Introduction

Heat kernel estimates

Geometric analysis on manifolds with ends
Satoshi Ishiwata (石渡 聡)
Yamagata University (山形大学)

March 4, 2021

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Yamagata University (山形大学)

Introduction

Heat kernel estimates

This talk is based on a joint work with
Alexander Grigory’an (Bielefeld)
andLaurent Saloﬀ-Coste (Cornell).
◦ Heat kernel estimates on connected sums of parabolic
manifolds, J. Math. Pures Appl. 113(2018), 155-194.
◦ Geometric analysis on manifolds with ends, Advances inAnalysis and Geometry, 3(2020), 325-343.
◦ Poincaré constant on manifolds with ends, in preparation.

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Yamagata University (山形大学)

Introduction

Heat kernel estimates

Definition (Manifold with ends (connected sum))
Let M1 , M2 , . . . , Mk be geodesically complete, non-compact
weighted manifolds of a same dimension.
We say that M is a manifold with k-ends M1 , . . . , Mk (andwrite M = M1 # · · · #Mk ) if there exists a compact setK ⊂ M so that M \K consists of k non-compact connectedcomponents E1 , . . . Ek so that each Ei is isometric to Mi \Ki ,exterior of a compact set Ki of Mi .
❊✐❊✐
▼✐

❑

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends❑✐

Yamagata University (山形大学)

Introduction

Heat kernel estimates

Our interest

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Yamagata University (山形大学)

Introduction

Heat kernel estimates

Our interest◦Stable global structure under the quasi-isometric modification.
In this talk, we focus on▶ Long time behavior of the heat kernel p(t, x, y) on M▶ We always assume that the heat kernel pi (t, x, y) on

each end Mi satisfies the Li-Yau type estimates(two-sided matching Gaussian estimates):(LY )Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

pi (t, x, y) ≈C√ e−bdi (x,y)/t .
Vi (x, t)
2

Yamagata University (山形大学)

Introduction

Heat kernel estimates

Typical examples
On M = Rn #Rn , n ≥ 3, for x ∈ M1 , y ∈ M2()d2 (x,y)111−btp(t, x, y) ≈ n/2+e.
t|x|n−2|y|n−2
On M = R2 #R2 , for x ∈ M1 , y ∈ M2 ,p(t, x, y)
()
√
21

log
t+log
t−log|x|log|y|2
 t log t√2 1 log e t e−b d (x,y)
tt log t|y|√≈d2 (x,y)1
log e|x|t e−b t t log( t)
d2 (x,y)1
 1+ log1|y| e−b ttlog |x|Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends√
if |x|, |y| ≤ t,√
if |y| ≤ t < |x|,√
if |x| ≤ t < |y|,√if |x|, |y| > t.

Yamagata University (山形大学)

Introduction

Heat kernel estimates

Typical examples
Let M = (R+ × S 1 )#R2 .

✷✰ ✂ ❙✶❑For x ∈ M1 , y ∈ M2
2√
1 −b d (x,y)
teif |y| > t,
 t()
√
√
|x|
11+ √
log e|y|t
if |x| , |y| ≤ t,p(t, x, y) ≍tt
√2√ 1 log e t e−b d (x,y)
t
if |x| > t ≥ |y| .
t|y|Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Yamagata University (山形大学)

Introduction

Heat kernel estimates

History of geometric analysis on manifolds with ends
1979 Kuz’menko, Molchanov
Failure of Liouville property on Rn #Rn , n ≥ 3.
1991 Grigor’yan, Saloﬀ-Coste (with Moser, Kusuoka, Stroocketc)(LY ) ⇐⇒ (P I) + (V D) ⇐⇒ (P HI)
1996 Benjamini, Chavel, Feldman
Bottleneck eﬀect on the heat kernel on Rn #Rn , n ≥ 3:
√For x ∈ M1 , y ∈ M2 with d(x, K) ≈ d(y, K) ≈ t,p(t, x, y) ≤Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Ctn+ϵ2for some ϵ > 0.
Yamagata University (山形大学)

Introduction

Heat kernel estimates2002 Grigor’yan and Saloﬀ-Coste
Estimate of the Dirichlet heat kernel exterior of acompact set
2002 Grigor’yan and Saloﬀ-Coste
Estimate of the first hitting probability to a compact set
2009 Grigor’yan and Saloﬀ-Coste
Heat kernel estimates on non-parabolic manifolds withends.
Mixed case (i.e. parabolic # non-parabolic) is reducedto a non-parabolic case by using Doob’s h-transform)2018 Grigor’yan, I and Saloﬀ-Coste
Heat kernel estimates on parabolic manifolds with ends.
Remark: M is called parabolic if the B.M. is recurrent.
Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Yamagata University (山形大学)

Introduction

Heat kernel estimates

Definition
A manifold M admits the Poincaré inequality if there exitsC, κ > 0 such that for any r > 0, x ∈ M and anyf ∈ C 1 (B(x, r)),∫∫2
2|f − fB(x,r) | dµ ≤ Cr|∇f |2 dµ,B(x,r)
1where fB(x,r) = µ(B(x,r))

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

B(x,κr)∫B(x,r) f dµ.

Yamagata University (山形大学)

Introduction

Heat kernel estimates

The connected sum M = M1 #M2 = Rn #Rn (n ≥ 3)
doesn’t satisfy the Poincaré inequality. Indeed, the function{1 x ∈ M1 ,f (x) =−1 x ∈ M2
implies that fB(o,r) = 0 for a central point o ∈ M and∫∫|f − fB(o,r) |2 dµ ≈ r n ,|∇f |2 dµ ≈ const.
B(o,r)

B(o,κr)which fails (PI).

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Yamagata University (山形大学)

Introduction

Heat kernel estimates

Heat kernel estimatespD
Ei (t, x, y): Dirichlet heat kernel on Ei = Mi \Ki ,τEi : First exit time to Ki ⊂ Mi .
Px (τEi ≤ t): First exit probability from x to Ki by time t.

Theorem 1 (Grigor’yan and Saloﬀ-Coste, 2009)
Let M = M1 # · · · #Mk and fix o ∈ K. For x ∈ Ei , y ∈ Ej ,p(t, x, y) ≈pD
Ei (t, x, y) + p(t, o, o)Px (τEi ≤ t)Py (τEj ≤ t)+∫ t()p(s, o, o)ds ∂t Px (τEi ≤ t)Py (τEj ≤ t)+Px (τEi ≤ t)∂t Py (τEj ≤ t) .
1

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Yamagata University (山形大学)

Introduction

Heat kernel estimates

Setting
We always assume that each end Mi satisfies (LY). If Mi isparabolic, we also assume that relatively connected annuli
condition (RCA) :Definition 2 (Relatively connected annuli condition (RCA))
Mi admits (RCA) for a fixed reference point oi ∈ Mi ifthere exists A > 1 and r0 > 0 such that ∀r > r0 ,Bi (oi , Ar)\Bi (oi , A−1 r)
is connected.

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Yamagata University (山形大学)

Introduction

Heat kernel estimates

On-diagonal estimates: non-parabolic case
For a fixed reference point oi ∈ Ki ⊂ Mi , let
Vi (r) = Vi (oi , r) = µi (Bi (oi , r)) and∫ r
sds
hi (r) :=.
1 Vi (s)Remark: hi (r) → ∞(r → ∞) ⇐⇒ M is parabolic.

Theorem 1 (Grigor’yan, Saloﬀ-Coste (2009))
Let M be a non-parabolic manifold with ends M1 , . . . , Mk . Thenp(t, o, o) ≈Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

1√√ .
mini Vi ( t)h2i ( t)Yamagata University (山形大学)

Introduction

Heat kernel estimates

On-diagonal estimates: parabolic caseDefinition Let Mi be parabolic.
Mi is called subcritical if hi (s) ≤ Cs2 /Vi (s) (∀s > 1).
Mi is called regular if there exist ε, η > 0 with η < ε/2 s.t.
( )ε( )2+ηtVi (t)tc≤(∀t > s > 0).
≤CsVi (s)sExampleVi (r) ≈ r α (log r)β is parabolic if α < 2 or α = 2, β ≤ 1.
Vi is subcritical if α < 2 and regular if α = 2, β ≤ 1.

Definition
Vm ≻ Vi if for all s > 0
CVm (s) ≥ Vi (s) and Vm (s)h2m (s) ≤ CVi (s)h2i (s).
Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Yamagata University (山形大学)

Introduction

Heat kernel estimates

On-diagonal estimates: parabolic case

Theorem 2 (Grigor’yan, I, Saloﬀ-Coste, in preparation)
Let M = M1 # · · · #Mk is parabolic. Assume also that▶ each end is either subcritical or regular.
▶ there exists m such that Vm ≻ Vi for all i = 1, . . . , k.

Then we havep(t, o, o) ≈Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends1√ .
Vm ( t)Yamagata University (山形大学)

Introduction

Heat kernel estimates

ExamplesLet M = Rn #Rn (n = 1, 2, . . .). Thenp(t, o, o) ≈1
tn/2.

Let M = (R+ × S 1 )#R2 . Thenp(t, o, o) ≈1t.

Let M = (R2 × S n−2 )#Rn with n ≥ 3. Thenh1 (r) ≈ log r andp(t, o, o) ≈Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

1t log2 t.

Yamagata University (山形大学)

Introduction

Heat kernel estimates

ExamplesLet M = R1 # · · · #Rk , where Ri is a model manifold withVi (o, r) ≈ r 2 (log r)βi satisfying
1 ≥ β1 ≥ β2 · · · ≥ βk .
Note that all ends are parabolic and, especially, regular.
Then1.
p(t, o, o) ≈t(log t)β1

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Yamagata University (山形大学)

Introduction

Heat kernel estimates

A highlight of the proof: non-parabolic case

Assume that M1 , . . . Mk are non-parabolic.
They prove the Faber-Krahn inequality which implies thatsup p(t, x, x) ≤x∈MC√ .
Vmin ( t)Since all ends are non-parabolic, for all i = 1, . . . , k,′p(t, o, o) ≥ cpD
Ei (t, oi , oi ) ≥ c pi (t, oi , oi ) ≥Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with endsc′′√ .
Vi ( t)Yamagata University (山形大学)

Introduction

Heat kernel estimates

A highlight of the proof: parabolic case
Assume that M1 , . . . Mk are parabolic. We estimate theintegrated resolvent kernel∫ ∫ ∞γλ (x) :=e−λt p(t, x, z)dzdt
K0

and its derivative γ̇λ in λ. They satisfy∆γλ − λγλ = −1K , ∆γ̇λ − λγ̇λ = −γλ .
In a open set A ⊂ M with K ⊂ A,(∆ − λ)(γλ − γλA ) = 0,where γλA (x) :=∫∫∞K0

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

e−λt pDA (t, x, z)dzdt
Yamagata University (山形大学)

Introduction

Heat kernel estimates

We apply comparison principle (a sort of maximal principle).
❆On ∂A,∑A(sup γλ )1∂Ai .
γλ − γλ ≤∂AiiThen we obtain in∑A,(sup γλ )hi ,γλ − γλA ≤i

❑

∂Ai

❊✐
❅❆✐

❅❑✐ ❂ ❅❊✐

where hi is the harmonic extension of 1∂Ai .
∑( i hi ≡ 1)
Since (∆ − λ)γλ = 0 in Ei , using the comparison principle again
on Ei , we obtainiγλ ≤ (sup γλ )(1 − ΦEλ ),∂EiwhereiΦEλ (x) = λSatoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

∫ ∞eλt Px (τEi > t)dt.

0Yamagata University (山形大学)

Introduction

Heat kernel estimates

By calculation,(sup γλ )∂Kk∑
i=1i
inf ΦEλ ≤ C.

∂Ai

Since
i
inf ΦEλ (x) ≥∂Ai

We obtain

Satoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with endschi ( √1 ),λ1sup γλ ≤ C min hi ( √ ).
i∂KλYamagata University (山形大学)

Introduction

Heat kernel estimates

By a similar argument, we obtain(k∑i(sup γ̇λ )inf ΦEλ ≤ C + (sup γλ )
∂Ai
∂K
∂K
i=1whereiΨEλ (x) =∫ ∞k∑)

iC+sup ΨE
λi=1 ∂Ai,

te−λt ∂t Px (τEi ≤ t)dt.

0

By the assumption of Mi , we obtaini
sup ΨEλ ≤∂AiC
λ2 Vi ( √1 )h2i ( √1 )λλwhich concludes
sup γ̇λ ≤ C∂Kmini h2i ( √1 )λ2 minSatoshi Ishiwata (石渡 聡)
Geometric analysis on manifolds with ends

Cλ= 2.
112λ mini Vi ( √1 )i Vi ( √ )hi ( √ )λλλYamagata University (山形大学)