Willmore surfaces and \(F\)-Willmore surfaces in space forms (Q1949026)

Let \(M^2\) be a compact Willmore surface in the \(n\)-dimensional space form \(\mathbb{N}^n(c)\) of constant curvature \(c\). Denote by \(\phi _{ij}^{\alpha}\) the trace-free part of the second fundamental form \(h=(h_{ij}^{\alpha})\), and by \(\mathbf H\) the mean curvature vector of \(M^2\). Let \(\Phi\) be the square of the length of \(\phi _{ij}^{\alpha}\) and \(H=|\mathbf{H}|\). The first main result of this paper is the following extension to the space of forms of the result obtained by the author and \textit{Y.-J. Hsu} [Taiwanese J. Math. 8, No. 3, 467--476 (2004; Zbl 1075.53052)]: The inequality \(\int _M \Phi(C(n)(c+\frac{H^2}{2})-\Phi)dv \leq 0\) holds, where \(c(n)=2\) when \(n=3\) and \(c(n)=\frac{4}{3}\) when \(n\geq 4\). In particular, if \(0\leq \Phi \leq C(n)(c+\frac{H^2}{2}),\) then either \(\Phi =0\) and \(M\) is a totally umbilical sphere, or \(\Phi = C(n)(c+\frac{H^2}{2})\). In the latter case, either \(M\) is the Clifford torus in \(S^3(c)\) of \(\mathbb{N}^n(c)\), or \(M\) is the Veronese surface in \(S^4(c)\) of \(\mathbb{N}^n(c)\) (Theorem 1.3). The \(F\)-Willmore functional of submanifold in space forms is defined by \(W_F(x)=\int _M F(\Phi )dv\), where the function \(F :[0,\infty )\rightarrow \mathbb{R}\) is \(\mathcal{C}^3\)-differentiable. A critical point of \(W_F(x)\) is called a \(F\)-Willmore submanifold. The second important result is the improvement of the integral inequality given by \textit{J. Liu} and \textit{H. Jian} [Front. Math. China 6, No. 5, 871--886 (2011; Zbl 1227.53074)]: Let \(M^2\) be a compact \(F\)-Willmore surface in the \(n\)-dimensional space form \(\mathbb{N}^n(c)\) of constant curvature \(c\). If \(F'(\Phi )\geq 0\), then \[ \int _M \{F''(\Phi)[\frac{1}{2}|\nabla \Phi|^2-\sum _{\alpha,i,j}\phi _{ij}^{\alpha}\Phi _{j}H_{i}^{\alpha}]+F(\Phi )H^2+F'(\Phi )(2c-K(n)\Phi )\Phi\}dv \leq 0. \] If \(F'(\Phi )\leq 0,\) then \[ \int _M \{F''(\Phi)[\frac{1}{2}|\nabla \Phi|^2-\sum _{\alpha,i,j}\phi _{ij}^{\alpha}\Phi _{j}H_{i}^{\alpha}]+F(\Phi )H^2+F'(\Phi )(2c-K(n)\Phi )\Phi\}dv \geq 0. \] The constant function \(K(n)=1\) when \(n=3\) and \(K(n)=\frac{3}{2}\) when \(n\geq 4\) (Theorem 1.5).