Abstract
In this paper, we get the sharpest known to date lower bounds for the minimal Green energy of the compact harmonic manifolds of any dimension. Our proof generalizes previous ad-hoc arguments for the most basic harmonic manifold, i.e. the sphere, extending it to the general case and remarkably simplifying both the conceptual approach and the computations.
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Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. First and third authors belong to the Universidad de Cantabria and are supported by Grant PID2020-113887GB-I00 funded by MCIN/ AEI /10.13039/501100011033. The second author belongs to the Universitat de Barcelona and has been partially supported by grant PID2021-123405NB-I00 by the Ministerio de Ciencia, Innovación y Universidades, Gobierno de España and by the Generalitat de Catalunya (project 2017 SGR 358). The third author has also been supported by Grant PRE2018-086103 funded by MCIN/AEI/10.13039/501100011033 and by ESF Investing in your future.
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Appendix
Appendix
1.1 A Some Properties of the Green Function
We recall some properties of \(G(\mathcal M;p,q)\) which hold in any compact manifold \(\mathcal M\). Green’s function is in some sense the inverse of the Laplace–Beltrami operator:
Proposition A.1
If \(f:\mathcal {M} \rightarrow \mathbb {R}\) is a continuous function with \(\int f =0\), then
is of class \(C^2\) in \(\mathcal {M}\) and satisfies \(\Delta u=f\).
Proof
See [2, Remark 2.3]. \(\square \)
The following result says that the Green function is a conditionally positive definite kernel.
Proposition A.2
Let \(\nu \) be any finite signed measure in \(\mathcal M\) with \(\nu (M)=0\). Then,
with equality if and only if \(\nu =0\).
Proof
See [2, p. 166, Def. 3.2] and [2, p. 175, Prop. 3.14]. \(\square \)
We also have the following result [12, p. 108, Lemma 5.3.1] that gives a closed formula for the expected value of the Green function when one of its entries lives in a ball.
Lemma A.3
Let \(\mathcal M=\mathbb S^n, \mathbb R\mathbb P^n, \mathbb C\mathbb P^n, \mathbb H\mathbb P^n\) or \(\mathbb O\mathbb P^2\).Then, for any \(p_0,p\in \mathcal {M}\),
-
If \(d_R(p_0,p)\ge a\), then
$$\begin{aligned} \frac{1}{V(a)}\int _{q\in B(p_0,a)}G(\mathcal {M}; p, q)\,dq=&\,G(\mathcal {M}; p, p_0)+K(\mathcal M,a). \end{aligned}$$ -
If \(d_R(p_0,p)< a\), then
$$\begin{aligned} \frac{1}{V(a)}\int _{q\in B(p_0,a)}G(\mathcal {M}; p, q)\,dq=&\,G(\mathcal {M}; p, p_0)+K(\mathcal M,a)\\&-\frac{1}{V(a)}\int _{d(p_0,p)}^av(r)\int _{d(p_0,p)}^r \frac{du}{v(u)}dr. \end{aligned}$$
In particular, for any \(p_0,p\in \mathcal {M}\),
Proof
We sketch a proof for completeness. For the first identity, multiplying by V(a) and computing the derivative with respect to a, it suffices to check that
It is clear that both sides of Eq. A.1 are equal as \(a\rightarrow 0\). We check that their derivatives also coincide. Call F(a) the left–hand term in Eq. A.1. Writing it down in normal coordinates with basepoint \(p_0\), we find that the derivative of the left–hand side equals
where N(q) is the unit vector orthogonal to \(S(p_0,a)\) at q and \(\nabla \) is the covariant derivative. From Green’s second identity, we get
Hence, the derivatives at both sides of Eq. A.1 are equal, proving Eq. A.1 and the first claim of the lemma in the case that \(d_R(p_0,p)<a\). The case \(d_R(p_0,p)=a\) follows from the continuity of both sides of the equality. Finally, if \(d_R(p_0,p)=t<a\) we can still compute the derivative using Green’s second identity, now to the other open set delimited by \(S(p_0,a)\) and using \(-N(q)\):
All in one, we have proved
The second claim in the lemma now follows, since
\(\square \)
1.2 B Closed Formulas for \(K(\mathcal M,a)\) and \(\Theta (\mathcal M,a)\)
Although we have not used them in our analysis or proofs above, in the cases \(\mathcal M=\mathbb{C}\mathbb{P}^n,\mathbb{H}\mathbb{P}^n,\mathbb{O}\mathbb{P}^2\) it is possible to produce exact formulas for these two functions. We summarize them in the following result.
Proposition B.1
Denoting \(S=\sin a\), we have:
Proof
These are all obtained directly from the definitions Eqs. 2.2 and 2.3, carefully computing all the indefinite integrals and using the explicit formulas given in Table 1. Once computed, their correctness can be checked by automatic differentiation. \(\square \)
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Beltrán, C., de la Torre, V. & Lizarte, F. Lower Bound for the Green Energy of Point Configurations in Harmonic Manifolds. Potential Anal (2023). https://doi.org/10.1007/s11118-023-10108-2
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DOI: https://doi.org/10.1007/s11118-023-10108-2