{"paper":{"title":"Applications of the duality between the Complex Monge-Amp\\`ere Equation and the Hele-Shaw flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CV","authors_text":"David Witt Nystrom, Julius Ross","submitted_at":"2015-09-09T07:55:58Z","abstract_excerpt":"We give two applications of the the duality between the complex Homogeneous Monge-Amp\\`ere Equation (HMAE) and the Hele-Shaw flow. First, we prove existence of smooth boundary data for which the weak solution to the Dirichlet problem for the HMAE over $\\mathbb P^1\\times \\overline{\\mathbb D}$ is not twice differentiable at a given collection of points, and also examples that are not twice differentiable along a set of codimension one in $\\mathbb{P}^1\\times \\partial \\mathbb{D}$. Second, we produce explicit families of smooth geodesic rays in the space of K\\\"ahler metrics on $\\mathbb P^1$ and on "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02665","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}