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You made several mistakes. First, you should apply the result with $R=1$ to the manifold $(M,R^{-2}g)$, so that the unit ball in this scaled metric equals the $R$-size ball in the metric $g$. Secon …

This does not answer your question, but it seems interesting to point out. A non-example would be any complex threefold diffeomorphic to the six-sphere (if such a manifold exists). To see this, firs …

[Edited] Such a manifold cannot exist. Indeed the small deformations of the "symmetric" Mukai-Umemura $3$-fold $X$ are described explicitly by Donaldson in this paper, pages 43-44. There he describe …

It seems worthwile to point out that it is not true that the vanishing of the integral first Chern class of a compact Kahler manifold implies that the canonical bundle is holomorphically trivial. It o …

Let me write your equations instead as $$\Theta^{(1)}_{i\bar{j}}=\lambda_1 g_{i\bar{j}},$$ $$\Theta^{(2)}_{i\bar{j}}=\lambda_2 g_{i\bar{j}},$$ where $\lambda_1,\lambda_2$ are real-valued functions. Th …

The first example of this phenomenon was discovered by Iku Nakamura in his 1975 paper Complex parallelisable manifolds and their small deformations, J. Differential Geom. 10 (1975), 85-112. The manif …

The answer is contained in the book by W. Fulton "Intersection theory", chapter 18 "Riemann-Roch for singular varieties".

There is an "elliptic" proof in this paper by Eminenti, La Nave, Mantegazza, see Theorem 3.1. It does still use Perelman's $\mathcal{W}$ entropy (but it does not use the Ricci flow, it just uses a min …

I just found this paper by B. Datta (later published in J. Indian Math. Soc. 60 (1994), no. 1-4, 171–190) that explains in details why one key equation in Hsiung's paper is wrong. See the whole discus …

This is theorem 7.9 in the book of Kobayashi-Nomizu "Foundations of Differential Geometry Vol.II". There the authors attribute it to Hawley and Igusa independently. These are probably the first papers …

If you assume that $M$ is compact and of dimension $4$ then Donaldson has conjectured that if $J$ is $\omega$-tame then $J$ is $\omega'$-compatible for some symplectic form $\omega'$, see Question 2 h …

I am not sure about the history of this result. You can find in Thierry Aubin's book "Some nonlinear problems in Riemannian geometry", Theorem 4.20. He attributes it to his own 1978 paper. The proof …

If $\Omega$ is a rational Kähler class, so $M$ is projective algebraic, then Donaldson was the first to prove the Chen-Hwang lower bound that you stated. In fact, in this case he proved more, namely t …

The calculation in local coordinates is not too hard and it works in any dimension $n$, namely if $c_1(M)=0$ then $\int_M c_2(M)\wedge\omega^{n-2}\geq 0$ for any Kahler metric $\omega$. Of course you …

For what it's worth, the Navier-Stokes equation on manifolds is also mentioned in this recent paper http://arxiv.org/pdf/1107.2698, see (1.16) there, in connection with another flow for vector fields …