{"paper":{"title":"Sobolev Metrics on Shape Space, II: Weighted Sobolev Metrics and Almost Local Metrics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Martin Bauer, Peter W. Michor, Philipp Harms","submitted_at":"2011-09-02T10:27:18Z","abstract_excerpt":"In continuation of [3] we discuss metrics of the form $$ G^P_f(h,k)=\\int_M \\sum_{i=0}^p\\Phi_i(\\Vol(f)) \\g((P_i)_fh,k) \\vol(f^*\\g) $$ on the space of immersions $\\Imm(M,N)$ and on shape space $B_i(M,N)=\\Imm(M,N)/\\on{Diff}(M)$. Here $(N,\\g)$ is a complete Riemannian manifold, $M$ is a compact manifold, $f:M\\to N$ is an immersion, $h$ and $k$ are tangent vectors to $f$ in the space of immersions, $f^*\\g$ is the induced Riemannian metric on $M$, $\\vol(f^*\\g)$ is the induced volume density on $M$, $\\Vol(f)=\\int_M\\vol(f^*\\g)$, $\\Phi_i$ are positive real-valued functions, and $(P_i)_f$ are operators "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0404","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"}