Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds
In this short note, we review some known results on the structure and regularity of spaces with lower Ricci curvature bounds. We present some known and new open questions about the next steps.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 104, 8 pages
Conjectures and Open Questions
on the Structure and Regularity of Spaces
with Lower Ricci Curvature Bounds
Aaron NABER
Department of Mathematics, Northwestern University, USA
E-mail: anaber@math.northwestern.edu
Received July 29, 2020, in final form October 11, 2020; Published online October 20, 2020
https://doi.org/10.3842/SIGMA.2020.104
Abstract. In this short note we review some known results on the structure and regularity of
spaces with lower Ricci curvature bounds. We present some known and new open questions
about next steps.
Key words: Ricci curvature; regularity
2020 Mathematics Subject Classification: 53C21; 53C23
1 Introduction to Ricci curvature and limits
1.1 What is Ricci curvature?
The Riemannian curvature tensor Rm of a Riemannian manifold
(
Mn, g
)
is a four tensor, that is
it takes three vector fields and gives a vector field. Classically it is defined as the antisymmetric
part of the Hessian:
Rm(X,Y )Z ≡ ∇2
X,Y Z −∇2
Y,XZ,
however from an analytic point of view one should interpret this four tensor Rm as the Hessian
of the Riemannian metric g. The actual Hessian of g is zero, essentially by definition, however
Rm is the correct moral replacement. In particular, one should expect that a manifold with
bounded curvature should behave much like a function on Rn with C2 bounds. Up to a choice
of coordinate chart, this is indeed the case.
The Ricci curvature is then defined as the two tensor obtained by tracing
Ric(X,Y ) ≡
∑
Ea
〈Rm(X,Ea)Ea, Y 〉,
where Ea is an orthonormal basis. If Rm is interpreted as the Hessian of the metric g, then
one should interpret the Ricci curvature as the Laplacian of the Riemannian metric. Clearly,
it is a highly nonlinear Laplacian and this moral only holds up to the diffeomorphism gauge
group, but it still puts into perspective why the Ricci curvature should play a central role in
so many situations. One can then similarly define from this the scalar curvature as the trace
R ≡
∑
Ea
Ric(Ea, Ea).
This paper is a contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov
on his 75th Birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Gromov.html
mailto:anaber@math.northwestern.edu
https://doi.org/10.3842/SIGMA.2020.104
https://www.emis.de/journals/SIGMA/Gromov.html
2 A. Naber
1.2 Limiting under lower Ricci curvature
The study of the structure and regularity of spaces with lower bounds on Ricci curvature is an
old topic. The possibility for making a systematic study of the structure of such spaces began
with the proof of a compactness theorem:
Theorem 1.1 (Gromov compactness theorem). Let
(
Mn
i , gi, pi
)
be a sequence of pointed Rie-
mannian manifolds such that Rici ≥ −λ uniformly. Then there exists a metric space (X, d, p)
with p ∈ X such that, after possibly passing to a subsequence, we have(
Mn, gi, pi
)
→ (X, d, p),
where the convergence is in the Gromov–Hausdorff topology.
This compactness theorem immediately begs the question of what type of metric spaces X
might exist as limits. Gromov’s theorem gives only that X is a metric space, without giving
any further refined structure on the space. However, one should expect much more in principle.
Though at this point one can really refine the following categories much more, we can roughly
break our interest in limits(
Mn, gi, pi
)
→ (X, d, p),
into two general groups:
� (lower Ricci noncollapsed)
(
Mn
i , gi, pi
)
satisfy Rici ≥ −λ and noncollapsing Vol(B1(pi)) >
v > 0,
� (lower Ricci collapsed)
(
Mn
i , gi, pi
)
satisfy Rici ≥ −λ and collapsing Vol(B1(pi))→ 0.
In each of these cases, one want to explore both the structure of the metric spaces (X, d) and
a priori regularity of the sequence
(
Mn
i , gi
)
. We will cover what is known in these cases and
present some open questions on future directions.
2 Lower Ricci curvature and noncollapsing
Let us discuss in this section the structure and regularity of limits under a uniform lower Ricci
curvature and noncollapsing assumption:(
Mn
i , gi, pi
)
→ (X, d, p) s.t. Rici ≥ −λ and Vol(B1(pi)) > v > 0.
The first real structure theorem for even the regular part of such spaces was presented by
Cheeger and Colding:
Theorem 2.1 (Cheeger–Colding [2]). Let
(
Mn
i , gi, pi
)
→ (X, d, p) satisfy
Rici ≥ −λ and Vol(B1(pi)) > v > 0,
then X is bi-Hölder to a manifold away from a set of codimension two.
The proof of the above is based on a Federer type stratification theory, which we review in
Section 2.2, as well as a Reifenberg type theorem in order to deal with the regularity of the
regular part. The Reifenberg argument necessarily only gives bi-Hölder control over the regular
set, and the first open question we mention is on whether this can be improved:
In fact, Gromov did also prove that X is a length space.
Conjectures and Open Questions on the Structure and Regularity of Spaces 3
Open Problem 2.2. Let
(
Mn
i , gi, pi
)
→ (X, d, p) satisfy Rici ≥ −λ and Vol(B1(pi)) > v > 0,
then is X bi-Lipschitz to a manifold away from a set of codimension two?
This has turned out to be a subtle question. In order to get some sense of this let us recall
the following example:
Example 2.3 (Colding–Naber [7]). There exists a limit space X, which is homeomorphic to Rn
and for which every tangent cone, see Section 2.1, is Rn. There is a distinguished point p ∈ X
such that for any x ∈ X if γx : [0, 1] → X is the geodesic from p, then for every pair x, y ∈ X
and any θ ∈ [0, 2π) we can find ti → 0 such that the angle between γx(ti) and γy(ti) converges
to θ. As a consequence, if u : B1(p) → Rn is a homeomorphism onto its image with u either
harmonic or a collection of distance functions, then u cannot be bi-Lipschitz.
The above example tells us that the maps one would typically attempt to use to build this
homeomorphism cannot be bi-Lipschitz.
The other remaining question is on the size of the set on which X is a manifold:
Conjecture 2.4. Let
(
Mn
i , gi, pi
)
→ (X, d, p) satisfy Rici ≥ −λ and Vol(B1(pi)) > v > 0,
then X is homeomorphic to a manifold away from a set of codimension three.
The challenge in the above conjecture is that if one only removes a set of codimension three,
then you must contend with some non-small singular points. These singularities should still all
be of the form of a cone over a sphere on this set, and hence homeomorphic to a manifold, but
this is still an open and challenging question.
2.1 Tangent cones of noncollapsed spaces
In order to discuss more refined structure of noncollapsed limits X we need to build the stratified
singular set. The two main ingredients in this is the introduction of tangent cones and of
symmetries of these tangent cones. Let us begin with a definition:
Definition 2.5 (tangent cones). Given a metric space (X, d) with x ∈ X, we say another metric
space Xx is a tangent cone of X at x ∈ X if there exists rj → 0 such that
(
X, r−1
j d, x
)
→ Xx.
Thus a tangent cone is representative of the infinitesimal behavior of a metric space X near
the point x. The tangent cone Xx is obtained by zooming onto smaller and smaller balls around x
and taking a limit.
It follows from Gromov’s compactness theorem that tangent cones exist for Ricci limits,
a result which holds whether or not the limit is noncollapsed. One of the most important
contributions of the works of Cheeger–Colding was the proof of the following:
Theorem 2.6 (tangent cones are metric cones [1]). Let
(
Mn
i , gi, pi
)
→ (X, d, p) satisfy Rici ≥
−λ and Vol(B1(pi)) > v > 0, then every tangent cone Xx is a metric cone Xx ≡ C(Yx) over Yx
with diamYx ≤ π.
The study of a stratification theory for essentially all nonlinear equations begins with a state-
ment analogous to the above. In most contexts the proof of the above statement is actually quite
simple, however it takes a good amount of deep analysis in the context of lower Ricci bounds.
It is known that tangent cones need not be unique at a given point. It follows from the
stratification theory that away from a set of codimension two every tangent cone is Rn, and
hence unique. A conjecture from [6] is that this uniqueness should still hold down to a set of
codimension three:
Nonlinear harmonic maps, minimal surfaces, Yang–Mills, etc.
4 A. Naber
Conjecture 2.7. Let
(
Mn
i , gi, pi
)
→ (X, d, p) satisfy Ric ≥ −(n− 1) and Vol(B1(pi)) > v > 0,
then the tangent cones of X are unique away from a set of codimension three.
To prove the above means controlling the top stratum of the singular set to a much larger
degree than currently exists. Recently it was proved in [3] that tangent cones are at least unique
away from a set of n− 2 measure zero. See Conjecture 2.16 on further refinements of the above
conjecture.
Besides tangent cones being nonunique, examples in [6] even provided examples where tangent
cones are not homeomorphic:
Example 2.8 (Colding–Naber). There exists a limit space X5 with p ∈ X such that tangent
cones at p are homeomorphic to both R5 and C
(
CP 2#CP 2
)
, depending on the blow up sequence.
It was conjectured this should not happen often:
Conjecture 2.9. Let
(
Mn
i , gi, pi
)
→ (X, d, p) satisfy Ric ≥ −(n− 1) and Vol(B1(pi)) > v > 0,
then the tangent cones of X are all homoemorphic at a fixed point away from a set of codimension
five.
In order to build the stratification of the singular set, we need to also discuss the symmetries
of tangent cones:
Definition 2.10 (symmetries). We say a metric space X is 0-symmetric if it is a cone space
X ≡ C(Y ). We say X is k-symmetric if X ≡ Rk × C(Y ) additionally splits off k-Euclidean
factors.
2.2 Stratification of the singular set
Now we can introduce the stratification of a noncollapsed Ricci limit space X, in the sense of
Federer:
Definition 2.11 (stratification). We define the k-stratum Sk(X) ⊆ X by
Sk(X) ≡
{
x ∈ X : no tangent cone at x is k + 1-symmetric
}
.
Note the subtlety in the above definition, which goes back to Federer. If x 6∈ Sk(X), then
this guarantees the existence of at least one tangent cone with k + 1-degrees of symmetry. We
have that this is a nested sequence
S0(X) ⊆ S1(X) ⊆ · · · ⊆ Sn−1(X) ⊆ X.
By applying Theorem 2.6 and the dimension reduction technique of Federer, [2] was able to show
dimSk ≤ k.
It is an important additional result of [2] that Sn−1(X) = Sn−2(X), which is the starting
point for Theorem 2.1. In terms of additional structure for the stratification of singular sets,
the main result is
Theorem 2.12 (Cheeger–Jiang–Naber [3]). Let
(
Mn
i , gi, pi
)
→ (X, d, p) satisfy Ric ≥ −λ and
Vol(B1(pi)) > v > 0, then
1) Sk(X) is k-rectifiable,
2) For Hk-a.e. x ∈ Sk(X) we have that every tangent cone is k-symmetric.
This was more a consequence of a broader structure theory on the rectifiability of the singular set.
Conjectures and Open Questions on the Structure and Regularity of Spaces 5
Remark 2.13. In order to prove the above one has to work with the quantitative stratifi-
cation Skε , first introduced in [4], which decomposes the stratification into nicer pieces. In
particular, one can show that Skε have finite k-dimensional measure.
Proving the above involves going beyond the dimension reduction technique of Federer. A con-
sequence of the second statement, applied to the top stratum, is that away from a set of n− 2
measure zero ’every’ tangent cone is of the form Rn−2 × C
(
S1
r
)
, where S1
r is a circle of radius
r ≤ 1.
The first statement, that Sk is k-rectifiable, says that Sk(X) has a k-manifold structure away
from a set of measure zero. This ’away from a set of measure zero’ statement turns out to be
sharp as one can build the following example:
Example 2.14 (Li–Na [9]). For k ≤ n − 2 there exists a limit
(
Mn
i , gi, pi
)
→ (X, d, p) where
seci ≥ 0 and Vol(B1(pi)) > v > 0 such that Sk is a k-rectifiable, k-cantor set.
The example tells us that an open manifold structure on Sk is not possible for k ≤ n − 2.
However, one might ask the following:
Conjecture 2.15. Let
(
Mn
i , gi, pi
)
→ (X, d, p) satisfy Ric ≥ −λ and Vol(B1(pi)) > v > 0,
then Sk is contained in a union of k-dimensional submanifolds. More precisely, there exist
a countable collection of bi-Hölder maps ϕi : B1
(
0k
)
→ M such that if Ŝk(X) ≡
⋃
i ϕi
(
B1
(
0k
))
then
1) Sk(X) ⊆ Ŝk(X),
2) for each x ∈ Ŝk(X) we have that every tangent cone is at least k-symmetric.
Note it is important that we only have an inclusion in (1) above as opposed to equality by
Example 2.14. A consequence of the above conjecture is the following:
Conjecture 2.16. Let
(
Mn
i , gi, pi
)
→ (X, d, p) satisfy Ric ≥ −λ and Vol(B1(pi)) > v > 0, then
for each k ≤ n− 2 there exists Ŝk with dim Ŝk ≤ k such that for each x ∈ X \ Ŝk we have that
every tangent cone is at least k + 1-symmetric.
Note that a positive answer to the above conjecture would solve both Conjecture 2.7 and be
a major step toward Conjecture 2.4. If the Mi are Kähler then the above two conjectures have
been solved by the very nice result in [10]. Indeed, in this case one has equality Sk(X) ≡ Ŝk(X)
as the complex analytic nature of the problem forces the singularities to be varieties.
2.3 Regularity of noncollapsed lower Ricci spaces and the energy identity
Estimates and apriori estimates are often times a major goal in the study of solutions of equa-
tions. On noncollapsed spaces with lower Ricci bounds the best one has at the moment is the
following:
Theorem 2.17 (Ji–Na [8]). Let
(
Mn, g, x
)
satisfy Ric ≥ −λ and Vol(B1(x)) > v > 0, then for
each 0 < p < 1 there exists C(n, λ, v, p) > 0 such that
B1(x)
|Ric|p < C.
The above estimate seems less than optimal, and one might predict that an L1 estimate on
the Ricci curvature is possible, which is the context of Yau’s conjecture:
a structure which is not possible in the real case by Example 2.14.
6 A. Naber
Conjecture 2.18 (noncollapsing Yau’s conjecture). Let
(
Mn, g, x
)
satisfy
Ric ≥ −λ and Vol(B1(p)) > v > 0,
then ∃ C(n, λ, v) > 0 such that
B1(x)
R < C.
Recall that if there is a lower bound on the Ricci curvature, then an L1 bound on the Ricci
and scalar curvature are equivalent. See Conjecture 3.5 for a more general version of the above.
Once an L1 estimate on the scalar curvature is established, one can start to ask questions
about the behavior of the scalar curvature R dvg as a measure. The best way to understand this
is to again consider sequences
(
Mn
j , gj , xj
)
→ (X, d, x) and let the measures
Rj dvgj → µ,
converge as measures to a limit on X. To understand the behavior of µ let mention an example:
Example 2.19. Let X = C
(
S1
r
)
be a two dimensional ice cream cone, which metrically is a cone
over a circle of radius r < 1. Using warped coordinates we can construct smooth manifolds Mj
which are isometric to X outside a ball Bj−1(xj) around the cone point, and have globally
nonnegative sectional curvature. It is clear that Rj = 0 outside Bj−1(xj), and from the Gauss–
Bonnet formula we can compute that
�
Mj
Rj = 2π(1− r). In particular, Rj dvgj → 2π(1− r)δ0
converges to a Dirac delta measure at the cone point.
From the above example we see that we should expect the scalar curvature measure to
concentrate along the top stratum of the singular set. Conjecturally, this is all that should
happen. To be precise recall from Theorem 2.12 that away from a set of n − 2 measure zero
there is a radius function rx such that at any x in the domain we have that the tangent cone at x
is unique and isometric to Rn−2 ×C
(
S1
rx
)
. In reference to other areas, particularly Yang–Mills,
we call the following the energy identity conjecture:
Conjecture 2.20 (energy identity). Let (Mn
j , gj , xj) → (X, d, x) satisfy Ricj ≥ −λ and the
noncollapsing condition Vol(B1(xj)) ≥ v > 0, then as measures the scalar curvature converges
Rj dvj → RXHn + 2θxHn−1
Σ + 2π(1− rx)Hn−2
S ,
where RX : X → R is a locally L1 function which is bounded from below, Σ is an n−1 rectifiable
set with θx ≤ 1, and 2π(1−rx)Hn−2
S is an n−2-rectifiable measure supported on the top stratum
of the singular set Sing(X).
Let us consider two basic examples of the above:
Example 2.21. Let X = C
(
S1
r
)
be a metric cone over a circle of radius r < 1, thus one can
picture X as an ice cream cone. We can smooth X to nonnegatively curved manifolds Mi by
rounding off the cone tip, see [9], and by Gauss–Bonnet one has Ri dvi → 2π(1− r)δ0, where δ0
is the dirac delta measure at the cone point.
Example 2.22. Let X̃ = D(0, 1) be a disk of radius 1 and let X be its doubling. Thus X
is a nonsmooth space whose tangent cones are everywhere R2. We can smooth X to nonnega-
tively curved manifolds Mi by rounding off the boundary, and by Gauss–Bonnet one then has
Ri dvi → 2H1
Σ, where Σ is the circle of radius 1.
Conjectures and Open Questions on the Structure and Regularity of Spaces 7
There would be an interesting corollary of the above Conjecture 2.20. We had mentioned
previously that Hn−2(S(X)) can be infinite, however when studying the quantitative stratifica-
tion Skε [3] we do have finiteness results on the measure. The following corollary of the above
conjecture would give effective understanding of this finiteness for the top stratum (see also [9]
for a more general version of the below in the Alexandrov case):
Corollary 2.23. Hn−2
(
Sε(X) ∩ B1
)
≤ C(n, v, λ)ε−1. If we denote εi ≡ 2−i then we have the
Dini type estimate∑
i
εiHn−2
((
Sεi \ Sεi+1
)
∩B1
)
≤ C(n, v, λ).
3 Lower Ricci curvature and collapsing
We now turn our attention to sequences of manifolds with lower Ricci curvature bounds which
are collapsing. In order to do this, it becomes particularly important to associate a new piece
of information to the limiting process, namely a measure. As the volume of the sequence is
tending to zero, one associates with a pointed manifold
(
Mn
i , gi, pi
)
the normalized measure
νi ≡ Vol(B1(pi))
−1 dvi. Then we can consider the measured Gromov–Hausforff limits(
Mn
i , gi, νi, pi
)
→ (X, d, ν, p) s.t. Rici ≥ −λ and Vol(B1(pi))→ 0.
It is first worth emphasizing the importance of even being able to study such a sequence.
The noncollapsing lower volume bound in the previous section is analogous to an upper bound
on the energy in the context of other nonlinear equations, for instance Yang–Mills and nonlinear
harmonic maps. In other contexts there are essentially no, or certainly at least very limited,
situations in which one can study sequences with energy blowing up, and no general theory. The
distinction in the context of lower Ricci curvature is that there is another rigidity which may be
exploited in order to produce regularity, namely the splitting theorem.
Currently, the most complete structural theorem about collapsed limits is obtained by com-
bining the results of [2] and [5]:
Theorem 3.1 (Cheeger–Colding [2], Colding–Naber [5]). Let
(
Mn
i , gi, νi, pi
)
→ (X, d, ν, p) with
Ric ≥ −λ and Vol(B1(pi))→ 0. Then there exists a unique k ≤ n such that X is k-rectifiable. In
particular, there is a ν-full measure set Rk(X) ⊆ X s.t. the tangent cones of x ∈ Rk are unique
and isometric to Rk. Further, ν ∩ Rk is absolutely continuous with respect to the Hausdorff
measure Hk ∩Rk.
The above allows us to associate a unique dimension to a limit space X. It is not clear however
that this dimension agrees with the Hausdorff dimension. Interestingly, along the regular set ν
and Hk are absolutely continuous, and thus the issue is due to the singular set S(X), which is
a ν-measure zero set:
Open Problem 3.2. Show the Hausdorff dim of X is same as the rectifiable dimension k from
Theorem 3.1.
The proof of Theorem 3.1 has two distinct parts. In [2] it is first proved that X =
⋃
kXk
decomposes into pieces which are each k-rectifiable. In order to prove the uniqueness of k, it is
then shown in [5] that tangent cones along geodesics change at a Hölder continuous rate. By then
finding geodesics which must intersect differentXk one eventually concludes a contradiction. The
Hölder rate is sharp when comparing how the geometries along a geodesic change. Conjecturally,
the volume ratio should be behaving even better:
8 A. Naber
Conjecture 3.3. Let γ : (−2, 2)→ X be a minimizing geodesic, then for s, t ∈ (−1, 1) we have
that
1) (Lipschitz rate) ν(Br(γ(s)))
ν(Br(γ(t))) ≤ C(n, λ)|t− s|,
2) (constant density) if X is noncollapsed, then lim
r→0
ν(Br(γ(s)))
ν(Br(γ(t))) = 1.
Theorem 3.1 gives a lot in terms of the analytical structure of X, but it lacks almost com-
pletely a topological understanding of X. Maybe the most important open question in this
direction is the following:
Open Problem 3.4. Let
(
Mn
i , gi, νi, pi
)
→ (X, d, ν, p) with Ric ≥ −λ and Vol(B1(pi)) → 0,
then is there an open subset of full ν-measure R(X) ⊆ X which is homeomorphic to a k-
manifold?
Finally, let us end by discussing the regularity of manifolds with lower Ricci bounds, poten-
tially including those with small volume. The main conjecture out there is one due to Yau, and
we state a local version of it below:
Conjecture 3.5 (local Yau conjecture). Let
(
Mn, g
)
satisfy Ric ≥ −λ, then
�
B1(p)R ≤ C(n, λ).
We discussed refinements of the above in the noncollapsing case in Section 2.3.
Acknowledgements
The second author was partially supported by the National Science Foundation Grant No. DMS-
1809011.
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1 Introduction to Ricci curvature and limits
1.1 What is Ricci curvature?
1.2 Limiting under lower Ricci curvature
2 Lower Ricci curvature and noncollapsing
2.1 Tangent cones of noncollapsed spaces
2.2 Stratification of the singular set
2.3 Regularity of noncollapsed lower Ricci spaces and the energy identity
3 Lower Ricci curvature and collapsing
References
|
| id | nasplib_isofts_kiev_ua-123456789-211016 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T10:37:42Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Naber, Aaron 2025-12-22T09:29:13Z 2020 Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds. Aaron Naber. SIGMA 16 (2020), 104, 8 pages 1815-0659 2020 Mathematics Subject Classification: 53C21; 53C23 arXiv:2010.10031 https://nasplib.isofts.kiev.ua/handle/123456789/211016 https://doi.org/10.3842/SIGMA.2020.104 In this short note, we review some known results on the structure and regularity of spaces with lower Ricci curvature bounds. We present some known and new open questions about the next steps. The second author was partially supported by the National Science Foundation Grant No. DMS1809011. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds Article published earlier |
| spellingShingle | Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds Naber, Aaron |
| title | Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds |
| title_full | Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds |
| title_fullStr | Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds |
| title_full_unstemmed | Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds |
| title_short | Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds |
| title_sort | conjectures and open questions on the structure and regularity of spaces with lower ricci curvature bounds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211016 |
| work_keys_str_mv | AT naberaaron conjecturesandopenquestionsonthestructureandregularityofspaceswithlowerriccicurvaturebounds |