Optimizing the portfolio in modern portfolio theory

Question

I am trying to understand some aspects of the modern portfolio theory, which has brought me to a point I don't fully understand. I would appreciate any hep/suggestions/references. Lets assume that the utility function, say $U:\mathbb{R} \times {\mathbb{R}^ + } \to \mathbb{R}$, is an increasing function of its first argument $\mu$ and a decreasing function of its second argument $\sigma$. These two arguments $\mu$ and $\sigma$ are both increasing functions of the weight variables $w_1$, $w_2$, ..., and $w_N$, when all of these weight variables are considered independent. Now, obviously one wants the additional constraint that $\sum_{n=1}^{N}{w_n}=1$. Now the question is, under these conditions, would the point maximizing $U$ w.r.t. all the weight variables (satisfying the additional normalizing constraint) necessarily lie on the manifold which minimizes the $\sigma$ function (in the wight variables space) for each $\mu$. In other words, since one could build some sort of manifold in the $w_n$ space by minimizing $\sigma\left(w_1,\ldots,w_N\right)$ for the constraints $\mu\left(w_1,\ldots,w_N\right)=\mu^{*}$ and $\sum_{n=1}^{N}{w_n}=1$, for all relevant values of $\mu^{*}$, does this manifold necessarily contain the optimizing point/points of the considered utility function?