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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?

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    $\begingroup$ The paper is Bounding curves in algebraic surfaces by genus and Chern numbers. Math. Res. Lett. 2 (1995), no. 6, 663-676 (the Caporaso-Harris-Mazur paper was accepted in revised form early 1995). However, Lu and Miyaoka prove only a weak form of the geometric Lang conjecture, namely that there are only finitely many smooth rational or elliptic curves on a surface of general type. $\endgroup$ – abx Jan 15 '18 at 16:18
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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 (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 of mine, and in its answer by Francesco Polizzi.

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