Smoothness and Poisson structures of Bridgeland moduli spaces on Poisson surfaces

Let X be a projective smooth holomorphic Poisson surface, in other words, whose anti-canonical divisor is effective. We show that moduli spaces of certain Bridgeland stable objects on X are smooth. Moreover, we construct Poisson structures on these moduli spaces.Comment: We would like to thank Sergey Mozgovoy for pointing out a mistake in the first and journal version of this paper. Our result only holds for $H$ that is numerically parallel to $K_X

Infinitely many solutions for the prescribed boundary mean curvature problem on BN\mathbb B^N

We consider the following prescribed boundary mean curvature problem in $\mathbb B^N$ with the Euclidean metric $-\Delta u =0$, $u>0$ in $B^N, \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} K(x) u^{N/(N-2)}$ on $S^{N-1}, where$K$is positive and rotationally symmetric on$\mathbb S^{N-1}$. We show that if${K}$has a local maximum point, then the equation has {\bf infinitely many positive} solutions, which are non-radial on$\mathbb S^{N-1}$.Comment: arXiv admin note: text overlap with arXiv:0804.4030 by other author