https://neckpinch.com/authors/kotaro-kawai/New arXiv papers by Kotaro Kawai, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/2103.13863/arxiv:2103.13863Thu, 25 Mar 2021 00:00:00 GMTKotaro Kawai, Hikaru Yamamoto — We can define the "volume" V for Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold X, which can be considered to be the "mirror" of the standard volume for submanifolds. This is called the Dirac-Born-Infeld (DBI) action in physics. In this paper, (1) we introduce the negative gradient flow of V, which we call the line bundle mean curvature flow. Then, we show the short-time existence and uniqueness of this flow. When X is Kähler, we relate the negative gradient of V to the angle function and deduce the mean curvature for Hermitian metrics on a holomorphic line bundle defined by Jacob and Yau. (2) We relate the functional V to a deformed Hermitian Yang–Mills (dHYM) connection, a deformed Donaldson–Thomas connection for a G₂-manifold (a G₂-dDT connection), a deformed Donaldson–Thomas connection for a rm Spin(7)-manifold (a rm Spin(7)-dDT connection), which are considered to be the "mirror" of special Lagrangian, (co)associative and Cayley submanifolds, respectively. When X is a compact rm Spin(7)-manifold, we prove the "mirror" of the Cayley equality, which implies the following. (a) Any rm Spin(7)-dDT connection is a global minimizer of V and its value is topological. (b) Any rm Spin(7)-dDT connection is flat on a flat line bundle. (c) If X is a product of S¹ and a compact G₂-manifold Y, any rm Spin(7)-dDT connection on the pullback of the Hermitian complex line bundle over Y is the pullback of a G₂-dDT connection modulo closed 1-forms. We also prove analogous statements for G₂-manifolds and Kähler manifolds of dimension 3 or 4.Mean curvature flowLagrangian mean curvature flowRicci-flat & Calabi–Yau3-manifoldsKähler geometryStability & uniquenessHarmonic maps & DeTurck trick