The Compactness Conclusion in X.M. Li’s Extended Myers’ Theorem

Time to “publish” my theorem of 1995/1996. First for the verbal statement:

A complete Riemannian manifold with strict stochasticly positive Ricci curvature is compact and has finite fundamental group.

(An exact statement and proof is due as soon I find time.)

Said theorem is one of many many many reasons why I droppped the math PhD… Another reason:

I consider myself having been publicly awarded a secret honorary doctorate by K.D. Elworthy at a colloquium of the Max Planck Institute for Maths, Bonn 1996:

While presenting a summa of his life’s work to the Institute (Hirzebruch sitting in front row, me in back) he asserted said theorem had been proven at the workshop on Dirichlet forms right that very week. My attempt of proof presented there, however, had turned out contradictory three days and nights before. (Half a year later I found out why – and soon happened on a true proof fitting 1 page (thus never published), based on a new ingenious result in logarithmic Sobolev inequalities.) After my confession, Elworthy still insisted on my scheduled talk at the workshop. So, I imprompted a talk on something that doesn’t work – quite rare in the world of mathematics. (Anton Thalmeier later wrote a paper on the problem.) The day before Elworthy surprisingly turned out the first and only person on this planet to fully grasp my Regensburg math diploma thesis – Within 10 minutes, while walking to the mensa. We almost lost our way while discussing…

{Update, late September 2010, early c21st: Haven’t yet resurrected computer cum floppy dings. The GIF image file seems missing from my USB sticks collection…}


2 Responses to The Compactness Conclusion in X.M. Li’s Extended Myers’ Theorem

  1. Phil Henshaw says:

    hmmm… would you be interested in a general theorem unifying the conservation laws in a single law of continuity? It seems to imply rather strongly that quantum mechanics and the rest of physics is necessarily descriptive rather than proscriptive, I think.

    You’ll see some coincidence in the date I first proposed it…

    • Hi Phil,
      alas I’ve got almost no practical experience with physics. I am/was a classic “pure” mathematician… Is there any relation to Noether’s theorem (which makes conservation laws correspond with symmetries)?


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