Background The singular metric has nonnegative scalar curvature Positive mass Theorem: Nonnegativity Positive mass Theorem: Rigidit

Positive mass theorem with low-regularity
Riemannian metrics
Weimin Sheng
with W. Jiang and H. Zhang
Zhejiang University

Pacific Rim Complex & Symplectic Geometry Conference
Aug. 1-5, 2022

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive mass Theorem: Nonnegativity Positive mass Theorem: Rigidit

1

Background
Positive Mass Theorem
Our theorem

2

The singular metric has nonnegative scalar curvature
Smoothing the metric
Weak nonnegative scalar curvature

3

Positive mass Theorem: Nonnegativity

4

Positive mass Theorem: Rigidity
Ricci flat away from the singular set
RCD space with nonnegative Ricci curvature

5

Scalar curvature in distribution with lower bounds
Scalar curvature lower bounds are preserved by Ricci flow

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Positive
mass
Mass
Theorem:
TheoremNonnegativity
Our theoremPositive mass Theorem: Rigidit

1

Background
Positive Mass Theorem
Our theorem

2

The singular metric has nonnegative scalar curvature
Smoothing the metric
Weak nonnegative scalar curvature

3

Positive mass Theorem: Nonnegativity

4

Positive mass Theorem: Rigidity
Ricci flat away from the singular set
RCD space with nonnegative Ricci curvature

5

Scalar curvature in distribution with lower bounds
Scalar curvature lower bounds are preserved by Ricci
flow
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Positive
mass
Mass
Theorem:
TheoremNonnegativity
Our theoremPositive mass Theorem: Rigidit

Positive Mass Theorem
One of the most famous results on manifolds with nonnegative
scalar curvature is the positive mass theorem proved by Scheon
and Yau: 1979, 1981 (for n < 8), 2017.
The ADM mass of each end of an n-dimensional
asymptotically flat manifold with nonnegative scalar curvature
is nonnegative.
If the ADM mass of an end is zero, then the manifold isisometric to the Euclidean space.
If the manifold is spin, Witten in 1981 proved the positive
mass theorem by a different method.
All the results are assumed to be smooth.

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Positive
mass
Mass
Theorem:
TheoremNonnegativity
Our theoremPositive mass Theorem: Rigidit

Problem-PMT with singularity

It is natural to ask:
If the manifold admits singularity in a subset ⌃, what is theconditions of ⌃ such that the positive mass theorem still holds?
It is necessary to assume that the metric is continuous:
In 2018, Shi and Tam has constructed an asymptotically flat metricwith a cone singularity and nonnegative scalar curvature, but withnegative ADM mass.

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Positive
mass
Mass
Theorem:
TheoremNonnegativity
Our theoremPositive mass Theorem: Rigidit

PMT with singularity
There are many results
Miao (2002) and Shi-Tam (2002) proved a PMT with Lipschitz
metric and ⌃ is a hypersurface satisfying certain conditions on
the mean curvatures of ⌃. Shi-Tam used this result to prove
the positivity of the Brown-York quasilocal mass.
McFeron and Székelyhidi (2012) used Ricci flow giving a new
proof of positivity of Miao’s result and proved the rigidity whenthe ADM mass is zero.
Dan Lee (2013) considered a PMT for (M n , g ) with bounded
1,pWloc -metric for n < p  1 and are smooth away from asingular set ⌃ with n2 (1 n/p )-dimensional Minkowski content
vanishing. This result was improved by Shi-Tam 2018 where
they showed that the singular set ⌃ only requires(n 2)-dimensional Minkowski content vanishing.
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Positive
mass
Mass
Theorem:
TheoremNonnegativity
Our theoremPositive mass Theorem: Rigidit

PMT with singularity

If (M n , g ) is spin, Lee and LeFloch (2015) were able to prove
1,na PMT for C 0 \ Wloc
metric, where the metric could be
singular. Their theorem can be applied to all previous results
for non-smooth metrics under the additional assumption thatthe manifold is spin.
Li-Mantoulidis 2019 considered bounded metric with skeleton
singularities along a codimensional 2 submanifolds. Without
further derivative assumptions on metric, they proved a PMT
in dimensional 3 with isolated singularity.

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Positive
mass
Mass
Theorem:
TheoremNonnegativity
Our theoremPositive mass Theorem: Rigidit

1

Background
Positive Mass Theorem
Our theorem

2

The singular metric has nonnegative scalar curvature
Smoothing the metric
Weak nonnegative scalar curvature

3

Positive mass Theorem: Nonnegativity

4

Positive mass Theorem: Rigidity
Ricci flat away from the singular set
RCD space with nonnegative Ricci curvature

5

Scalar curvature in distribution with lower bounds
Scalar curvature lower bounds are preserved by Ricci
flow
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Positive
mass
Mass
Theorem:
TheoremNonnegativity
Our theoremPositive mass Theorem: Rigidit

Our theorem

We improve and recover some results of Shi-Tam 2018, Lee 2013
and Lee-LeFloch 2015.
Theorem
1, p

Let M n (n 3) be a smooth manifold. Let g 2 C 0 \ Wloc (M )(n  p  1) be a complete asymptotically flat metric on M.
Assume g is smooth away from a bounded closed subset ⌃ with
p
n p 1
H
(⌃) < 1 if n  p < 1 or Hn 1 (⌃) = 0 if p = 1, assume
that Rg 0 on M \ ⌃. Then the ADM mass of g of each end isnonnegative. Moreover, the ADM mass of one end is zero if andonly if when (M , g ) is isometric to Euclidean space.

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Positive
mass
Mass
Theorem:
TheoremNonnegativity
Our theoremPositive mass Theorem: Rigidit

Remarks

RemarkFor the rigidity part, we will show such space has nonnegative
Ricci curvature in RCD sense provided that the mass is zero. The
rigidity would follow by the volume rigidity of nonnegative Ricci
curvature.
RemarkFor the case p = 1, the condition Hn 1 (⌃) = 0 is optimal, since
one can construct counterexample if ⌃ is a hypersurface. In
particular, this confirms a conjecture of Lee (2013).

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

1

Background
Positive Mass Theorem
Our theorem

2

The singular metric has nonnegative scalar curvature
Smoothing the metric
Weak nonnegative scalar curvature

3

Positive mass Theorem: Nonnegativity

4

Positive mass Theorem: Rigidity
Ricci flat away from the singular set
RCD space with nonnegative Ricci curvature

5

Scalar curvature in distribution with lower bounds
Scalar curvature lower bounds are preserved by Ricci
flow
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

Some fundament definitions
Definition (asymptotically flat)
Let M be a smooth n-manifold, and g be a C 0 metric on M. We saythat (M , g ) is asymptotically flat if 9K ⇢ M a compact subset, s. t.
g is C 2 on M \K , M \K has finite many components, says ⌃l ,l = 1, . . . , p, and for each component ⌃l , 9 smooth diffeomorphismnl : ⌃l ! R \ B, B a ball, s. t. if we see l as a coordinate systemon ⌃l , thengij
ij = O (|x | )gij ,k = O (|x |gij ,kl = O (|x |

1

2

)

),

where > (n 2)/2 some constant, and the commas denotepartial derivatives. We also call each components ⌃l as an end ofM.
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

Definition (ADM mass)
Given an asymptotically flat manifold (M , g ), we define the ADM
mass of each end ⌃l as the limit

limr !1 2(n

1
1)

!n 1

Z

n
X

Sr i ,j =1(gij ,igii ,j )⌫j d µ,where Sr : the coordinate sphere in (⌃l , l ) of radius r, ⌫: the unit
outward normal vector of Sr , !n 1 : the volume of the unit(n 1)-dim sphere, d µ: the volume form✓ofmetric.
◆qSr in Euclidean
In a given end ⌃l , we will denote r =⇤l⌃ni=1 (x i )2 .

Bartnik: this limit exists provided that the scalar curvature of g is
integrable, and it is a geometric invariant.
We only check one end of M, other ends holds in the same way
We choose an arbitrary end, and denote the mass by m(g ).
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

We fix a smooth background metric h on M, C 1 h  g  Ch for
C > 1, and h = g outside some compact subset of M. All theconvergences are taken w.r.t. h. r̃: the covariant derivative taken
w. r. t. h.
Definition
1,p

Let M n smooth, g: C 0 \ Wloc metric on M, h: a fixed smooth metricas above. For a family of smooth function {f }, we say that f
converge to a function f locally in W 1,p -norm, if 8✏ > 0, 80 < r < 1,
9 0 > 0, s. t. 8 2 (0, 0 ), x 2 M, we haveZ

Br (x )

|f

f |p d µh < ✏,Z

Br ( x )|r̃fr̃f |p d µh < ✏,the norm are taken with respect to h. f converge to a function f
locally in C 0 -norm, if 8✏ > 0, 8r > 0, 9 0 > 0, s. t.
8 2 (0, 0 ), x 2 M, we have supBr (x ) |f f |d µh < ✏.
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

Definition
1,ng be C 0 \ Wloc
metric on M, and h is a fixed smooth metric as
above. We can define the scalar curvature distribution as Lee and
LeFloch 2015,hRg , 'i :=ZM

V · r̃ '

d µg
d µh

!

+ F'

d µg
d µh

!

d µh

(1)

for any compactly supported ' 2 W 1,n/(n 1) , where ”·” w.r.t. h, V : avector field, F: a scalar field,
1 klkij := g2
V k := g ij kij⇣r̃ g + r̃ g
Og
F := trg Ric

i jl

g ik

j il

⌘

r̃l gij ,j
ij k `=gg (r̃j gi `ji

(2)r̃` gij ),⇣r̃k g ij kij + r̃k g ik iji + g ij

and µh : the Lebesgue measure w.r.t. h.
Weimin Sheng with W. Jiang and H. Zhangk `k ` ijk `j ` ik

⌘

(3)

,

(4)

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

By Lee-Lefloch, hRg , 'i is independent of h,RhRg , 'i = M Rg 'd µh when g is C 2 and Rg is defined inclassical sense.

We say that g has weakly nonnegative scalar curvature if for
any nonnegative test function ', hRg , 'i 0.

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

Smoothing the metric
The following mollification lemma could be found in Grant-Tassottin2014 Lemma 4.1( for W 2, 2 version), our version could be proved in
the same manner.
Lemma (2.1)
1,nLet M n smooth manifold, g 2 C 0 \ Wloc
on M, then 9 smooth
metrics {g }, > 0, s. t. g converge to g locally both in C 0 -norm
and in W 1,n norm. Moreover, if g is smooth away from a compact
subset, then we can take g such that g = g on M \ K somecompact set K independent of .

RemarkSince we take g such that g = g on M \ K some compact set K
independent of , by the definition of ADM mass, m(g ) = m(g ).
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

From this mollification, the scalar curvature distribution has an
approximation.
Lemma (2.2)
1, n
Let M n be smooth, g 2 C 0 \ Wloc
on M. Suppose that the L 2
Sobolev constant of (M , g ) has an upper bound Cs . Let {g } themollification of g. Suppose g = g on M \ K some compact set K .
Then|hRg , u2 i

hRg , u2 i| 

( )

Z

M

|ru|2 d µg , 8u 2 C01 (M ),where lim !0 ( ) = 0. Here ( ) depends only on the Sobolev
constant Cs and the W 1,n -norm of |g g |.
The L 2 Sobolev constant condition means: 8u 2 C01 (M ),

⇣R

2n
n 2

⌘ nn 2Ru d µg Cs M |ru|2 d µg . This always holds forasymptotically flat manifolds.
M

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

RemarkIn the same condition of Lemma 2.2, we can calculate in the samemanner and get that|hRg , ui
where

hRg , ui| 

( )

Z

n

M

|ru| n 1 d µg( ) is independent of u andWeimin Sheng with W. Jiang and H. Zhang! nn 1, 8u 2 C01 (M ),( ) ! 0 as

! 0.

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

1

Background
Positive Mass Theorem
Our theorem

2

The singular metric has nonnegative scalar curvature
Smoothing the metric
Weak nonnegative scalar curvature

3

Positive mass Theorem: Nonnegativity

4

Positive mass Theorem: Rigidity
Ricci flat away from the singular set
RCD space with nonnegative Ricci curvature

5

Scalar curvature in distribution with lower bounds
Scalar curvature lower bounds are preserved by Ricci
flow
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Smoothingmass theTheorem:
metric Nonnegativity
Weak nonnegative
Positivescalarmass
curvature
Theorem: Rigidit

Weak nonnegative scalar curvature
Under the same assumption about ⌃ as in Theorem, we can checkthat Rg is weakly nonnegative.
Lemma (2.3)
1,pLet M n be a smooth manifold with g 2 C 0 \ Wloc (M ) withn  p  1. Assume g is smooth away from a closed subset ⌃ withp
Hn p 1 (⌃) < 1 if n  p < 1 or Hn 1 (⌃) = 0 if p = 1, andassume Rg 0 on M \ ⌃, then hRg , ui 0 for any nonnegative,
compactly supported u 2 W 1,p /(p 1) .
Construct a cut-off function ⌘✏ 0 is key (Cheeger 2003) butpn p 1standard, based on the condition H(⌃) < 1 if n  p < 1 orHn 1 (⌃) = 0 if p = 1.
三1 in Bg E

应iihR , (1giihRg , ui = hRg , ⌘✏ ui + hRg , (1 ⌘✏ )ui.
Z⌘✏ )ui =Rg (1 ⌘✏ )ud µg 0, lim |hRg , ⌘✏ ui| = 0.
M \⌃Weimin Sheng with W. Jiang and H. Zhang✏!0

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive mass Theorem: Nonnegativity Positive mass Theorem: Rigidit

Schoen-Yau’s Lemma
Let us recall a lemma essentially proved in Schoen-Yau 1979.
Lemma (3.1)
Let (M , g̃ ) be a complete smooth asymptotically flat manifold, f , h
be smooth functions with compact support on M, then there exists
an ✏ > 0, such that if f satisfies

ZMf ⇠ 2 d µg̃

✏

Z

M

|r⇠|2 d µg̃ , 8⇠ 2 C01 (M ),then the equationg̃ v

fv = h(5)

has a solution v satisfying v = O (r 2 n ) as r ! 1. Moreover,v = A /r n 2 + !,where A is a constant, ! = O (r 1 n ) and |@!| = O (r n ).
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive mass Theorem: Nonnegativity Positive mass Theorem: Rigidit

The proof is almost the same as Schoen-Yau 1979.
1, p
Now under assumption of Theorem (g 2 C 0 \ Wloc (M ), n  p  1,
· · · ), we consider the equation (see also Li-Mantoulidis 2019)(cn '2 Rg u = 0
limr !1 u = 1
g u

on

M

,

(6)

where ' : M ! [0, 1] be a smooth cut-off function such that ' = 1
on K , ' = 0 outside some neighborhood of K , a compact set
K ⌃, cn = 4(nn 21) .
Corollary (3.2)
There exists 0 > 0 such that the equation (6) has a positive
solution for all 2 (0, 0 ). Moreover, we have
u =1+

A
r

+ !,
n 2

u.tt

where A is a constant, ! = O (r 1 n ) and |@!| = O (r n ).
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive mass Theorem: Nonnegativity Positive mass Theorem: Rigidit

Now, we define the conformal metricsg̃ = u4n 2

gThen the standard conformal transformation formula shows
1R̃g = cn u

=u

1

n +2
n 2

n +2
n 2

(

( Rg

g u

cn R g u )'2 Rg )

0,since ' = 1 on K and Rg = R (g )Weimin Sheng with W. Jiang and H. Zhang

0 on M \K .

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive mass Theorem: Nonnegativity Positive mass Theorem: Rigidit

Lemma (3.4)
The lower limit of mass of g is no less than the lower limit of massof g̃ .
By the definition of mass, we can calculate straightforwardly andget the following equation (see P. Miao 2002),m(g̃ ) = m(g ) + (nWe can show
1A =(2 n)!n
and

Z ⇣
M

2

1)A

2

(7)

2

⌘

|rg u | + cn ' Rg u d µg

lim+ A  0.
!0

Thus

lim m(g )lim m(g̃ ).

!0+

!0+

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive mass Theorem: Nonnegativity Positive mass Theorem: Rigidit

Proof of the inequality part of Theorem
Proof of the inequality part of Theorem.
Since g = g on M \K , we havem (g ) = m (g )Since g̃ has nonnegative scalar curvature, by the classicalpositive mass theorem, we havem(g̃ )

0Thus by Lemma 3.4, we havem (g )

0⇤
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Ricci flat
mass
away
Theorem:from theNonnegativity
singular set RCD
Positive
space
mass
with
Theorem:nonnegative
RigiditR

1

Background
Positive Mass Theorem
Our theorem

2

The singular metric has nonnegative scalar curvature
Smoothing the metric
Weak nonnegative scalar curvature

3

Positive mass Theorem: Nonnegativity

4

Positive mass Theorem: Rigidity
Ricci flat away from the singular set
RCD space with nonnegative Ricci curvature

5

Scalar curvature in distribution with lower bounds
Scalar curvature lower bounds are preserved by Ricci
flow
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Ricci flat
mass
away
Theorem:from theNonnegativity
singular set RCD
Positive
space
mass
with
Theorem:nonnegative
RigiditR

Outline of the proof of rigidity partLet us outline the idea of the proof.
First, we follow the idea of Shi-Tam 2018 to show that the
manifold is Ricci flat away from ⌃.
Then we show that the manifold has nonnegative Ricci
curvature in RCD sense.
Noting that the manifold is asymptotically flat, by volume
convergence and volume comparison, we get the rigidity
result.

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Ricci flat
mass
away
Theorem:from theNonnegativity
singular set RCD
Positive
space
mass
with
Theorem:nonnegative
RigiditR

We first prove that m(g ) = 0 implies Ricci curvature vanishingaway from the singular set ⌃.
Lemma (4.1)
Assume as Theorem. If m(g ) = 0, then Ricg ⌘ 0 on M \⌃.
The idea of the argument comes from Shi-Tam 2018. We canargue it by a contradiction through a local perturbation.

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Ricci flat
mass
away
Theorem:from theNonnegativity
singular set RCD
Positive
space
mass
with
Theorem:nonnegative
RigiditR

1

Background
Positive Mass Theorem
Our theorem

2

The singular metric has nonnegative scalar curvature
Smoothing the metric
Weak nonnegative scalar curvature

3

Positive mass Theorem: Nonnegativity

4

Positive mass Theorem: Rigidity
Ricci flat away from the singular set
RCD space with nonnegative Ricci curvature

5

Scalar curvature in distribution with lower bounds
Scalar curvature lower bounds are preserved by Ricci
flow
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Ricci flat
mass
away
Theorem:from theNonnegativity
singular set RCD
Positive
space
mass
with
Theorem:nonnegative
RigiditR

RCD space
We show that the singular space has nonnegative Ricci curvaturein RCD sense providing the manifold is Ricci flat away from thesingular set.
Theorem (4.2)
1, p

Let (M n , g ) (n 3) be a smooth manifold with g 2 C 0 \ Wloc (M )with n  p  1. Assume g is smooth and Ricci flat away from apn p 1
closed subset ⌃ with H(⌃) < 1 when n  p < 1 andHn 1 (⌃) = 0 when p = 1, and assume g is asymptotically flat.
Then (M n , g ) as a metric measure space with Lebesgue measure
has nonnegative Ricci curvature in the sense of RCD.

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Ricci flat
mass
away
Theorem:from theNonnegativity
singular set RCD
Positive
space
mass
with
Theorem:nonnegative
RigiditR

Definition (4.3 RCD Ricci lower bound)
Let K be some real constant. For a Riemannian manifold (M n , g )with g 2 C 0 (M ), we say that it is a RCD(K , n) space, or say it has
Ricci curvature not less than K in the sense of RCD, if(1) it is infinitesimally Hilbertian,(2) for some C > 0, and some point p 2 M, it holds2µg (Br (x ))  e Cr , for any r > 0, where µg is Lebesgue
measure taken w. r. t. g,(3) for any f 2 W 1,2 (M ) satisfying |rf | 2 L 1 (M ), it admits a
Lipschitz representative f̃ with Lip(f̃ )  krf kL 1 (M ) ,(4) 8f 2 D ( ) with f 2 W 1,2 (M ), 8' 2 L 1 (M ) \ D ( ) with' 0, ' 2 L 1 (M ), the Bochner ineq
1
2

Z

M

|rf |

2

'd µg

1
n

Z

M

2

( f ) 'd µg +holds.
Weimin Sheng with W. Jiang and H. Zhang

Z

M

⇣

2

⌘

' hrf , r f i + K |rf | d µg

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Ricci flat
mass
away
Theorem:from theNonnegativity
singular set RCD
Positive
space
mass
with
Theorem:nonnegative
RigiditR

RemarkFor a Riemannian manifold (M n , g ) with volume measure andg 2 C 0 (M ), (1) and (3) in Definition 4.3 hold automatically.
If g is asymptotically flat, then (2) holds.

Therefore, to prove Theorem 4.2, it only needs to check a
weak Bochner inequality (see Proposition 4.4).
Proposition (4.4 Weak Bochner inequality)
Assume as Theorem 4.2. Then for any u 2 D ( ) such thatu 2 W 1,2 , and any nonnegative bounded test function ' 2 D ( )with | '| 2 L 1 , the following Bochner inequality holds
1
2

Z

M

'|ru|2 d µg

ZM

'hr u, ruid µg

Weimin Sheng with W. Jiang and H. Zhang

1
n

Z

M

'( u)2 d µg . (8)Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Ricci flat
mass
away
Theorem:from theNonnegativity
singular set RCD
Positive
space
mass
with
Theorem:nonnegative
RigiditR

Outline of the Proof of Prop 4.4
Step 1 The classical Bochner inequality holds pointwisely awayfrom the singular set for smooth functions :
1
|ru|2
2

1
( u)2 + hr u, rui, onn

M \ ⌃, 8u 2 C 1 (M ).

To get the weak Bochner inequality for u 2 C 1 (M ), we may
multiply the cut-off function ✏ , times a test function ' to theinequality above, and integrat it, take limit as ✏ ! 0+ .
Step 2 For u 2 D ( ) with u 2 W 1,2 , we construct a smoothapproximation of u, and take limit for the W-Bochner inequality forthe smooth approximation. In order to ensure the limit hold, thesmooth approx. must satisfy some good property, which does nothold for standard mollification. We use the heat kernel to construct
the approximation and get the gradient estimate. This is the mostdifficult part.
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Ricci flat
mass
away
Theorem:from theNonnegativity
singular set RCD
Positive
space
mass
with
Theorem:nonnegative
RigiditR

Proof of Theorem 4.2.
Noting the Bochner inequality proved in Proposition 4.4, (M n , g )with Lebesgue measure is an RCD space with nonnegative Ricci
curvature.
⇤
Proof of rigidity part of Theorem .
Now the rigidity part of Theorem follows immediately from
Theorem 4.2 and the volume stability theorem of RCD space(Lott-Villani 2009; Sturm2006). Actually, by Lemma 4.1andTheorem 4.2, the manifold (M n , g ) has nonnegative Ricci curvaturein RCD sense. (To be continued.)⇤Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Ricci flat
mass
away
Theorem:from theNonnegativity
singular set RCD
Positive
space
mass
with
Theorem:nonnegative
RigiditR

Proof of rigidity part of Theorem
Continue.
Take any point x 2 M, noting that the manifold is asymptoticallyflat, we have

Vol(BR (x ))
= 1,
R !1
!n R nlim(9)

By volume comparison of RCD space with nonnegative Ricci
curvature, we have

Vol(BR (x )) = !n R n .

(10)

By volume rigidity Theorem by De Philippis-Gigli-Nicola 2018 , we
get BR (x ) is isometric to BR (0n ) ⇢ Rn . This implies M is isometricto Rn .
⇤
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Scalarmass
curvature
Theorem:lowerNonnegativity
bounds are preserved
Positiveby
mass
RicciTheorem:
flow
Rigidit1

Background
Positive Mass Theorem
Our theorem

2

The singular metric has nonnegative scalar curvature
Smoothing the metric
Weak nonnegative scalar curvature

3

Positive mass Theorem: Nonnegativity

4

Positive mass Theorem: Rigidity
Ricci flat away from the singular set
RCD space with nonnegative Ricci curvature

5

Scalar curvature in distribution with lower bounds
Scalar curvature lower bounds are preserved by Ricci
flow
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Scalarmass
curvature
Theorem:lowerNonnegativity
bounds are preserved
Positiveby
mass
RicciTheorem:
flow
Rigidit

Scalar curvature lower bounds preserved by Ricci flow
Theorem (5.1)
M n compact, g 2 W 1,p (M ) \ C 2 (M \ ⌃), (n < p  1),
n

p
p 1

(⌃) < 1 if n < p < 1 or Hn 1 (⌃) = 0 if p = 1. Rg a on
M \ ⌃ for some constant a. Let g (t ), t 2 (0, T0 ) be the Ricci flowfrom initial metric g. Then Rg(t ) a on M, 8t 2 (0, T0 ).
H

RemarkHere the Ricci flow g (t ), t 2 (0, T0 ) from C 0 initial metric gmeans that g (t ) satisfies the Ricci flow equation for t 2 (0, T0 )and limt !0+ dGH ((M , g (t )), (M , g )) = 0(M. Simon 2002) For a continuous metric g, there exists a flowg̃ (t ), t 2 (0, T0 ) equivalent to the Ricci flow modulo the actionof diffeomorphisms and C 0 converge to g.
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Scalarmass
curvature
Theorem:lowerNonnegativity
bounds are preserved
Positiveby
mass
RicciTheorem:
flow
Rigidit

The relation between nonnegative and positive scalar
curvature
Theorem (5.2)
M n (nnpp 13) compact, g 2 W 1,p (M ) \ C 2 (M \ ⌃), (n < p  1),(⌃) < 1 if n < p < 1 or Hn 1 (⌃) = 0 if p = 1. Rg 0 on
M \ ⌃. Then either (M , g ) is isometric to a Ricci flat manifold or M
admits a metric with positive scalar curvature (Rg > 0).
H

RemarkIf p = 1, the condition Hn 1 (⌃) = 0 is optimal. In fact, if ⌃ is a
hypersurface, then there exists counterexamples: we canprescribe arbitrary scalar curvature on M \ ⌃ if M is a torus.

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Scalarmass
curvature
Theorem:lowerNonnegativity
bounds are preserved
Positiveby
mass
RicciTheorem:
flow
RigiditThis result is connected with the following
Conjecture (R. Schoen)
Let g be a C 0 metric on a compact manifold M which is smoothaway from a submanifold ⌃ ⇢ M with codim(⌃ ⇢ M ) 3 andRg 0 on M \ ⌃, then either g smoothly extends to a Ricci flat
metric on M, or M admits a metric with Rg > 0.
Theorem (Li - Mantoulidis 2019)
Schoen’s conjecture holds if n = 3 and ⌃ is a finite set.
Theorem 5.2 is a corollary of Theorem 5.1 and following
Theorem (Bourguignon, before 1975)
Let (M , g ) be a compact Riemannian manifold with nonnegative
scalar curvature (Rg 0), then either g is Ricci flat, or M admits ametric with positive scalar curvature (Rg > 0).
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Scalarmass
curvature
Theorem:lowerNonnegativity
bounds are preserved
Positiveby
mass
RicciTheorem:
flow
RigiditTorus rigidity
One of the most well known results related to scalar curvature is:
Theorem (Torus rigidity theorem, Schoen, Yau, Gromov, Lawson)
Any metrics on torus with nonnegative scalar curvature is flat.
RemarkIn 1979, Schoen and Yau proved this theorem in dimensions 3
to 7.
In 1980, Gromov and Lawson proved this theorem in all
dimensions.
It would be useful if we strengthen this theorem replacing thetorus by the connected sum of a torus and an arbitrarycompact manifold of the same dimensional. Schoen and Yau
used minimal surface approach and proved it in 2017.
Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Scalarmass
curvature
Theorem:lowerNonnegativity
bounds are preserved
Positiveby
mass
RicciTheorem:
flow
Rigidit

We have
Theorem (5.3)
p

M = T n , g 2 W 1,p (M ) \ C 2 (M \ ⌃), (n < p  1), Hn p 1 (⌃) < 1 if
n < p < 1 or Hn 1 (⌃) = 0 if p = 1. Rg 0 on M \ ⌃. Then (M , g )is isometric to a flat torus.
RemarkIf p = 1, the condition Hn 1 (⌃) = 0 is optimal. In fact, if ⌃ is a
hypersurface, then there exists counterexamples: we canprescribe arbitrary scalar curvature on M \ ⌃ if M is a torus.

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics

Background The singular metric has nonnegative scalar curvature Positive
Scalarmass
curvature
Theorem:lowerNonnegativity
bounds are preserved
Positiveby
mass
RicciTheorem:
flow
RigiditThanks

Weimin Sheng with W. Jiang and H. Zhang

Positive mass theorem with low-regularity Riemannian metrics