Tema 1

domain of the polar coordinates, the Laplace operator is given by (3.4), which we rewrite as ∆ = ∂ 2 ∂2ρ+m(ρ, θ) ∂ ∂ρ+ ∆Sρ. (15.1)

Here ∆Sρ is the Laplace operator on the sphere ∂B(o, ρ) and m(ρ, θ) is a smooth

function onR+×Sd−1, which will be of primary interest for us. In fact, its geometric meaning is the mean curvature of the sphere∂B(o, ρ) in the radial direction (see [69], [27], [169]).

We will compare the manifold M with a model manifold Mψ introduced in

Section 3.2. Let us equip by a hat all notation related toMψ. In particular, we set c

M =Mψ. By (3.6), we have the following expression for a Laplace operator onMc b ∆ = ∂ 2 ∂2ρ+mb(ρ) ∂ ∂ρ+∆bSρ, (15.2) where b m= (d−1)ψ 0 ψ. It is important thatmb(ρ) does not depend onθ.

Let us consider also the following Schr¨odinger operators onM andMc: L= ∆−q(x)

and

b

L=∆b −bq(ρ),

where the functionsqand bqare non-negative and continuous.

Theorem 15.1. LetM be a geodesically complete non-compact manifold, ando∈ M.

(i) Assume that, for all (ρ, θ)in the domain of the polar coordinates centered at o, with ρbeing large enough,

m(ρ, θ)≤mb(ρ) and q(ρ, θ)≥qb(ρ). (15.3) If the equationLu= 0 has a non-zero bounded solution on all of Mc, then so does Lub = 0.

(ii) Let obe a pole. Assume that, for all large enoughρand all θ,

m(ρ, θ)≥mb(ρ) and q(ρ, θ)≤qb(ρ). (15.4) If the equationLub = 0 has a non-zero bounded solution on all of M, then so does Lu= 0.

o / ρ α ν Uk Cut(o) θ ρ=lk( ) ρ=l( ) Uk

Figure 32. Approximation of the cut locus by a smooth hypersurface

For the case when o is a pole, this theorem was proved in [76]. The technical difficulties which arise due to the cut locus can be handled by the method of Cheeger and Yau [27] developed in the context of comparison of heat kernels (see also [193], [192], [169, Section I.1]).

Proof. (i)Assume that the only bounded solution for the equationLub = 0 isu≡0. Take someR >0 and define the function v(ρ) on [R,∞) to be the solution to the Cauchy problem

v00+mvb 0−bqv= 0, v(R) = 0, v0(R) = 1.

Then Lvb = 0 in Ω :=b Mc\Bb(o, R) and, by the maximum principle, the function v(ρ) is monotone increasing. By Theorem 13.6, v must be unbounded, whence v(ρ) → ∞ as ρ → ∞ (the converse to Corollary 13.7 for spherically symmetric manifolds).

Let us consider nowv(ρ) as a function onM. Due to (15.3) andv0 ≥0,we have in Ω\Cut(o) (where Ω := M\B(o, R))

Lv=v00+mv0−qv≤v00+mvb 0−qvb =Lvb = 0.

IfCut(o) is empty, then, by Corollary 13.7, this finishes the proof becauseLv≤0 in Ω andv(x)→ ∞as x→ ∞.

Let Cut(o) be non-empty. We will show that Lv ≤0 is still true in Ω in the sense of distribution (which is enough for Corollary 13.7). More precisely, let us prove that, for any non-negative test function φ∈C₀∞(Ω),

hLv, φi:=−

Z M

(∇v∇φ+qvφ)dµ≤0. (15.5) For any unit vectorθ∈To(M),let us definel(θ) to be the length of the geodesics

which starts atoin the directionθand ends atCut(o) (see Figure 32). SinceCut(o) is a closed set, the functionl(θ) is lower semi-continuous. Letlk(θ) be an increasing

sequence of smooth positive functions on the unit sphere which converges tol(θ) as k→ ∞. Denote byUkthe set of points (ρ, θ)∈M\Cut(o) such thatρ < lk(θ). The

boundary ∂Uk is a smooth hypersurface given by the equationρ=lk(θ). Clearly,

the sequence{Uk}is increasing and S

kUk=M\Cut(o).

We have, by the Green formula (2.6),

Z U_k (∇v∇φ+qvφ)dµ= Z U_k (−∆vφ+qvφ)dµ+ Z ∂U_k ∂v ∂νφ , (15.6)

whereνis the unit outward normal vector field on∂Ωk. The first term on the right

hand side of (15.6) is non-negative because

Lv≤0 on Uk∩suppφ⊂Ω\Cut(o).

We claim that the second term is also non-negative. Indeed, the normal ν forms an acute angleαwith the radial direction ∂

∂ρ, and ∇v=v0(ρ) ∂ ∂ρ, whence ∂v ∂ν =ν∇v=ν(v 0_(ρ)∂ ∂ρ) =v 0_{(ρ) cosα≥0.} Thus, (15.6) yields Z Uk (∇v∇φ+qvφ)dµ≥0.

Ask→ ∞,we can replace hereUkbyM\Cut(o).Finally,Cut(o) has the measure

zero, whence we conclude (15.5).

(ii)By Theorem 13.6, there is also apositive bounded solutionutoLub = 0 on

c

M. By symmetrizing it, we can assume thatu=u(ρ).By the maximum principle, u(ρ) is increasing. Fix some R > 0, put v(ρ) = u(ρ)−u(R) and observe that

b

Lv≥0.

The functionvis positive in the region Ω :={ρ > R}, and, byv0 ≥0 and (15.4), we have Lv ≥ Lvb ≥ 0 in Ω. Therefore, v is a q-subharmonic function, which is bounded, positive in Ω and vanishing on∂Ω, whence we see that Ω is q-massive. By Theorem 13.6, there is a non-zero bounded solution to Lu= 0 onM.

Combining Theorem 15.1 with the criteria for parabolicity and stochastic com- pleteness in terms of the Liouville properties for the Schr¨odinger equations (see the beginning of Section 13.2) and with Propositions 3.1, 3.2, we obtain the following statement.

Corollary 15.2. Let M be a geodesically complete non-compact manifold, o∈M