https://neckpinch.com/authors/tatsuya-miura/New arXiv papers by Tatsuya Miura, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2606.04586/arxiv:2606.04586Wed, 03 Jun 2026 00:00:00 GMTTatsuya Miura, Fabian Rupp — We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along proper oriented mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. In fact, our result covers a class of singular calibrations and singular initial data. Even in the smooth setting, this provides a new finite variational framework for mean curvature flow beyond the finite-volume regime. As a main application, we establish a general dynamical rigidity theorem for calibrated cones in arbitrary codimension: no singular calibrated cone can be desingularized by a proper oriented mean curvature flow. Our framework further yields novel rigidity theorems for solitons and convergence for two-dimensional immortal flows.Mean curvature flowSurfacesWeak & singular flowsStability & uniquenessLong-time existence & convergence https://neckpinch.com/papers/2410.06183/arxiv:2410.06183Tue, 08 Oct 2024 00:00:00 GMTTatsuya Miura — For a class of area-preserving curvature flows of closed planar curves, we prove that every immortal solution becomes asymptotically circular without any additional assumptions on initial data. As a particular corollary, every solution of zero enclosed area blows up in finite time. This settles an open problem posed by Escher–Ito in 2005 for Gage's area-preserving curve shortening flow, and moreover extends it to the surface diffusion flow of arbitrary order. We also establish a general existence theorem for nontrivial immortal solutions under almost circularity and rotational symmetry.Curve shorteningOther curvature flowsVolume-preserving & constrained flowsSingularitiesLong-time existence & convergenceRotational symmetry & warped products