Approximating coarse Ricci curvature on submanifolds of Euclidean space
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Antonio G. Ache
Abstract
For an embedded submanifold Σ ⊂ ℝN, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds Σ in the same way. For this purpose, we derive asymptotics for the approximation of the Ricci curvature proposed in [2]. Specifically, we prove Proposition 3.2 in [2].
A Integrals of monomials
We have encountered some integrals of products of monomials and Gaussian functions on balls centered at the origin of ℝd. It is important to point out that these integrals will be computed with respect to the Lebesgue measure of the Euclidean metric. We now explain how to compute these integrals in a systematic way.
Lemma A.1
Let z = (z1, …, zd) ∈ ℝdand consider a subset {i1, …, ip} of {1, …, d}. Given positive integers αi1, …, αip we have
where ωαi1,…,αipis defined by ωαi1…αip = 0 if some αij is odd, and
A similar computation that we will need is
where
which implies
and by scaling we have the identity
Note also that using
From the above we observe that if
for some constant C0 > 0, and similarly
We now state the main lemma in this section.
Lemma A.2
(Odd-cancellation). Let {i1, …, ip} be a subset of {1, …, d} and let αi1, …, αip be positive integers and let α = αi1 + … + αip. Then for any ε > 0 we have
If all integers αij, …, αip are even then
If at least one integer αii, …, αip is odd then
A.1 Proofs of the two lemmas
Proof of Lemma A.1
The computation of the integral on the left-hand side of (A.1) is in fact equivalent to computing a moment of order α = αi1 + … + αip of a multivariate normal random vector. More specifically, let ξ1, …, ξd be independent and identically distributed (i.i.d.) random variables and assume that the common distribution of these random variables is 𝓝(0, 1), i.e., a normal random variable with mean 0 and variance 1. In this case, the random vector ξ = (ξ1, …, ξd) has a multivariate normal distribution with mean vector μ = (0, …, 0) and co-variance matrix Id. Using that the components of the random vector ξ are independent we have
where 𝔼 is the expectation with respect to the multivariate normal distribution 𝓝(μ, Id). In particular we may view each factor
Observe that by symmetry, if αij is odd we have
and differentiating both sides k times and evaluating at s = 1 we obtain
We obtain
Proof of Lemma A.2
We start by writing
By scaling, i.e. using
noting that (A.5) follows from Lemma A.1. We now estimate II. Observe that from equation (A.2) we have
This proves the lemma. □
B Proof of Expansion Lemma 3.12
Proof of Lemma 3.12
Recall the following expansions, cf. equation (3.18):
and
Then
by definition of ϕ(z). We now look at the expansion of the integrand, Taylor expanding w̃(x, z) in terms of z at z = 0:
We group these by order of polynomial in z, and order of remainder. We distinguish between terms that have the factor 1/(2t) and those that do not. First,
Next, we list all homogeneous terms containing the factor 1/(2t)
and the remainder terms
With this setup (B.1a) becomes
We deal with these in turn. First, applying Lemma A.2 we have
We repeat this for
At this point we pause and, collecting terms, observe the fact that
In other words, going back to (B.23), what is needed to complete the proof is to show that
First we argue that
Note that the expansion of P̃8 involves 40 terms, before subtracting the lower order terms, so we elect to not express these explicitly. However, for fixed x, P̃8 is a polynomial in z with a factor of 1/t, which can be made explicit in terms of J4w, J5ϕ and J4dμ. Each term has order at least 8 in z. Using the change of variable
Now we turn to the remainder terms:
We will explicitly deal with Rϕ(x, z, t), the remaining 7 terms can be dealt with in a similar fashion:
Recall
and (3.17)
We then get
where Cϕ is a constant that depends on bounds on J6ϕ. Scaling, we obtain the following upper bound:
Now clearly, the first term (in the sense by Lemma A.2) is a sum of terms with order at least 3 in t and dependence on J4w and J4dμ. The second term has a single term of order 2 in t, namely
The remainder of the terms (again using the estimates in Lemma A.2) are terms of higher order. We have shown (B.33). The analysis of each of the remainder terms of the form
is similar: In each case we can use bounds such as (B.35) or (B.36) and in others we can use bounds such as
We note that after integration, the terms (B.16), (B.17) and (B.18) determine the least decaying terms. In fact, each of these terms decays like O(t5/2), but (B.16) contains a term of order 6 in z with a factor of 1/(2t) which of course comes down to a term of order O(t2) after rescaling. This is why this term needed special consideration and why we see the term w̃(x, 0)Φ(t2; ϕ) in the expansion. In fact, the term (B.16) is equivalent to w̃(x, 0)Φ(t2; ϕ) in the sense of inequality (3.21). □
Acknowledgements
The authors are grateful to the anonymous referees who provided input and comments that greatly helped to improve the presentation of this article.
Communicated by: F. Duzaar
Funding: The first author was partially supported by a postdoctoral fellowship of the National Science Foundation, award No. DMS-1204742. The second author is partially supported by NSF Grant DMS-1438359.
References
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Semistable Higgs bundles on elliptic surfaces
- On the rigidity of harmonic-Ricci solitons
- On Ricci negative derivations
- Approximating coarse Ricci curvature on submanifolds of Euclidean space
- Convex cones spanned by regular polytopes
- On the geography of line arrangements
- Towards resolving Keller’s cube tiling conjecture in dimension seven
Artikel in diesem Heft
- Frontmatter
- Semistable Higgs bundles on elliptic surfaces
- On the rigidity of harmonic-Ricci solitons
- On Ricci negative derivations
- Approximating coarse Ricci curvature on submanifolds of Euclidean space
- Convex cones spanned by regular polytopes
- On the geography of line arrangements
- Towards resolving Keller’s cube tiling conjecture in dimension seven
