Some examples of open manifolds withpositive Ricci curvature
Jiayin Pan (joint work with Guofang Wei)
University of California, Santa Cruz

Pacific Rim Complex and Symplectic Geometry Conference,
Kyoto, 2022

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Part 1. In a nutshell

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Past constructions
Question
For open (complete and non-compact) manifolds, what are thedifferences between nonnegative/positive sectional curvature andnonnegative/positive Ricci curvature?
There are lots of examples addressing the above question (e.g.,constructions by Sha-Yang, Menguy, Perelman, etc.)
For this talk, we look at Nabbonand’s example and some related
constructions.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Past constructions
Question
For open (complete and non-compact) manifolds, what are thedifferences between nonnegative/positive sectional curvature andnonnegative/positive Ricci curvature?
There are lots of examples addressing the above question (e.g.,constructions by Sha-Yang, Menguy, Perelman, etc.)
For this talk, we look at Nabbonand’s example and some related
constructions.
Nabbonand’s examples (1980): Rk × S 1 (k ≥ 3) admits a
Riemannian metric with Ric > 0.
Cf: If M has sec > 0, then M is diffeomorphic to Rn .

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Past constructions
Question
For open (complete and non-compact) manifolds, what are thedifferences between nonnegative/positive sectional curvature andnonnegative/positive Ricci curvature?
There are lots of examples addressing the above question (e.g.,constructions by Sha-Yang, Menguy, Perelman, etc.)
For this talk, we look at Nabbonand’s example and some related
constructions.
Nabbonand’s examples (1980): Rk × S 1 (k ≥ 3) admits a
Riemannian metric with Ric > 0.
Cf: If M has sec > 0, then M is diffeomorphic to Rn .
Wei’s examples (1988): Rk × N, where N is a nilmanifold and klarge, admits a Riemannian metric with Ric > 0.
Cf: If M has sec ≥ 0, then π1 (M) is virtually abelian.
Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

New observations

By investigating the universal covers of related constructions, wegive negative answers to the following open problems (Pan-Wei
2021, 2022).

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

New observations

By investigating the universal covers of related constructions, wegive negative answers to the following open problems (Pan-Wei
2021, 2022).
Open Problem
For an open manifold with nonnegative Ricci curvature, is it truethat its Busemann function at a point must be proper?
Open Problem
For a Ricci limit space, is it true that the Hausdorff dimension ofits singular set cannot exceed that of the regular set?
Both are true under the sectional curvature condition.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Part 1. In a nutshell
Part 2. Past constructions

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Nabonnard/Bergery’s examples (1980/1986)
Let (N, g0 ) be a closed manifold of Ric(g0 ) ≥ 0. Doubly warped
product on Rk × N.
M = [0, ∞) ×f S k−1 ×h N,

dr 2 + f 2 (r )ds 2 + h2 (r )g0

has Ric > 0 for suitable (implicit) f (r ), h(r ), and large k.
f odd, h even, f ′ (0) = 1, h(0) > 0, ...
h(r ) decreases → 0 or → c > 0 as r → +∞.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Wei’s examples (1988)

Let Ne be a simply connected nilpotent Lie group and let Γ be ae N = N/Γe admits a family of metrics {gr }r ≥0 withlattice in N.
|sec(gr )| ≤Jiayin Pan (joint work with Guofang Wei)c.
1 + r2

Examples with positive Ricci curvature

Wei’s examples (1988)

Let Ne be a simply connected nilpotent Lie group and let Γ be ae N = N/Γe admits a family of metrics {gr }r ≥0 withlattice in N.
c.
1 + r2|sec(gr )| ≤Then for a suitable function f (r ), the warped product
M = [0, ∞) ×f S k−1 × Nr ,dr 2 + f 2 (r )ds 2 + gr

has Ric > 0 when p is large.
diam(Nr ) decays at a rate of r −α (α > 0).
π1 (M) = π1 (N) is not virtually abelian.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureChoice of warping function h
Doubly warped product
M = [0, ∞) ×f S k−1 ×h S 1 ,dr 2 + f 2 (r )ds 2 + h2 (r )g0

with Ric > 0.
Wei’s construction, a polynomial decay h:f (r ) = r (1 + r 2 )−1/4 ,h(r ) = (1 + r 2 )−α .

A logarithm decay h:√f (r ) =ln 2 · r

1/2

ln(2 + r 2 )Jiayin Pan (joint work with Guofang Wei),h(r ) = ln−α (2 + r 2 ).

Examples with positive Ricci curvature

Part 1. In a nutshell
Part 2. Past Constructions
Part 3. New observations
3.1 Busemann functions
3.2 Ricci limit spaces

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature3.1 Busemann function
Let M be an open Riemannian manifold.
Busemann function associated with a unit speed ray c:bc (x) = lim t − d(x, c(t)).
t→+∞

We can view bc (x) as (the negative of) a renormalized distancefrom x to infinity along c.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature3.1 Busemann function
Let M be an open Riemannian manifold.
Busemann function associated with a unit speed ray c:bc (x) = lim t − d(x, c(t)).
t→+∞

We can view bc (x) as (the negative of) a renormalized distancefrom x to infinity along c.
Busemann function at a point p ∈ M:
bp (x) = sup bc (x),cwhere sup is taken over all rays starting at p.
For convenience, we use the term Busemann function to refer tothe one at some point.
Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Busemann function and nonnegative curvature

Theorem (Cheeger-Gromoll 1970)
If M has sec ≥ 0, then the Busemann function is convex andproper.
Remark: This is one of the key steps in the proof of
Cheeger-Gromoll’s soul theorem...also, properness follows fromconvexity.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Busemann function and nonnegative curvature

Theorem (Cheeger-Gromoll 1970)
If M has sec ≥ 0, then the Busemann function is convex andproper.
Remark: This is one of the key steps in the proof of
Cheeger-Gromoll’s soul theorem...also, properness follows fromconvexity.
Theorem (Cheeger-Gromoll 1970)
If M has Ric ≥ 0, then the Busemann function is subharmonic.
Remark: This is one of the key steps in the proof of
Cheeger-Gromoll’s splitting theorem.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureIs it proper?
Open Problem (since the time of splitting theorem, per private
communication with Cheeger):
QuestionIs the Busemann function always proper when M has Ric ≥ 0?
Remark: The properness of bp does not depend on the choice ofp ∈ M.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureIs it proper?
Open Problem (since the time of splitting theorem, per private
communication with Cheeger):
QuestionIs the Busemann function always proper when M has Ric ≥ 0?
Remark: The properness of bp does not depend on the choice ofp ∈ M.
Theorem
Let M be an open manifold with Ric ≥ 0. Then the Busemannfunction is proper when
(1) M has Euclidean volume growth (Shen 1996), or(2) M has linear volume growth (Sormani 1998).

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureAnswer

Theorem (Pan-Wei 2022)
Given any integer n ≥ 4, there is an open n-manifold with positive
Ricci curvature and a non-proper Busemann function.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureAnswer

Theorem (Pan-Wei 2022)
Given any integer n ≥ 4, there is an open n-manifold with positive
Ricci curvature and a non-proper Busemann function.
In Sormani’s work (linear vol growth) and Shen’s work (Euclidean
vol growth), any asymptotic cone of the manifold is polar.
Question: What if we also assume that all asymptotic cones of M
are polar, is the Busemann function proper?

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureAnswer

Theorem (Pan-Wei 2022)
Given any integer n ≥ 4, there is an open n-manifold with positive
Ricci curvature and a non-proper Busemann function.
In Sormani’s work (linear vol growth) and Shen’s work (Euclidean
vol growth), any asymptotic cone of the manifold is polar.
Question: What if we also assume that all asymptotic cones of M
are polar, is the Busemann function proper?
Theorem (Pan-Wei 2022)
There is an open manifold with positive Ricci curvature, a uniquepolar asymptotic cone, and a non-proper Busemann function.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Look at the covers

Bottom-line:
e of a doubly warped
We will use the Riemannian universal cover M
k−1
1
product M = [0, ∞) ×f S
×h S with suitable functions f and has our examples.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Look at the covers

Bottom-line:
e of a doubly warped
We will use the Riemannian universal cover M
k−1
1
product M = [0, ∞) ×f S
×h S with suitable functions f and has our examples.
e has a non-proper Busemann
When h decreases to 0 as r → ∞, Mfunction.
e has a unique polar asymptotic
When h has logarithm decay, Mcone, as the standard half-plane, and a non-proper Busemannfunction.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Look at the covers

Bottom-line:
e of a doubly warped
We will use the Riemannian universal cover M
k−1
1
product M = [0, ∞) ×f S
×h S with suitable functions f and has our examples.
e has a non-proper Busemann
When h decreases to 0 as r → ∞, Mfunction.
e has a unique polar asymptotic
When h has logarithm decay, Mcone, as the standard half-plane, and a non-proper Busemannfunction.
To prove the Busemann function at a chosen base point p̃ ise starting at p̃ ...firstly
non-proper, we need to find all the rays in M
find all geodesics in M starting at p.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureGeodesics in M
M = [0, ∞) ×f S k−1 ×h S 1 ...write each point in M as (r , x, y )...fix
a point p = (0, x, y ) at r = 0 as our base point.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureGeodesics in M
M = [0, ∞) ×f S k−1 ×h S 1 ...write each point in M as (r , x, y )...fix
a point p = (0, x, y ) at r = 0 as our base point.
For each x ∈ S k−1 , we define a submanifold
C (x) = {(r , ±x, y )|r ≥ 0, y ∈ S 1 }.
It is totally geodesic and geodesically complete.
The induced Riemannian metric on C (x) is isometric to the surface
revolution R ×h S 1 .

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureGeodesics in M
M = [0, ∞) ×f S k−1 ×h S 1 ...write each point in M as (r , x, y )...fix
a point p = (0, x, y ) at r = 0 as our base point.
For each x ∈ S k−1 , we define a submanifold
C (x) = {(r , ±x, y )|r ≥ 0, y ∈ S 1 }.
It is totally geodesic and geodesically complete.
The induced Riemannian metric on C (x) is isometric to the surface
revolution R ×h S 1 .
Sx∈S k−1 Tp C (x) = Tp M...equivalent to study geodesics in R ×h S 1 .

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureGeodesics in M
M = [0, ∞) ×f S k−1 ×h S 1 ...write each point in M as (r , x, y )...fix
a point p = (0, x, y ) at r = 0 as our base point.
For each x ∈ S k−1 , we define a submanifold
C (x) = {(r , ±x, y )|r ≥ 0, y ∈ S 1 }.
It is totally geodesic and geodesically complete.
The induced Riemannian metric on C (x) is isometric to the surface
revolution R ×h S 1 .
Sx∈S k−1 Tp C (x) = Tp M...equivalent to study geodesics in R ×h S 1 .
(By classical differential geometry and our choice of h) a geodesic
in R ×h S 1 starting at p = (0, y ) has one of the following forms:(1) a radial ray,(2) a closed geodesic as the circle {r = 0},(3) a bounded geodesic oscillating between {r ≥ 0} and {r ≤ 0}
indefinitely.
Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureRays in the universal cover of Me starting at p̃.
Let c be a unit speed ray M(By the above classification and the splitting theorem,) c must
project to a radial ray c̄(t) = (r (t), x, y ) in M for some x ∈ S k−1
and y ∈ S 1 .

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureRays in the universal cover of Me starting at p̃.
Let c be a unit speed ray M(By the above classification and the splitting theorem,) c must
project to a radial ray c̄(t) = (r (t), x, y ) in M for some x ∈ S k−1
and y ∈ S 1 .
Let γ be a generator of Γ = π1 (M, p) ≃ Z. For all l ∈ Z:d(γ l p̃, c(t)) ≥ d(p, c̄(t)) = t; d(γ l p̃, c(t)) ≤ t + |l| · 2πh(t).
⇒ bc (γ l p̃) = 0.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureRays in the universal cover of Me starting at p̃.
Let c be a unit speed ray M(By the above classification and the splitting theorem,) c must
project to a radial ray c̄(t) = (r (t), x, y ) in M for some x ∈ S k−1
and y ∈ S 1 .
Let γ be a generator of Γ = π1 (M, p) ≃ Z. For all l ∈ Z:d(γ l p̃, c(t)) ≥ d(p, c̄(t)) = t; d(γ l p̃, c(t)) ≤ t + |l| · 2πh(t).
⇒ bc (γ l p̃) = 0.
⇒ bp̃ (γ l p̃) = supc bc (γ l p̃) = 0.
Thus the level set bp̃−1 (0) contains the entire orbit Γp̃, which isunbounded. That’s it.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature3.2 Ricci limit spaces
The Gromov-Hausdorff distance between two metric spaces
measures how they look alike.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature3.2 Ricci limit spaces
The Gromov-Hausdorff distance between two metric spaces
measures how they look alike.
Gromov’s precompactness theorem
Let (Mi , pi ) be a sequence of complete Riemannian n-manifolds of
Ric ≥ −(n − 1), then (Mi , pi ) has a Gromov-Hausdorff convergent
subsequence with limit as a length metric space (X , p).
The limit (X , p) is referred as a Ricci limit space.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature3.2 Ricci limit spaces
The Gromov-Hausdorff distance between two metric spaces
measures how they look alike.
Gromov’s precompactness theorem
Let (Mi , pi ) be a sequence of complete Riemannian n-manifolds of
Ric ≥ −(n − 1), then (Mi , pi ) has a Gromov-Hausdorff convergent
subsequence with limit as a length metric space (X , p).
The limit (X , p) is referred as a Ricci limit space.
For a Ricci limit space X , a point x ∈ X is called
k-regular, if every tangent cone at x is isometric to Rk .
singular, if otherwise.
Rk : the set of k-regular points.
S: the set of singular points.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Regularity theory of Ricci limit spaces
Rectifiable dimension (Cheeger-Colding 2000, Colding-Naber 2012)
Let X be a Ricci limit space. There is a unique integer k ∈ [0, n]such that (among many other properties,) Rk has full ν-measure
for any renormalized limit measure ν. Moreover, dimH (Rk ) = k.
This integer k is called the rectifiable dimension of X .

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Regularity theory of Ricci limit spaces
Rectifiable dimension (Cheeger-Colding 2000, Colding-Naber 2012)
Let X be a Ricci limit space. There is a unique integer k ∈ [0, n]such that (among many other properties,) Rk has full ν-measure
for any renormalized limit measure ν. Moreover, dimH (Rk ) = k.
This integer k is called the rectifiable dimension of X .
Theorem (Cheeger-Colding 1996)
Let X be a noncollapsing Ricci limit space. Then both its
rectfiable dimension and Hausdorff dimension equal n. Moreover,dimH (S) ≤ n − 2.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Regularity theory of Ricci limit spaces
Rectifiable dimension (Cheeger-Colding 2000, Colding-Naber 2012)
Let X be a Ricci limit space. There is a unique integer k ∈ [0, n]such that (among many other properties,) Rk has full ν-measure
for any renormalized limit measure ν. Moreover, dimH (Rk ) = k.
This integer k is called the rectifiable dimension of X .
Theorem (Cheeger-Colding 1996)
Let X be a noncollapsing Ricci limit space. Then both its
rectfiable dimension and Hausdorff dimension equal n. Moreover,dimH (S) ≤ n − 2.
Question (Cheeger-Colding 1996)
For a collapsing Ricci limit space, is it true that the Hausdorff
dimension of the singular set cannot exceed that of the regular set?
Equivalently, Rectifiable dimension=Hausdorff dimension?
Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureAnswer

Theorem (Pan-Wei 2021)
Given any β > 0, for n sufficiently large (depending on β), there isan open n-manifold N with Ric > 0, whose asymptotic cone (Y , y )satisfies the following:(1) Y = S ∪ R2 , where S is the singular set and R2 is the set of
2-regular points;(2) S has Hausdorff dimension 1 + β.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureAnswer

Theorem (Pan-Wei 2021)
Given any β > 0, for n sufficiently large (depending on β), there isan open n-manifold N with Ric > 0, whose asymptotic cone (Y , y )satisfies the following:(1) Y = S ∪ R2 , where S is the singular set and R2 is the set of
2-regular points;(2) S has Hausdorff dimension 1 + β.
When β > 1, dimH Y = dimH S > dimH R.
Limit measure and any dimensional Hausdorff measure may not be
mutually absolutely continuous.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureAsymptotic Cones
Let (M, p) be an open manifold of Ric ≥ 0. For any ri → ∞,passing to a subsequence if necessary, we obtain the pointed
Gromov-Hausdorff convergence:GH(ri−1 M, p) −→ (Z , z).
We call (Z , z) an asymptotic cone of M.
Example: The asymptotic cone of a flat cylinder is a line.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureAsymptotic Cones
Let (M, p) be an open manifold of Ric ≥ 0. For any ri → ∞,passing to a subsequence if necessary, we obtain the pointed
Gromov-Hausdorff convergence:GH(ri−1 M, p) −→ (Z , z).
We call (Z , z) an asymptotic cone of M.
Example: The asymptotic cone of a flat cylinder is a line.
Fact: When M has sec ≥ 0, its asymptotic cone (Z , z) is unique asa metric cone with vertex z.
In general, when Ric ≥ 0, its asymptotic cone may not be uniqueand may not be polar at the base point.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Equivariant Asymptotic Conese p̃) with Γ = π1 (M, p)-action, we can
For the universal cover (M,obtain the equivariant pointed GH convergence:GHe p̃, Γ) −−−−→(ri−1 M,πy(Y , y , G )

π
y

GH

(ri−1 M, p) −−−−→ (Z , z) = (Y /G , ȳ ),where G is a closed subgroup of Isom(Y ). We call (Y , y , G ) ane Γ).
equivariant asymptotic cone of (M,Example: Flat cylinder...

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Equivariant Asymptotic Conese p̃) with Γ = π1 (M, p)-action, we can
For the universal cover (M,obtain the equivariant pointed GH convergence:GHe p̃, Γ) −−−−→(ri−1 M,πy(Y , y , G )

π
y

GH

(ri−1 M, p) −−−−→ (Z , z) = (Y /G , ȳ ),where G is a closed subgroup of Isom(Y ). We call (Y , y , G ) ane Γ).
equivariant asymptotic cone of (M,Example: Flat cylinder...
e Γ)Fact: When M has sec ≥ 0, equivariant asymptotic cone of (M,kis unique as (C (X ), v , G ), where Gv is isometric to an R -factor.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Look at the covers

Bottom-line:
e of a doubly warped
We will use the Riemannian universal cover M
k−1
1
product M = [0, ∞) ×f S
×h S with suitable functions f and has our examples.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Look at the covers

Bottom-line:
e of a doubly warped
We will use the Riemannian universal cover M
k−1
1
product M = [0, ∞) ×f S
×h S with suitable functions f and has our examples.
e Γ), Y has
For any equivariant asymptotic cone (Y , y , G ) of (M,rectifiable dimension 2; the orbit Gy is the singular set and ishomeomorphic to R.
When h has polynomial decay ∼ r −β , the orbit Gy has Hausdorff
dimension 1 + β.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Look at the covers

Bottom-line:
e of a doubly warped
We will use the Riemannian universal cover M
k−1
1
product M = [0, ∞) ×f S
×h S with suitable functions f and has our examples.
e Γ), Y has
For any equivariant asymptotic cone (Y , y , G ) of (M,rectifiable dimension 2; the orbit Gy is the singular set and ishomeomorphic to R.
When h has polynomial decay ∼ r −β , the orbit Gy has Hausdorff
dimension 1 + β.
e
To prove this, we need to study the π1 (M, p)-action on M.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Distance estimate
f (r ) = r (1 + r 2 )−1/4 , h(r ) = (1 + r 2 )−α as in Wei’s construction.
Let p ∈ M at r = 0 and γ be a generator of π1 (M, p) = Z.
Let cl be a minimal representing loop of γ l .

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

Distance estimate
f (r ) = r (1 + r 2 )−1/4 , h(r ) = (1 + r 2 )−α as in Wei’s construction.
Let p ∈ M at r = 0 and γ be a generator of π1 (M, p) = Z.
Let cl be a minimal representing loop of γ l .

Because h is decreasing, cl goes further into the end as l increases;
this also shortens the length of cl .
A simple, but crucial, estimate: As l → ∞,1length(cl ) ∼ l 1+2α ,1size(cl ) ∼ l 1+2α .
Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureAsymptotic conesGHe p̃, Γ) −−−−→ (Y , y , G )(ri−1 M,


π
π
y
y
GH

(ri−1 M, p) −−−−→

Jiayin Pan (joint work with Guofang Wei)

(Z , z).

Examples with positive Ricci curvatureAsymptotic conesGHe p̃, Γ) −−−−→ (Y , y , G )(ri−1 M,


π
π
y
y
GH

(ri−1 M, p) −−−−→ (Z , z).
√
Recall that f (r ) ∼ r and h(r ) ∼ r −2α , thus Z is a half-line[0, ∞) with z = 0.
Z = Y /G ...Y as attaching orbits to Z = [0, ∞)...G = R and Y is
homeomorphic to R × [0, ∞).
Blank space for drawing:

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

The orbit Gy

Let g ∈ G with d(gy , y ) = 1. Then we can naturally assign every
point in Gy a coordinate (b, 0), where b ∈ R, such that g b ycorresponds to (b, 0).

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

The orbit Gy

Let g ∈ G with d(gy , y ) = 1. Then we can naturally assign every
point in Gy a coordinate (b, 0), where b ∈ R, such that g b ycorresponds to (b, 0).
The (extrinsic) distance on Gy = R × {0} satisfies
1
1
C1 · |b1 − b2 | 1+2α ≤ d((b1 , 0), (b2 , 0)) ≤ C2 · |b1 − b2 | 1+2α
for all b1 , b2 ∈ R, where C1 , C2 > 0 only depend on α.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature

The orbit Gy

Let g ∈ G with d(gy , y ) = 1. Then we can naturally assign every
point in Gy a coordinate (b, 0), where b ∈ R, such that g b ycorresponds to (b, 0).
The (extrinsic) distance on Gy = R × {0} satisfies
1
1
C1 · |b1 − b2 | 1+2α ≤ d((b1 , 0), (b2 , 0)) ≤ C2 · |b1 − b2 | 1+2α
for all b1 , b2 ∈ R, where C1 , C2 > 0 only depend on α.
Gy is not geodesic; in fact, the intrinsic distance between any two
distinct points in Gy is infinity.
(Gy , d) has Hausdorff dimension 1 + 2α.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureSome remarks
With some further estimates, we can show that any pointz ∈ Y − Gy is 2-regular.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureSome remarks
With some further estimates, we can show that any pointz ∈ Y − Gy is 2-regular.
e p̃), Y is non-polar at y .
For any asymptotic cone (Y , y ) of (M,The first examples of non-polar asymptotic cones are constructedby Menguy.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureSome remarks
With some further estimates, we can show that any pointz ∈ Y − Gy is 2-regular.
e p̃), Y is non-polar at y .
For any asymptotic cone (Y , y ) of (M,The first examples of non-polar asymptotic cones are constructedby Menguy.
As α → 0 in h(r ) = (1 + r 2 )−α , the eGH distance between(Y , y , G ) and (R × [0, ∞), (0, 0), R) tends to 0, where R acts astranslations on R × [0, ∞).

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvatureSome remarks
With some further estimates, we can show that any pointz ∈ Y − Gy is 2-regular.
e p̃), Y is non-polar at y .
For any asymptotic cone (Y , y ) of (M,The first examples of non-polar asymptotic cones are constructedby Menguy.
As α → 0 in h(r ) = (1 + r 2 )−α , the eGH distance between(Y , y , G ) and (R × [0, ∞), (0, 0), R) tends to 0, where R acts astranslations on R × [0, ∞).
If h(r ) has logarithm decay ∼ ln−α (r ) or it → c > 0 as r → ∞,then (Y , y ) splits isometrically as R × [0, ∞). Thus logarithm
decaying h gives examples with polar asymptotic cones andnon-proper Busemann functions.

Jiayin Pan (joint work with Guofang Wei)

Examples with positive Ricci curvature