Notes on the relative Yamabe invariant (Q2752891)

In this survey article, the author reports recent results about the relative Yamabe invariant of a compact connected manifold with boundary. Let \(W\) be a compact connected \(n\)-manifold with nonempty boundary \(\partial W = M\) (possibly disconnected), and \(n = \dim(W) \geq 3\). For a conformal class \(C \in {\mathcal C} (M)\), let \(\overline C\) be a conformal class on \(W\) such that its restriction to \(M\) is \(C\), i. e., \(\partial \overline C = C\). For such a conformal class \(\overline C\), we consider the subclass \({\overline C}^0\) of \(\overline C\) defined by NEWLINE\[NEWLINE {\overline C}^0 = \{\overline g \in \overline C\mid H_{\overline g} = 0\text{ along }M\}, NEWLINE\]NEWLINE where \(H_{\overline g}\) denotes the mean curvature of \(M\) with respect to the metric \(\overline g\). The \textit{relative Yamabe constant} \(Y_{\overline C} (W, M;C)\) of the pair \((\overline C, C)\) is defined by NEWLINE\[NEWLINE Y_{\overline C} (W, M;C) = \inf_{\overline g \in {\overline C}^0} \frac{\int_W s_{\overline g} dv_{\overline g}} {\text{Vol}_{\overline g}(W)^{(n-2)/n}}, NEWLINE\]NEWLINE where \(s_{\overline g}\) denotes the scalar curvature of \(\overline g\). For a conformal class \(C\) of \(M\), we also define the \textit{relative Yamabe invariants} of the triple \((W, M;C)\) and the pair \((W,M)\), respectively, by NEWLINE\[NEWLINE Y(W, M;C) =\sup_{\overline C} Y_{\overline C}(W, M;C)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE Y(W, M)=\sup_CY(W, M;C).NEWLINE\]NEWLINE A similar argument as in a closed (compact without boundary) manifold shows for any \((W, M;\overline C, C)\) NEWLINE\[NEWLINEY_{\overline C}(W, M;C)\leq Y_{[\hbar]}(S^n_+,S^{n-1};[h]) = n(n-1) \text{Vol}_{\overline h}(S^n_+)^{2/n}, NEWLINE\]NEWLINE where \(S^n_+\) denotes the round \(n\)-hemisphere with the standard metric \(\overline h\) and \(S^{n-1} \subset S^n_+\) the equator with \(h = {\overline h}|_{S^{n-1}}\). Thus the relative Yamabe invariants are well defined and NEWLINE\[NEWLINE Y(W, M;C), Y(W,M) \leq n (n-1) \text{Vol}_{\overline h}(S^n_+). NEWLINE\]NEWLINE One can notice that \(Y(W,M;C) > 0\) if and only if any metric \(g \in C\) can be extended conformally to a positive scalar curvature metric \(\overline g\) on \(W\) with the minimal boundary condition \(H_{\overline g} = 0\). Also \(Y(W,M) > 0\) if and only if there exists a positive scalar curvature metric \(\overline g \) on \(W\) with the minimal boundary condition. The author summarizes fundamental properties on relative Yamabe constants and relative Yamabe invariants.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00049].