Minimal hypersurfaces asymptotic to quadratic cones in \({\mathbb{R}}^{n+1}\)
Set \[ C^{p,q}={\mathbb{R}}^{p+1}\times {\mathbb{R}}^{q+1}\cap \{(x,y):q| x|^ 2=p| y|^ 2\}, \] where p and q are positive integers with \(p+q=n-1\). The authors prove a classification theorem for minimal hypersurfaces (no necessarily area-minimizing) which are asymptotic to \(C^{p,q}\) near infinity, provided \(C^{p,q}\) is area-minimizing. (It is known from other work that \(C^{p,q}\) is area-minimizing in case \(\min \{pq,p+q\}>5.)\) In fact, assuming \(C^{p,q}\) is area-minimizing, they show there are, case \(p\neq q\), exactly 3 and, in case \(p=q\), exactly 2 minimal hypersurfaces asymptotic to \(C^{p,q}\) at infinity. Let C be a minimal cone which is smooth away from the origin. The authors set up a framework in which they can consider a hypersurface having C as an asymptote at infinity as a graph of a function defined on C. The mean curvature of such a graph is given by an elliptic, quasi-linear, divergence form second order operator \({\mathcal M}: C^{2,\alpha}\to C^{0,\alpha}\), the linearization of which is known as the Jacobi operator (here \(C^{r,\alpha}\) denotes the function space). The proof involves a careful analysis of spectrum of the Jacobi operator. In the case \(C=C^{p,q}\), this is facilitated by the fact that \[ C^{p,q}\cap S^ n=[p/(n-1)]^{1/2}S^ p\times [q/(n-1)]^{1/2}S^ q, \] where \(S^ d\) is the unit sphere in \({\mathbb{R}}^{d+1}\), so the completely understood spectral decomposition of \(L^ 2(S^ d)\), for the Laplacian, can be used.
scientific article; zbMATH DE number 3979898