Some topics from submanifold theory and geometric partial differential equations (Q2868618)

This very nice paper consists of three parts.NEWLINENEWLINENEWLINEIn the first part, the author studies certain geometrically constrained variational problems related to the volume functional and to the Willmore functional as well as rigidity results for Willmore surfaces. Recall that if \(\Sigma\) is a surface and \(f:\Sigma\longrightarrow(N,h)\) an immersion, where \((N,h)\) is a Riemannian manifold, the Willmore functional of the immersion \(f\) is defined as NEWLINE\[NEWLINEW(f)=\frac14\int_\Sigma(|H_f|^2+K_f)d\mu_f,NEWLINE\]NEWLINE where \(H_f\) is the mean curvature vector of \(f\), \(K_f\) is the Gauss curvature of \(f_*(T\Sigma)\) with respect to \(h\), and \(d\mu_f\) is the area element of \(\Sigma\) induced by \(f\). A \textit{Willmore immersion} is a critical point of the functional \(W\) under compactly supported variations. He proves a gap theorem for contact stationary Legendrian surfaces in \(\mathbb{S}^5\) which in particular shows that, with \(B\) denoting the second fundamental form of the surface in \(\mathbb{S}^5\), contact stationary Legendrian surfaces in \(\mathbb{S}^5\) with \(0\leq |B|^2\leq 2\) must be either totally geodesic or generalized Clifford tori. He next studies the critical points of \(W\) among Lagrangian surfaces in \(\mathbb{C}^2\) and among Legendrian surfaces in \(\mathbb{S}^5.\) In the former case, he proves a curvature estimate under a smallness assumption in \(L^2\) of the second fundamental form. In both cases, he introduces a flow method for which he shows short time existence. Finally, he shows a Bernstein-type result, namely, that an entire Willmore graph in \(\mathbb{R}^3\) with square-integrable mean curvature is a plane. In the second part, the author studies biharmonic submanifolds in nonpositively curved manifolds, and biminimal submanifolds, giving a variety of sufficient conditions for biharmonic submanifolds to be minimal and for biminimal submanifolds to be minimal. In the third and final part, he at first studies the concentration compactness for second order geometric PDEs, by first considering the bubble tree structure for sequences of approximated harmonic maps with uniformly bounded energy and tension fields satisfying certain assumptions around a singular point, and proving both the energy identity and bubble tree convergence. He considers the quantization of sequences that solve a singular Liouville equation on a domain of \(\mathbb{R}^2\) with an integral constraint, obtaining, as an application, a new proof of the quantization result for solutions of the mean field equation on a Riemann surface.