A compactness theorem for surfaces with \(L_ p\)-bounded second fundamental form
Let M be a compact connected surface in \(R^ 3\). For an integer \(p>2\), we write \(L_{2,p}(M,R^ 3)\) for the usual Sobolev space of maps of M into \(R^ 3\) whose generalized derivatives up to order 2 are p- integrable. We remark \(L_{2,p}(M,R^ 3)\subset C^ 1(M,R^ 3)\) by the Sobolev inequality. Set \(Imm_{2,p}(M,R^ 3)=\{\psi \in L_{2,p}(M,R^ 3),\) \(\psi\) is an immersion\(\}\). For \(\psi \in Imm_{2,p}(M,R^ 3)\), we write \(dV_{\psi}\) (resp. \(\Pi_{\psi})\) for the induced area element (resp. the second fundamental form) of the immersion \(\psi\). Moreover set Area\((\psi)=\int_{M}dV_{\psi}\) and \(E_ p(\psi)=\int_{M}| \Pi_{\psi}|^ p dV_{\psi}.\) Then the following is proved: Compactness theorem. For given constants A, E and \(p>2\), set \(\Omega =\{\psi \in Imm_{2,p}(M,R^ 3)\); Area\((\psi)<A,\) \(E_ p| \psi | <E\) and \(\int_{M}\psi dV=0\}\). Then for any sequence \(\{\psi^ n\}\) in \(\Omega\), there exists a sequence of surface diffeomorphisms \(\{\phi^ n\}\) such that a subsequence of \(\{\psi^ n\circ \phi^ n\}\) converges weakly in \(L_{2,p}(M,R^ 3)\) (so especially in \(C^ 1\)-topology) to some \(\psi \in Imm_{2,p}(M,R^ 3)\). Here the diffeomorphisms \(\phi^ i\) have the same smoothness as \(\{\Psi^ i\}_{i=1,2,...}.\) As an application of the above result, we have: Existence theorem. For \(p>2\), the functional \(E_ p\) achieves its infimum within each component of \(Imm_{2,p}\cap Area^{-1}(1)\), i.e., in each regular homotopy class of unit area \(L_{2,p}\)-immersions of M into \(R^ 3\).
scientific article; zbMATH DE number 3899793