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Let $(M_i,g_i,x_i)$ be a sequence of stochastically complete pointed Riemannanian manifolds ($x_i$ being the marked point on $M_i$) of injectivity radius uniformly bounded from below, Gromov-Haussdorf converging to an other stochastically complete manifold $(M_{\infty},g_{\infty},x_{\infty})$ (of same dimension from the assumption on the injectivity radius). I am interested in the convergence of the associated family of heat kernels $p_i(x,y,t)$ to the heat kernel of the limit manifold $p_{\infty}(x,y,t)$.

More precisely;

We fixe $R,t > 0$ and we denote by $B(x,R)$ the ball centered at $x$ on radius $R$, do we have uniform convergence over the product $B(x_i,R) \times B(x_i,R)$ of the function $p_i(\cdot,\cdot,t)$ to the function $p_{\infty}(\cdot, \cdot,t)$ defined on the product $B(x_{\infty},R) \times B(x_{\infty},R)$ as $i \to \infty$ ?

The special case I am ultimately looking for concerns a sequence of Galois covers of a given compact manifold, converging to some limit Galois cover. As a toy model, I checked it for the sequence of metric circles $\mathbb{R}/ (i\mathbb{Z})$ using some Fourier analysis, and it seems to be true in this case.

Thank you for your reading.

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  • $\begingroup$ If you mean convergence uniform in $x$, I suspect the answer is false in general for hyperbolic manifolds. $\endgroup$
    – Bombyx mori
    Jul 13, 2018 at 9:13
  • $\begingroup$ Uniform in $x$, but the last remaining in some definite neighbourhood of the marked point $x_i$, which is less asking than a uniform convergence over the whole manifold if this is what you were thinking of, were you? $\endgroup$
    – user50806
    Jul 13, 2018 at 9:17
  • $\begingroup$ Yes, I was thinking about some bound uniform in $x$. Since you assumed scholastically completeness, maybe this property will be useful. $\endgroup$
    – Bombyx mori
    Jul 13, 2018 at 9:22
  • $\begingroup$ I am particularly interested in covers of a given compact hyperbolic manifold and it seems to me that is the setting you are comfortable with, is it by any chance easier in this case ? $\endgroup$
    – user50806
    Jul 13, 2018 at 9:26
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    $\begingroup$ My guess is no, because even for Einstein manifolds, the Gromov-Hausdauff convergence may produce non-Einstein manifolds (perhaps?) with singular metric. I am not a big expert on Brownian motion, so I do not know how this relates to stochastic completeness. $\endgroup$
    – Bombyx mori
    Jul 13, 2018 at 10:20

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As said, this holds if the manifolds have Ricci curvature uniformly bounded from below. Perhaps the quickest reference for this convergence is my paper

https://link.springer.com/article/10.1007/s00526-009-0303-9

which is also the one where, to the best of my knowledge, this topic has been first studied. Notice that here the convergence of the heat flow, not kernel, is studied. And only in the compact case.

This approach has been generalized in the more recent

https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/pdv047

Let me say few words to put this result in context. What plays a key role in both papers is

1) the Jordan-Kinderlehrer-Otto interpretation of the heat flow as gradient flow of the Boltzmann entropy w.r.t. the quadratic Kantorovich (Wasserstein) distance

2) the Lott-Sturm-Villani approach to lower Ricci curvature bounds in terms of $K$-convexity of the Boltzmann entropy w.r.t. the same distance.

As proved by Lott-Sturm-Villani, mGH-convergence of metric measure spaces implies De Giorgi's $\Gamma$-convergence of the Boltzmann entropy. $\Gamma$-convergence is a kind of `zeroth order' convergence, so to say, and per se is not sufficient to pass to the limit in the gradient flow equation, which is first-order in nature. But if one imposes a uniform second order bound, like a uniform convexity assumption, then gradient flows pass to the limit (by analogy: pointwise convergence of functions does not imply any kind of pointwise convergence of derivatives in general, but if all the functions are convex then - up to deal with non-differentiability points - this becomes true). For more on this topic, see also Sandier-Serfaty's paper

https://onlinelibrary.wiley.com/doi/full/10.1002/cpa.20046

Finally, let me point out that mGH-convergence of manifolds with Ricci curvature bounded from below have been extensively studied by Cheeger-Colding in the 90's: at that time the `technology' of optimal transport was not available, but nevertheless they proved several results about convergence of Dirichlet forms, see in particular

https://projecteuclid.org/euclid.jdg/1214342146

There is nothing specific about heat kernel, but my impression is that this is the case only because they were not aiming at that kind of result

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