In their paper "Uniformity of rational points", Caporaso, Harris, and Mazur asserted the following:


"The Geometric Lang conjecture has been proved for all surfaces with $c\_1^2 > c\_2$ ([B]), and has recently been announced for all surfaces ([LM])."


The referece [LM] in that paper (link here: <http://www.ams.org/journals/jams/1997-10-01/S0894-0347-97-00195-1/S0894-0347-97-00195-1.pdf>) is a preprint by S. Lu and M. Miyaoka, with no title. Their paper was published in 1997, so surely this preprint should be published by now, but I am unable to find it. Does anyone know the actual title of the paper, and perhaps what journal it is published in?



Geometric Lang conjecture - reference
abx's comment was made while I was writing this, but I am posting it as an answer anyway.


There has not been a proof of this conjecture of Lang, which remains a wide open problem. Lu and Miyaoka's paper to which Caporaso, Harris and Mazur refer has to be [this one](https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1995/0002/0006/MRL-1995-0002-0006-a001.pdf) (*Bounding curves in algebraic surfaces by genus and Chern numbers*, IMRN, 1995) and its sequel cited as [6] in higher dimensions. However, the main result of that paper, cf. Corollary 1 and the announced *Theorem 0*, has actually an (unnatural) smoothness condition on the rational or elliptic curves involved. 


For a comparison you may recall that in the arithmetic case, for arithmetic surfaces (algebraic curves), Vojta has proved a weaker form of his conjecture that has an *arithmetic discriminant* term $d\_a(P)$ in place of a field discriminant term and which, in contrast to the Vojta conjecture, is far too weak to be adequate for deriving any consequence about the $abc$ conjecture.


A related problem and another partial result on the geometric Lang conjecture can be found in [this question](https://mathoverflow.net/questions/173547/generalizations-of-de-franchis-and-function-field-mordell) of mine, and in its answer by Francesco Polizzi. 


