2024/03/29 by Yifan Guo, Guo, Yifan
Computer Science · #Analysis of PDEs (math.AP) #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2404.00115
openalex publication_date 2024/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We are interested in finding a nonlinear polynomial P on ℝn that solves the minimal surface equation. Even though no explicit solution is found in this article, we investigate constraints that a polynomial solution must obey. We first prove a structure theorem on such polynomials. We show that the highest degree term Pm must factor as pkQm where k is odd, p is irreducible, and Qm≥ 0 on ℝn with \Qm=0\⊂\p=0\∩\∇ p=0\. Moreover, the level sets of Pm are all area-minimizing and the unique tangent cone of graph P at infinity is \p=0\×ℝ. If k≥ 3, we know further that lower order terms down to some degree are divisible by p. We also show that P must contain terms of both high and low degree. In particular, it cannot be homogeneous. As a consequence of the structure theorem, we get degree estimates for polynomial solutions. We have deg P≥ 4 by ruling out cubic polynomial solutions. Using an extended eigenvalue estimate on the Jacobi operator by Zhu \citezhu2018first, we are able to show that μn-< deg p +k-1deg Qm< μn+ where μn^±=(n-1±√((n-3)2-4(n-2)))/(2). Finally, we prove that \p=0\ cannot be an isoparametric minimal cone. We also show that for a nonlinear polynomial solution on ℝ8, we have deg p=3 and that \p=0\ is an area-minimizing but not strictly minimizing cone in ℝ8. These results give strong restrictions on possible polynomial solutions to the minimal surface equation.