:0)1− h (cid:1)L respectively, where h ∈ [0,w(Ω)) is arbitrary. Hence, by domain w(Ω) monotonicity of the Dirichlet eigenvalues, we have that (cid:18) h (cid:19)−2 λ(Ω)≤π2h−2+π2 1− L−2. w(Ω) Minimising the right-hand side above with respect to h gives that (w(Ω)L2)1/3 h= . 1+(cid:0) L (cid:1)2/3 w(Ω) It follows that π2 (cid:18) (cid:18)w(Ω)(cid:19)2/3(cid:19)3 λ(Ω)≤ 1+ . (30) w(Ω)2 L As w(Ω)≤L we obtain by (30) that π2 (cid:32) (cid:18)w(Ω)(cid:19)2/3(cid:33) λ(Ω)≤ 1+7 . (31) w(Ω)2 L In order to complete the proof we need the following. Lemma 5. If Ω is an open, bounded, convex set in R2, and if L is the length of the longest line segment in the closure of Ω, perpendicular to z z , then 1 2 diam(Ω)≤3L. (32) Proof. Let d ,d ∈ ∂Ω such that |d −d | = diam(Ω). We denote the projec- 1 2 1 2 tions of d ,d onto the line through z ,z by e ,e respectively. Let z be the 1 2 1 2 1 2 intersection of the lines through z ,z and d ,d respectively. Then, by the 1 2 1 2 maximality of L, we have that |d −e | ≤ L,|d −e | ≤ L. Furthermore, by 1 1 2 2 convexity, |e −z|+|e −z|≤w(Ω). Hence, 1 2 |d −d |≤|d −e |+|e −z|+|d −e |+|e −z|≤2L+w(Ω)≤3L. 1 2 1 1 1 2 2 2 9 By (31), we have that π2 (cid:32) (cid:18) w(Ω) (cid:19)2/3(cid:33) λ(Ω)≤ 1+7·32/3 . w(Ω)2 diam(Ω) This implies Theorem 2 by (29). (cid:3) 4 Proof of Theorem 3 We denote by d:M×M (cid:55)→R+ the geodesic distance associated to (M,g). For x∈M, R>0, B(x;R)={y ∈M :d(x,y) 2 and 0 < D < 2 there exist 2 1 constants 00, 1 2 e−d(x,y)2/(2D1t) e−d(x,y)2/(2D2t) C ≤p (x,y;t)≤C . 1(|B(x;t1/2)||B(y;t1/2)|)1/2 M 2(|B(x;t1/2)||B(y;t1/2)|)1/2 (34) Finally, since by (33) the measure of any geodesic ball with radius r is bounded polynomially in r, the theorems of Grigor’yan in [6] imply stochastic complete- ness. That is, for all x∈M, and all t>0, (cid:90) dyp (x,y;t)=1. M M Proof of Theorem 3. We choose D = 1, D = 3 in (34), and define the corre- 1 2 sponding number K =max{C ,C−1}. Then 2 1 K−1e−d(x,y)2/(2t) ≤(|B(x;t1/2)||B(y;t1/2)|)1/2p (x,y;t)≤Ke−d(x,y)2/(6t). M (35) Letq ∈M bearbitrary,andletR>0besuchthatΩ(q;R):=B(q;R)∩Ω(cid:54)= ∅. The spectrum of the Dirichlet Laplacian acting in L2(Ω(q;R)) is discrete. Denote the bottom of this spectrum by λ(Ω(q;R)). Then λ(Ω(q;R)) ≥ λ(Ω). Bythespectraltheorem,monotonicityofDirichletheatkernels,andtheLi-Yau bound (35), we have that p (x,x;t)≤e−tλ(Ω(q;R))/2p (x,x;t/2) Ω(q;R) Ω(q;R) ≤e−tλ(Ω(q;R))/2p (x,x;t/2) M ≤Ke−tλ(Ω(q;R))/2|B(x;(t/2)1/2)|−1. (36) 10

Spectral bounds for the torsion function

Similar Spectral bounds for the torsion function