1
$\begingroup$

Most formulations of Varadhan's formula

$$\lim _{t \to 0_+} 4t \log p_t(x,y) = -d(x,y)^2$$

that I have encountered do not specify where $(x,y)$ lives, so until today I imagined that $(x,y) \in M \times M$. It seems that this is not the case, though, since Berger in "A Panoramic View of Riemannian Geometry" explicitly states (Theorem 168, page 424) that the formula is valid only for $x$ within the injectivity radius at $y$, and that outside this subset wild things happen, citing "Short Time Behaviour of the Heat kernel and its Asymptotic Derivatives" by Malliavin and Stroock. So I found the aforementioned article and, surprise!, the authors claim that Varadhan's formula holds for all $(x,y) \in M \times M$ (formula 0.3, second page of the article)!

Now, I am completely confused. Could you please clarify for me what is the correct, rigorous, formal statement of Varadhan's formula on Riemannian manifolds? (To make things worse, if I recall correctly, Varadhan's result was originally obtained for open subsets of $\mathbb R^n$.)

(Let me add that the heat equation differs by a factor of $\frac 1 2$ in front of the Laplacian between the two references above, but the versions of Varadhan's formula are identical, which makes me believe that Berger is at fault here.)

On a funnier note, is differential geometry the branch with the sloppiest texts in mathematics, or can one get even lower? I'm having troubles with a mistake in texts by Chavel and by Donnely, and I'm amazed about how these guys claim (false) things without bothering to prove them (which would detect their mistakes before publication). I am even more amazed at the referees who let these pass...

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ It seems that Malliavin and Stroock Corollary 2.29 shows where one must restrict to stay away from the cut locus. It is when taking derivatives that it becomes important which is discussed on p.552. So it seems Berger has made a slightly sloppy remark, but it's not so amazing such a thing could happen, and is most certainly not limited to DG. I challenge you to find every minor error in any text book! Perhaps your frustration has lead to over exaggerating somewhat? $\endgroup$
    – Paul Bryan
    Commented Nov 2, 2017 at 0:59
  • $\begingroup$ @PaulBryan: Corollary 2.29 concerns the derivatives of $\log p_t$. Is it obvious to you that the domain given therein is the same for $\log p_t$? If so, then Berger was right and formula 0.3 of Malliavin and Stroock is the faulty one. And indeed, I am a bit frustrated, take a look at the discussion found here: mistakes, if not caught, tend to propagate and invalidate a whole work (I was about to base my work on wrong results). Strange how you consider this "minor". $\endgroup$
    – Alex M.
    Commented Nov 2, 2017 at 9:12
  • 1
    $\begingroup$ Right. As I said, it seems to be when taking derivatives that the "wild behaviour" occurs. I don't know the result well enough to say with certainty that Varadahn's formula holds on the entire $M \times M$. As you pointed out, Malliavin and Stroock state that it does hold, while Berger merely says the formula is valid inside the cut locus (which is consistent with Malliavin and Stroock) and that wild things happen near the cut locus. Berger does not seem to actually assert that Varadahn's formula does not hold on all $M \times M$, though a cursory reading may suggest that is being asserted. $\endgroup$
    – Paul Bryan
    Commented Nov 2, 2017 at 11:03
  • 1
    $\begingroup$ What I think is "minor" is that Berger appears to make an aside remark about the subject that could be open to interpretation. My copy does not seem to have section and Google Books Preview doesn't show me the subsequent pages so I can't be sure Berger's remark is only an aside and not used in his book in a major way. Consulting the reference clarifies that Varadahn's formula holds on $M \times M$ and the "wild behaviour" is seen in the derivatives of $\rho_t$. Hardly deserving of being called "sloppy"! Much worse occurs throughout all of mathematics simply because people are not perfect. $\endgroup$
    – Paul Bryan
    Commented Nov 2, 2017 at 11:09
  • 1
    $\begingroup$ @Raziel: Yes, see the latest comment by Paul Bryan: Varadhan's formula is valid on the whole of $M \times M$. Berger jumps from Varadhan's formula to another one that involves derivatives, and for this one you'll have to remove the cut locus. $\endgroup$
    – Alex M.
    Commented Mar 21, 2018 at 18:27

0

Reset to default

You must log in to answer this question.