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A well-known theorem by Varadhan relates the logarithm of the heat kernel on a manifold and the geodesic distance function. In particular, if $d(x,y)$ is geodesic distance from $x$ to $y$ and $p(t,x,y)$ is the heat kernel, his theorem shows $$\lim_{t\rightarrow0} -2t\log p(t,x,y)=d(x,y)^2.$$ In other words, you can recover geodesic distance from short-time behavior of the heat kernel.

My question is: Can we say anything about the function $-2t\log p(t,x,y)$ when $t>0$ is fixed?

For instance: Can it satisfy the triangle inequality (approximately)? How does it relate to $d$ or the geometry of the surface? Is it concave or convex? Can we say anything about its critical points?

Thanks!

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    $\begingroup$ If your manifold is closed, then the heat equation drives a function towards its average and so $p(t,x,y)\to V^{-1}$ as $t\to\infty$, where $V$ is the volume of the manifold. If you want to say something about $-2t\log p$ for any fixed $t>0$, your statement should remain true in the limit $t\to\infty$ when $p$ becomes independent of $x$ and $y$. Also, if $t$ is fixed and $x$ is very close to $y$, we have $p(x,t,y)>1$ and your "distance" is negative. Can you be more specific about what you would like to have? $\endgroup$
    – Joonas Ilmavirta
    Commented Oct 12, 2014 at 7:29
  • $\begingroup$ there is a recent paper on the long time behavior of heat kernel, see msc.tsinghua.edu.cn/~gxu/Public/large%20time%20behavior.pdf $\endgroup$
    – littlelittlelittle
    Commented Oct 14, 2014 at 14:32
  • $\begingroup$ @Joonas, I'll admit my question is a bit vague. Basically, a few algorithms in geometry processing use the heat kernel and justify it via this Varadhan paper. This of course is a perfectly acceptable argument, but I'm hoping to understand what happens as you deviate from the $t\rightarrow0$ case. $\endgroup$
    – Justin
    Commented Oct 15, 2014 at 18:31
  • $\begingroup$ The article linked by @littlelittlelittle has moved to msc.tsinghua.edu.cn/~gxu/public/JDG.pdf. Nevertheless, the reader is warned that it doesn't address the question asked by the OP. $\endgroup$
    – Alex M.
    Commented Oct 19, 2016 at 8:24

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