Table Of ContentSpectral bounds for the torsion function
M. van den Berg
School of Mathematics, University of Bristol
University Walk, Bristol BS8 1TW
United Kingdom
7
[email protected]
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30 March 2017
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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 de-
noted 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)<R}. For a measurable set A⊂M
we denote by |A| its Lebesgue measure. The Bishop-Gromov Theorem (see
[4]) states that if M is a complete, non-compact, m-dimensional, Riemannian
manifold with non-negative Ricci curvature, then for p ∈ M, the map r (cid:55)→
|B(p;r)| is monotone decreasing. In particular
rm
|B(p;r )| (cid:18)r (cid:19)m
2 ≤ 2 , 0<r ≤r . (33)
|B(p;r )| r 1 2
1 1
Corollary 3.1 and Theorem 4.1 in [8], imply that if M is complete with non-
negative Ricci curvature, then for any D > 2 and 0 < D < 2 there exist
2 1
constants 0<C ≤C <∞ such that for all x∈M, y ∈M, t>0,
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
